Hi there, welcome back to my YouTube channel. This is Daniel Rosal here. So a couple of nights ago. I did a demonstration failover failing from my primary internet connection over to my cellular Backup connection that I just got set up and I did a couple of videos on How to do that for anybody else interested in backup internet and how it actually works. So basically what I got was a TP link router. Now, this is something I picked up a couple of weeks back It's one of the more basic TP link routers, but it is a 4G cellular router It's got a SIM card slot in it got a couple of antennas and it picks up 4G onto that SIM card and the way I have this setup is as a as a backup or failover Router so the kind of network config as I described in the previous YouTube video is that my ISP's router is going into the one port of this cellular router and It's passing along the ISP ISP connectivity and it also changed a few settings so that it knows that it's running as a backup So that the cellular connection it can give is not the primary It's just a failover connection So if the ISP line your internet service provider your internet your home internet if that goes down The cellular will come in so I demonstrated that that worked which was nice And good, but it didn't work quite as swiftly as I kind of had pictured So I thought it was like an instant thing that one second you are Get you are just using your home internet then you took it out and then within a second and If you ask anybody who knows about home networking I've learned they'll tell you that it never works exactly that instantaneously So I thought it would be interesting as an experiment just to repeat that failover test Except this time to try to measure the length of time Now I did a previous take of this video that almost ended in disaster because I use the DNS Disabling switch on my router Which was not a good idea at all because so don't don't do that Don't disable the DNS button on your router because my ISP had to like Do something remotely to get it back. I think in any event it was not smart So what I'm going to do this time is as I did before like physically remove the DSL from the from the ISP router just to simulate a Complete drop-off and connectivity and we're going to see how long it takes to kick in now I have found a useful website and it's called my is my internet your internet is working Is my internet working comm so what I'm actually going to do Is I'm going to just take myself out of this here and I'm going to run over to the other side of the room Pull out the DSL cabling from the from a router and as soon as that has happened I'm going to come back here and hit start so it's not going to be a perfect process or be a couple of seconds of of You know that it'll be so we'll just have to take two seconds off the results to account for the time it took for me to Wheel across the room in my chair. So there is an auto test function So I go to now set that to two seconds so every two seconds it's going to check my internet to see that it's up and The second thing I'm doing here is what is my what is my browser.com has a what's your ISP? Testing facility and I'm going to just do a quick refresh just to show you guys It's refreshed that I'm with partner communications. That's my ISP so Internet's good internet stable internet's up and now You're listening to me wheel wheel wheel across the room And I don't want to take out the RJ 11 cabling the SL line to my ISP rather three two one Okay Speed is connected and they should start saying no indeed it does so We have we have done the big deed. Oh the timer the timer stop watch Well, I guess I can measure later in the video. So this is going to be like 10 seconds often for it's not late So, let's see right like let's see 15 seconds so far And we're still not having internet Let's see I think the last time I got like 30 seconds for the failover so that really wasn't bad I mean how often does this happen? Not that often But it's not like two seconds even even without that time where I was like forgot to start this on time So what should be happening now is the Here we go. Okay, so 35 seconds was what I recorded but I also Fail to start this on time, right? So I need to add it's very late in the morning My brain is not working so well. I need to add like a bit more time to To that to actually account so I would say about like 45 seconds for that failover to happen And if I do a refresh here, it should say something like I forget it. It's like X phone or X 0 18 It's not really the name of the phone company. It's like they're a pairing company X phone. There we go X phone 0 18 So now we know we are using the We're using the line of The cellular So I'm going to go what I'm going to go ahead and do now is repeat the process. So 40 let's say 45 seconds On failover to cellular. Now. Let's see how long it's going to take to get us back to the ISP line. So I'm going to now Do my wheelie trick again reset the stopwatch and Let's see how long this one takes so stopwatch is going to be clear zeroed and Weaving across the room again this time getting the DSL cable and I'll tell you the moment it's in. Oh, okay. It's safe. So this time let's get the stopwatch It's saying yes, but it's going to be saying yes anyway because there is internet So the more trying test is going to be or the more telling test will be when this switch is back to partner. So X phone Still X phone 14 seconds in Still X phone, I'm just going to turn the writer array back to its usual config still actual still X phone 30 seconds, so if it was 45 seconds on the way out, I wonder is that like just what it takes For the writer to detect these configuration changes and swap over. So I'm just checking the DSL is back up on the writer So the writer has it the writer's passed me along and now we're just waiting on the writer to actually take note of its shifted connectivity circumstances and This is actually taking longer. So it's taking me interestingly enough taking longer to fail back Over to ISP than it was to fail Across in the other direction Well, that's not really the right word fail I now I think we're gonna get we're gonna get partner because it's taking longer, but we're already into one minute and 20 seconds I'm gonna just open another one and I think this is what happened the last time is like it goes down for 15 seconds while it's doing its like change of network thing What is my browser calm? Or it could actually be the website I'm just going to another No, yes, okay. I'm on another website now and it's saying that I'm on Partners, so I think this is a coincident. There we go partner. Okay, so it was more like one minute and 30 seconds to fail back To the ISP from the sider So I guess this was like it's somewhat interesting test I guess the the what I've learned is that it takes give or take 40 seconds for From the time that your primary internet goes completely down or at least on this TP link device through to the point that it feeds in the cellular and you get your backup line running and When your primary internet comes that it comes back It's going to take like or on this on this particular configuration with my ISP my cellular And the way I wired and the TP router I'm using which again is the TP link 300 megabit per second TL MR 100 with all those variables It took approximately a little bit over a minute on the return site. So it actually took slightly longer to Provision the ISP line back than it did to do the failover. So that was interesting Anyway, that was the test of how long the failover took I hope this video is interesting if you'd like to get more videos from me Please feel free to subscribe to this YouTube channel


The Legend of Zelda Breath of the Wild is getting closer and closer to launch, but the closer we get, the more unanswered questions we end up with. With more and more footage being shown, you would expect some of these answers to clear themselves up, but it seems to be having the opposite effect. I'm GameOverJesse, and here with me is RMFH Gaming to give our thoughts on the top 5 questions fans seem to have about Breath of the Wild. 1. How Many Dungeons Will There Be? This one has been on everyone's minds since rumors first began coming forth about Breath of the Wild. We first heard that the game will feature around 100 mini-dungeons and 4 traditional dungeons. First reactions to this were, are you nuts 100 mini-dungeons? That would be insane, seeing as most games have far less, that includes mini-dungeons. On the other side, only 4 main dungeons might leave fans wanting more. But don't fret, Majora's Mask only had 4 dungeons, and it is looked at as one of the best in the series. However, rumors are rumors, and as far too many people say, should be taken with a grain of salt. But half of this rumor has been confirmed. During E3, and several times since, Nintendo has confirmed that there will be 100 mini-dungeons or what we know as Shrines. Now we just play the waiting game to see how many traditional dungeons we will get. Besides, playing 100 Shrines with the same aesthetic and blue atmosphere would get a bit repetitive for someone going from shrine to shrine, and we would need a good amount of time in the overworld or lengthy traditional themed dungeons to kill that repetitiveness. If it turns out that we do only have 4 main dungeons, what I would like to see happen is the player being forced to return to an already completed dungeon to explore or unlock a new path inside of it. Think about it like this, in Ocarina of Time, Dodongo's Cavern is inside the bottom of Death Mountain, and on top of Death Mountain, deep inside, we have the Fire Temple. Imagine if instead of two completely separate dungeons, we just have one massive connected dungeon. You may require a certain item like bombs to make your way throughout the first part, then after you conquer this part of the dungeon, you continue on your journey and come across an NPC in a village or a certain item later in the game and in all of the sudden clicks. While inside of the first section of the dungeon, you see a wall that you would have been able to claw shot or blow up, but since you didn't have the required item at the time, you simply ignored it. So now you return to this same dungeon, take a second or possibly third path that is now unlocked. This path could be far more dangerous and filled with even more complex puzzles or more powerful enemies. Maybe that thing we thought was the main boss in the first section of the dungeon, in this case King Dodongo, turns out to just be one of many several mini-bosses and once we make it to the actual main boss of the dungeon, we are greeted by a flaming dragon to take on inside a room surrounding us with lava. Now, a more recent rumor suggests that each territory, like the Great Plateau, will have its own village and dungeon. This kind of falls into common sense, as major areas in past Zelda games have their own dungeon, and it has been half a year or longer since the original rumor of 100 mini-dungeons and 4 main dungeons was leaked. So if it did happen to be 100% true at that time, Nintendo has had a lot of time to work on new dungeons. Remember, Majora's Mask was completed in a year. Yes, it reused assets like character models and items from Ocarina of Time, but it also brought an entirely new overworld, dungeons, villages, towns, storyline, one of the most complex side quests out of any game that I've ever played, and tons of new items and mass. All in just a year's time. Number 2. The Floating Island. Many have speculated on what this exactly is. Is it simply the remains of an ancient civilization, similar to the city in the sky from Twilight Princess, in which it will only offer a simple dungeon to explore, or is it more important than that? Could it be an entire floating village like Skyloft and Skyward Sword, full of NPCs, side quests, shops, and possibly its own dungeon? After all, if the guardians and other evil creatures are taking over Hyrule, it would make sense for some of the villagers to escape back up to the sky as they once did before the events of Skyward Sword. Or I could just be overthinking things and this could simply be Beetle's new airship. Instead of having a tiny boat or a small airship, he now literally has an entire sky mall. For those getting mad at this idea, it's just a joke and the truth is, we have no idea what this is or how we would even get to it. For example, will it land and let us simply climb up? Some people suggest that you may use the heat from Death Mountain to fly there using the paraglider. Could we discover a new rune to give us an ability that allows us to get there? Will we simply claw shot our way there? Or my personal favorite? Will we see the return of loft wings to make our way to the city in the sky? 3. The Master Sword and other non-breakable items Featured prominently in the artwork is the Master Sword, but I don't want to give the sword all of the spotlight. To start, we see that weapons, shields, and other items can all be broken giving them limited use. This isn't the first time we've seen this though. This goes as far back as Ocarina of Time and maybe even further. So what comes to mind first is the Deku Shield, which can easily be burned, and the Giant's Knife, which is broken only after a few hits, opening up the side quest to obtain the unbreakable Big Goron Sword. This game also introduced clothes to give you status abilities like withstanding the heat in Death Mountain, with the Goron's Tunic, or breathing underwater with the Zora's Tunic. However, the closest we have seen item durability matching that of Breath of the Wilds is in Skyward Sword. The very last home Zelda title before Breath of the Wild. In this game we are introduced to three separate lines of shields that have their own unique abilities and can each be upgraded to become even more durable. Then we get the Hylian Shield. But it's more of a bonus item, it isn't really part of the main quest. So this makes me think, will we see similar items in Breath of the Wild? Will we see items that are indestructible? Perhaps taking something like the Fire Rod and upgrading a few times until it just can't break? Or maybe the unbreakable items are the ones found as dungeon items? Now this takes us back to the Master Sword. We see that it's already in a weakened state from the trailer. Maybe we're to restore it to its full power in the game to defeat Calamity Ganon and while restoring it, it becomes one of these unbreakable items. Perhaps we also discover the Hylian Shield somewhere as well, and it, as in Skyward Sword, is unbreakable. The idea of having breakable items is a great addition to the game. It makes it so we're not constantly using the same item and makes us become familiar with the huge variety of items the game has to offer. If we're using a one-handed sword in Shield and the sword breaks, suddenly we're forced to use something like a two-handed axe and we're left to play in a different way as we would have before, as we now have no way of defending ourselves. But to play the endgame with an unbreakable sword, shield, and other items would make it easier for those going back to 100% the game. After all, how else are we supposed to shield-surf down the entirety of Deathmountain while being chased by this giant guardian? Number 4, Overworld Bosses. Something kind of new to the Zelda series, we will have several mini-bosses in the Overworld. So far we have seen this giant horned beast, whatever it may be, an even larger rock creature known as Step-Talus, this giant King Bokoblin, this even larger Stalfos, and this truly gigantic guardian scaling the side of Deathmountain. My question here is, are these all just what we would consider to be mini-bosses? Or could any of them be a traditional main boss? And does this open the door to full boss fights in the Overworld, where defeating one would be akin to beating a boss in a traditional dungeon? Like for example beating Morpha in Ocarina of Time and it restoring water to a one stride up Lake Hylia, or changing the season from winter to spring in Majora's Mask. If we fight these mini-bosses or bosses in the Overworld, will they simply respawn if we leave and come back, or will it actually make an impact on the game in a memorable way? I'm sure the Step-Talus and large horned beasts are a dime a dozen, but surely defeating this giant Stalfos or this larger than life guardian will have some effect on the game, even if it's just restoring a village that was once being terrorized by these beasts. This could open the game up for more side quests and give you a sense of actually saving someone and having an impact on the overall Overworld. After all, Miyamoto is quoted during an interview with saying, I can't talk much about it, but one of the things we're working on right now is that as you play, the world will change and be affected by what you choose to do. Number 5 Overworld Dungeons Imagine a dungeon that wasn't necessarily a dungeon, but acted as one. My buddy Nate, who runs Zeldin, brought this up. Imagine walking in an area akin to the Lost Woods. You start in an open field, and as you would go deeper and deeper into the wooded area, you start seeing small, fun little Overworld puzzles and the occasional group of enemies. Then once you are even deeper into the wooded area, you all of the sudden are confronted by Overworld puzzles that are actually making you stop and think on how to complete them, and the enemies you face are becoming harder and harder, when all of the sudden you're face to face with this giant Stalfos, or another similar boss. Then without realizing it, you just gradually completed a dungeon, but you never actually entered a dungeon. You were just trying to make it from point A to point B, it just turns out that the path you took took you through a dungeon without even giving you a hint. You were still outside, the Overworld music didn't really change, there was no loading screen like when entering a house or a shrine. Now imagine this concept carried over to a desert area, a water area, or even death mountain, and not just limited to the forest. What's up everyone, Jesse here, and I wanted to say thank you for checking out this video. If you have any of your own theories or questions about Birth of the Wild, leave them in the comments below for us to discuss, and we may feature them in a future video. If you're a fan of Zelda theories or videos similar to this, you can check out a great video here on Zelda Informers Channel about how Birth of the Wild may merge the split Zelda timelines, or you can head on over to my channel, Game Over Jesse, and subscribe for Nintendo and Zelda news, rumors, theories, and discussions featuring other YouTubers like HMK, Vortexia Gaming, Zelda Master, Macintyre Productions, and the rest of the Chamber of Sages.


Item Number – SCP-537 Object Class – SAFE Special Containment Procedures – SCP-537 is to be kept in a soundproof room. No additional containment measures are required, although SCP-537 has requested. One poster of the Muppet Movie – Granted Weekly dusting and cleaning of its record – Granted Daily one-hour visits from Foundation personnel – Denied Weekly one-hour to converse with Foundation personnel sent to clean its record following cleaning – Under consideration – A vinyl record of the soundtrack from the Muppet Movie – Denied – It can play the entire soundtrack without need of the actual record Description – Recovered from a pawn shop to which it had reportedly been brought by a young man in February of 1981. SCP-537 bears the appearance of a completely normal, albeit tattered, gramophone, with no visible deviations or abnormalities. The vinyl record, SCP-537-2, beneath its needle, has only one circular groove that does not move inward so that the needle never reaches the center of the turntable. The needle is abnormally difficult to lift, but can be lifted by hand with some effort. The needle will snap back down the moment it is released. SCP-537 is without a doubt intelligent and demonstrates the ability to both see and hear. It is not yet known how. It communicates by playing songs. SCP-537 is able to play any song that can be found on an existing vinyl record and can do so at will. SCP-537 plays songs all the way through and will never interrupt a song unless its needle is forced up, in which case it will resume the song from the point it left off at the next time it is able to play music. Forcing SCP-537 to interrupt a song is discouraged, as doing so seems to cause it to become irritated and respond petulantly to testing for up to several days. SCP-537 can raise and lower its needle on its own. It can answer yes or no questions quickly by raising and lowering the needle once for yes and twice for no. It moves its needle in a similar fashion to express numbers, usually used to refer to other SCPs. Attempts to teach SCP-537 Morse code as a way of accelerating communications have failed, owing to SCP-537's inability to read or spell. It does not matter what record is on SCP-537's turntable. It can play whichever songs it pleases regardless. It does, however, prefer to have SCP-537-2 as it seems unable to reset its needle alone and will eventually be unable to continue playing. SCP-537-2, when played back on a normal turntable, is blank. SCP-537 appears to enjoy interacting with humans and thoroughly enjoys being interviewed. Its personality is often described as pompous and flamboyant, usually greeting people with will common from the musical cabaret, yet distinctly childish. It is prone to tantrums when offended, which usually consist of it playing angry music at maximum volume non-stop until brought back to containment. It seems more trusting of younger employees and treats older people with cautious disrespect. When asked about its favorite song, SCP-537 invariably responds by playing The Rainbow Connection from the Muppet Movie. Question No. 537-1 Test for Special Properties in SCP-537-2 Performed by Dr. P. P. SCP-537-2 was played in a regular turntable. The record is apparently blank. A vinyl record, consisting of one perfectly circular groove along which the needle would travel, was made for the purpose of this experiment. When given to SCP-537 to play, though the replica record was identical, SCP-537 apparently realized that it had not been given SCP-537-2 and reacted angrily, raising its needle, and refusing to lower it, until given its own record back. SCP-537 continues to refuse to communicate with Dr. P. Document No. 537-2 Experiment 537-43-1 SCP-43, and a second functioning turntable, were introduced to SCP-537, overseen by Agent R. Transcript follows. SCP-43 plays to track 29. Heavy breathing is heard. Agent, he's here now 537. Say hello. SCP-537 plays Will Common from Cabaret. SCP-43, during song. Hello? What's that sound? SCP-537 makes a scratching sound, but continues playing Will Common. SCP-43, during song. This is the record player you told me about. Is there a way to shut it up? Agent, during song. 537 can play another song until it finishes the one it's playing. SCP-43, during song. How annoying. Song ends. SCP-43, right then. That was... lovely. I'm just supposed to... converse with you. So, how are you, I suppose? SCP-537 begins playing Sunshine Lollipops and Rainbows by Leslie Gore. SCP-43, oh, wonderful. Song ends. SCP-43, right. Let's talk about me then. Is there something you want to know about me? SCP-537 raises and lowers its needle, denoting an affirmative answer. SCP-43, and what would you like to know? SCP-537 begins playing Here Comes The Sun by The Beatles. SCP-43, during song. Why are you playing that song? What do you think I... SCP-537 raises its volume. SCP-43, during song. Inaudible, underneath music. Song ends. SCP-537 immediately begins playing the same song again. SCP-43, shouting, during song. Stop it! Stop it! SCP-537 raises volume. SCP-43, shouting, during song. Data expunged. SCP-43 is removed from the room as per its own request. SCP-43, unsurprisingly, was not willing to discuss its reaction to SCP-537, simply referring to it as... that cheeky thing, and mentioning that it's the type that needs to learn to be careful if it doesn't want to data expunged. Document number 537-2. Experiment 537-43-2. SCP-43 was placed, after reluctantly agreeing to do so, under SCP-537's needle. When asked how it was feeling, SCP-537 played the first eight tracks of the white album by The Beatles. It should be noted that it is unusual for SCP-537 to play a song on the record it is turning, let alone several songs in the proper order. After the eighth track, rather than the record stopping or revolution nine playing, the record began to skip. A loud noise, described as somewhere between a record scratch and a shriek, was heard and did not stop until the needle was forced up. SCP-43 refused to speak of the incident, saying only that it never wanted to be near SCP-537 again. SCP-537 reacted similarly. When asked its opinion of SCP-43, it responded by playing George Carlin's routine, seven words you can never say on television. Since Experiment 537-43-2, SCP-537 has flatly and consistently refused to play any song by The Beatles, responding each time by raising and lowering its needle twice. When asked multiple times to do so, SCP-537 reacts angrily. Dr. Addendum B. Speculation as to whether or not SCP-537 is capable of playing songs from vinyls released after its construction, the date of which is unknown, has been settled as of a recent interview which had ended by playing Caring Is Creepy from O. Inverted World by The Shins, an album released on vinyl in June of 2004. This is interpreted as SCP-537's way of expressing its apathy towards the interview. Lesson complete. If you missed the previous orientation, go watch SCP-536, Physical Law Testing Chamber, right now. Or for the complete course, watch this playlist.


What's going on guys got Kyle here late in sports cards, and we are doing a seven box personal for Jeffrey H We'll all be the brand new 2022 prison football Good luck Jeffrey. Let's do it. Let's see some big cards Whoo, good luck Jeffrey been a while since I read for you Jeffrey. Oh See what we can do man. Man. It's back used to do a lot of Instagram for Jeffrey. Oh, yeah. Oh, yeah Let's do it bud Start off. We got a silver rookie throwback Desmond Ritter nice one Jeffrey and a David a job of blue ice Ravens rookie to 99 Calvin Austin Second pack gold. Whoo. Come on Jeffrey. Come on Jeffrey. Let's see it a dolphin, maybe Gold rookie Skyler Thompson There you go Jeffrey We'll mag that for you. Gold QB rookie. We'll take it. That is numbered six of ten Nice one there man. We'll definitely mag that Nice gold to start Another Thompson got silver rookie tyquan Thornton and Devonte Smith. That's orange way to 60 Yeah, Higley is the personal for Jeffrey that spiller gonna be Wes Welker to 249 the guy's seen him at all yet Patriots Regas and Drake London Yeah, I appreciate that Higley Good looks Higley emergent Willis We got green and a green and yellow hyper Raheem Oster Trayvon Walker Jackson our first auto Silver Devonte Parker and auto it's gonna be Fred Warner Silver auto there veteran auto for the Niners absolute beast Nice one there Niners and right behind it another beast for the Niners Nick Bosa It's a 125. We have Hype insert Bateman and Dexter Lawrence purple cracked ice 225 silver LT and Blue ice for shod penny 99 Wilson rookie Randy Hello, what up Randy? How you doing man? Silver Rondale more and we have rookie auto to 149 Kate Aughton Little Bucks rookie there Jameson Williams No, it's only seven boxes Bebo About half a case just a little over got brilliance chase Red Wave CJ 2k Jones in a Brock Purdy base there. He's doing it now for you illegal He's picking up whoever you wasn't supposed to David Robertson got fireworks blue ice Lamar Jackson to 99 and Khalil Shakir orange to 249. Are you ready? hobby bias Emergent Damian Pierce and Alan Robinson purple power to 75 or 49. Sorry Traylon Burks rookie All right box to Jeff, let's do it. Good luck. Good luck Donnie Biden. So many yes Two trips of like Yes, never run out Do you like the trash on this side of the other side this side? Yeah, I'm a lefty. Oh, yeah Now of course, it's the only one Silver Daryl Green We have auto Jay Jetta nice silver auto there Vikings veteran probably the best receiver in the league Blue Wave Brandon cooks to 199 Brandon cooks Oh, I'll let him know Got silver Raji Jordan poyer red wave 149 Hutchinson that's a tough debate emergent Malik Willis red wave Tyrion Davis price nice color match there for the Niners to 149 Austin and Jones silver emergent pickings Steelers and purple ice that is Harrison Smith It's re-signed for the Vikings heard that earlier Good pick up there. They're keeping Looks like a RPA here. Good luck. Good luck Silver Jim Kelly RPA is gonna beat a 99 Kenny pick it. Boom. There you go, Jeffrey Solid one there on the Kenny to glove RPA We'll mag that for you Jeffrey really nice one there. It's a 99. Congrats on that one Drake one and rookie pick it. We've seen all day. Really? I think so pick it all Nice fireworks McCaffrey and Blue Wave Antonio Gates It's a 199 the Kenny pick a new recruit insert blue ice Tolbert rookie for the Cowboys. It's a 99 fireworks Breeze Hall rookie Blue Ice Chris Jones have orange Ramon Dre Stevenson 249 Gare Wilson rookie lock down Cameron Hayward and DJ more hyper. Oh nice silver rookie here sauce My sauce for you there Jeffrey got Quentin Nelson To 125 Skyler Thompson brilliance Josh Allen Got Shannon chart That is purple crack tights at 225. All right, Jeffrey. That doesn't for box two. Let's do box number three. Good luck Got a seven box personal here for you You'll get you going Silver Zack Thomas and Purple Power Amani Wari Wari To 49 and Garrett Wilson rookie silver rookie Christian Watson Nice one there and Pat Surtain red wave. Can you pick a base? 1.1. I'll give it a go nice brilliance cracked ice Justin Herbert Look nice with some color To 499 and a Kyler Gordon rookie As bear is also in 99 got height Justin Fields oranges Marquise Brown All right, let's keep it going Jeffrey. Good luck. Good luck. See a big rookie Maybe a case hit emergent Damian Pierce Got Jordan Davis that is orange wave to 60 Nice rookie there. Absolutely bees for the Eagles Appreciate that bevo. What's up smack? How you doing got brilliant Dalvin Cook DeAndre That's hyper to 175 Where you guys think Hopkins goes literally just gonna think he's going somewhere literally just gonna ask that don't know where I got No idea Minnesota dude Yeah, that would be not cool Packers fans would have a real bad week in joku silver and Nice one here. That's gonna be rookie green Scope Otto Jameson Williams There you go. Lions rookie receiver Came on strong let into the year only had a couple games with some injuries. There you go Really nice Jamo there Jeffrey got hutch also for the Lions So smack. Yeah, man. It's been fun. Good day on rich break this morning Pulled a bunch of heat out of prism and then Sam was a continuing that We got Carson wins blue ice Skyler Thompson Yeah, Dallas gets D-Huff. That's I don't think that could mess that up You just throw it up. He's down there somewhere Brilliant scooper cup Tyler Lockett green Scope Trayvon Walker height Bateman and Purple ice McCordy Just retired for the Pats Yeah, I'm trying to think of where the least place I'd want the Andre Hawkins to go. He'll probably go there Steelers Got Corey Davis silver and Vet Otto this is gonna go to the Broncos Jerry Judy. Let's go into Jeffrey, but he plays for the Broncos Yeah, Paul is really solid. Yeah, ooh nice silver throwback Herbert beautiful card and Blue Wave Jalen Johnson Lave and cross All right, Jeffrey box four, let's see a big big hit here, man Trying to try to catch up to those two fine. I'd say that yeah Make sure it's a box. You feel my pain. Yeah, all right, Tom Brady I'm good with that though for Jeffrey I'll take even touching it got silver Kyler Elam Chris Jones hyper Desmond Ritter rookie. Yeah, I think my favorite. Hey yesterday was a Sauce gardener snake skin at least that I pulled sweet Drake London rookie and Desmond Ritter nice one here Jeffrey. That's gonna be orange to 249 for the Falcons QB We'll mag that for you. Nice QB there. Yeah, right? Yeah, it is guy. It is seven box personal Love to see it And we have a camo For a rookie emergent is pick it and the camo is a rookie It is Dachston Hill Number to 15 for the Bengals and it's 15 of 15 the book in Nice rookie green or force camo. This is so sweet pick it rookie. Oh, yeah, I love this the Select snakes against you the green and green and black Yeah, I wish they were kind of numbered in in prism, but I mean case hits You got silver emergent Walker and Alex Smith purple George Pickens rookie Yeah, I really like all I'm gonna all same around to They got quite the quite the tandem going on over there There's a league winner here for me this year really Antonio Gibson blue ice to 99 Yeah, he he helped me I didn't win anything Certainly helped when he was healthy his beast Yeah, it's been a lot better this year for sure on the centering Almost every box is just about dead on. Yeah, Jamar chase silver Yeah, I can't Anderson to 149 or sorry to 49 another RPA. Let's go Jeffrey. See another Kenny got silver Micah There we go silver RPA this time it's sauce Gardner There you go Jeffrey doing well on the On the couple boxes so far. It's a 99 Sauce Gardner RPA That's on that Jeffrey. Let's keep it going Want to see a case hit for you? Yeah, he's doing well man. It's a sweet. Love to see it. Brilliant's borough Orange to 249 Zeke Hamilton and Dean fireworks is Alan purple ice Zach Thomas. Oh, have you smack really I feel like we haven't run into any here yet Keep my fingers crossed on that one then Got silver McKinney And it's a vet auto Gonna be George Kittle Nice one there Good vet to get nice kiddo, man, we've got waller cracked ice 225 definitely not a fan of McDaniels seems silver Dylan Got Joe mixing blue wave. We'll leave Willis base there Lockdown TJ Watt. Oh Nice Patrick Mahomes blue wave It's to 199 MVP saw space All right, Jeffrey is box number five. Let's keep it going Good luck. Good luck. Come on. Let's go manga. Let's go color blast Need it Emergent backwards that is crystal lava. I thought that was an auto and JK Dobbins That is purple power to 49 JK all day. I'm not sure on that one smack Seems like one or two. Oh We have a profile's I believe This is very short printed here Emmett Smith for the Cowboys boom Have not seen one of these yet Really nice card here Emmett Smith for the Cowboys Can't remember what his cuz force kept calling it a bunch of different stuff Couldn't remember what it was called. I think this is profiles Really nice one there. We'll mag that for you Jeffrey. Haven't seen that yet. So that's got to be a tough one to pull Got orange to 249 rookie Jake Jackson Tyson Chandler and Sam Howell Ty Chandler Tyson wait a second profiles. Yeah, thanks you Josh. I thought I thought I was that sweet Got silver Jeremy Chin. Oh, here's the auto. I was looking for silver rookie auto Chris Olavae There you go, Jeffrey nice one there for the Saints rookie Really solid year He actually caught a touchdown and had a concussion on the same play That was pretty insane Bailey zappy What a pole on that's that. Yeah, I know cuz I had a fantasy. I needed the touchdown. I needed the touchdown I was like you better have held on to Fireworks my homes have Marshall Falk purple ice Joshua I have been I have been snagging some Olaves for sure though. That's kind of that's definitely been happening That's a prison break Might add something else that is Jamar. Oh, here's a Robert Woods The Bobby trees. I believe the 25 Navy camel. Those are sweet. Both the camels are so sick. They're auto here. Good luck silver Tom Brady nice one there and We got a JK all-day auto for the Raven silver not numbered there Yeah, Pacheco base green scope. There you go Bailey zappy there rookie throwback and Our green scope is Josh Pascal rookie for the Lions number to 75 silver Antonio Gates purple Shaq Leonard emergent John Dotson Johan and Sam Williams rookie purple ice for the Cowboys silver rookie Brian Robinson and Our orange the 249 is AJ Dylan Ken Walker rookie. It's like another auto here. Good luck Jeff Oh, never mind. It's just another backwards height That is Trevor Lawrence this time an orange wave to 60 Justin Herbert Nice Herbie there. You're at 98% on Hawke Wars. Oh my lord How much have you just walked around Many people have you talked to someone imploded on me brilliance Rogers Blue ice Matt Ryan there you go. Hey, Glee Blue ice to 99. All right, Jeffrey. We got two more for you. It's me box number six. Good luck Brand-new 2022 prism. Oh, you lost a few hours. Oh, man That's not good. Yeah, silver prism break Lawrence And DJ more hyper Kenny pick it rookie.  I remember you and Brad always talking about Final Fantasy. I know you're a grinder silver rookie new recruit Malik Willis and Jay Jetta Is purple ice so 225? silver lambjack Look away Josh Auto to 149 is gonna be red wave Chris Olave Nice one there Jeff To a lava autos at these couple boxes It's 149 you already maxed all the levels to each How far you off cam I got you Damian Pierce instant impact. These are a little shorter printed this year for sure Harder to come by. Ooh nice Sam Howell there rookie QB Number to 249 saw the one there Jeff believe he's QB1. I mean, I think he's QB only one right now George Pickens they got rid of wince and the Heineke. I don't know if the enemy's plan is there sign Chad Henney Somebody's familiar with Got Shannon shark skill Is the 99 Jordan Davis Jedi survivor should be sweet. Yeah, you're a Star Wars guy right now somewhat yeah Like I'll watch them like watching the Mandalorian right now. Like yeah, so Joe mixing. Oh Here we go Jeff Kenny pick it base blue ice to 99 boom Let's go Jeff. Very nice. Got the Kenny pick it RPA and now the base to 99 Congrats Jeff that's sweet We'll mag that for you for sure Kenny too close Oh, they picked up Jacobi Wow, I didn't know that. Oh, yeah, that's right Jacobi solid Got scary Terry blue wave Silver is Kevin Byrd auto. This is gonna be Brandon cooks Silver auto Hamilton and Dean good luck Jeffrey. Let's keep it going for you Silver rookie is Danny Gray and we got a forest camo coming It is Corey Davis Rookie for the Jets on the forest camo to one or it's a 15. You got a saw space emergent Trayvon and D hop to 249 D hope Desmond Ritter base. Yeah, I'm gonna go with no on that one as well bevo Yeah got Aaron Donald Silver and Hyper Antonio Gates seen a lot of colored gates here. What's up JT? Yeah, man We got a Kenny pick it to 99 just a second ago and a can you pick it RPA? We also got the the profile short print Haven't seen one of those yet for Emmett Smith pretty sweet card new recruits Traylon Birks and Nicoby Dean rookie as purple ice to 225 Malik Willis rookie base All right, one last box for you here Jeffrey is box number seven. Good luck. Good luck Purple ice rookie this is Danny Gray For the Niners got Matt Corral rookie base. We have breeze halt new recruits Marshall Falk hyper how's everything been JT been a while man?  Got silver track dorsito. Oh sweet one here We have Michael Vic flashback auto on the green scope That's the 75 nice one there Mike Vic 11 of 75 Got Chris Godwin red wave. Yeah, come on snakeskin. I'm down. I want a manga. I haven't got one of those yet emergent Ken Walker We have purple Antoine Winfield the peace sign on Tyreek Pretty cool Pretty cool Peace them right out of Kansas City Got silver Hunter Renfrow Nice mall here Jamal Williams is green scope to 75 gray and woods 15 touchdowns seven against the Bucks Lockdown Shaq Leonard and Another green scope this one is David O'Jabbo Rookie for Sam's Ravens to 75 Kind of looks like Tiafo Got Alec Pierce Silver rookie Rashad white Got orange Dante Hall 249 Yeah, guys grab your spots for tomorrow. I got Sam in the morning and force at night Forest Fridays do not want to miss Isaiah likely to 249 Ryan D. He's always come in and say forest on Friday. Nothing better something like that Friday for us Friday. It's nothing better Some of those right fireworks lamb. We got Robert Woods purple ice to 225 Okay, York Bobby trees Bobby trees fireworks Tay Adams Got McClourn to 249 Bailey zappy rookie All right, two packs left Jeffrey. Good luck. Good luck This is a seven box personal for you Silver is JK Dobbins We have rookie Otto number to 25 forest camo Jalen Tolbert Not bad there for the Cowboys rookie receiver that is one of 25 Got a Willis base a height Bateman and Trav Kelsey nice one there on the blue wave to 199 Wide receiver one in the league. All right, Jeffrey that does it for your seven boxes We'll recap these magda hits real quick. All right. All right We had Kenny picket blue ice to 99 Sick one there Jeffrey 38 and 99 Have a lava rookie Otto to 99 That it or sorry to 149 red wave rookie Otto have Desmond Ritter to 249 rookie base Jameson Williams green scope Otto to 75 We have a sauce Gardner RP 8 and 99 Nice sauce there. We have a Kenny picket RP 8 and 99 We got gold to 10 Skyler Thompson nice rookie QB there on the true gold 6 of 10 and We have portraits Emmett Smith Very short printed one there. All right. Nice hits Jeffrey. We will get those Get those sorted out and we'll get them right out to you. Thank you so much, man


Hello, welcome to this lecture on bio-mathematics. Currently, in this section we are discussing statistics relevant to biology and we discussed about averages, standard deviation and some kind of distributions. So, we said the data, the simplest thing that we can learn from data is average and then standard deviation and in fact, to present the data instead of presenting a set of numbers, we can present it as a distribution which will have more meaning, which will convey much more than a set of numbers would convey. So, now, we discussed various distributions, we discussed distribution and said that typically many things in nature fall into something called normal distribution in quotes. So, now, today we will understand what normal distribution is. So, we just said in the last lecture that normal distribution has the form p of x is equal to a e power minus x minus x average whole square with some constant b. So, this is this particular form, normal distribution as this particular form. We will try and understand more about this in this lecture. We will pretty much understand more, learn more thing about this. So, the title of the lecture is, so this is in statistics itself and the specific title of today's lecture is understanding normal distribution. In this lecture, we will try and understand pretty much everything about normal distribution. Like whatever you need to know, know about normal distribution, we will try and understand this lecture. At the end of this lecture, I hope you will, you will know pretty much, you will be very confident to deal with normal distribution and pretty much learn everything about normal distribution that one should know. So, now, what do we, what is the simplest thing that we should learn about normal distribution? So, let us say you are given a normal distribution, which is like a normal distribution, which has this bell like shape. What do, what can you know from the, from a first look? What do you know from a first look? What all things you can learn from this? You know that the probability is high around 150. So, we had yesterday discussed, let us say this is above the height of a, this is the probability distribution of the height of students in a particular class. So, if x is 150, let us say this is 150 centimeters. So, this then, if you know that the probability has a peak, the maximum number of students will have around, heights around 150. And as we go to either side, the probability of finding students with lesser than 150 and more than 150 kind of decreases. So, this is something, which we can know from the first look as we discussed last time. But what more can we know from just looking at it? Can you tell, we look at here, can you tell what is the average? By looking at the curve, can you tell what is the average or what is the standard deviation? Well, you should be able to do this or you should be able to do pretty. So, you can at least roughly say by just, just looking without doing any measurements. Just by looking at the axis of the graph and then the range of the graph, you can tell and just looking at the graph itself, you can tell roughly what is the standard deviation and pretty much accurately what is the average. So, that is the question here. Can you tell what is the average and what is the standard deviation? At the end of this lecture, you should be able to do this. Now, we know that this normal distribution is a Gaussian function. It is a mathematical function is called a Gaussian. It has this particular form A e power minus B x square. So, this is this distribution, this if you plot this A e power minus B x square for some a given value of A and given value of B, you will get this curve. So, there are just two parameters that describe this normal distribution. One is, so we will discuss about that, but just by plotting this function, you will get this Gaussian, this particular bell shape. By plotting this using a, either you can take numbers, take a value for A and B and for different values of x, you can plot it and you will get this. But in many cases, you must have seen that normal distribution is presented in this particular way. Have a look at here. It might be, you might have seen that it is presented in a way that p of x is 1 by 2, 1 by 2 pi sigma square exponential x minus x 0 whole square by 2 sigma square. You must have seen this. So, what is this relation between this and this? It is clear that A is 1 by 2 pi sigma square and B is 1 over 2 sigma square and x 0 is 0. So, that is what the relation between, by comparing this and this, we can know that 1 by 2 pi sigma square is written as A, 1 over 2 sigma square is written as B and x 0 0. So, why this particular form of A and B? Why is A is 1 by 2? A can be anything, but why is it related to the sigma? Here the sigma is standard deviation. So, for normal distribution, sigma is standard deviation. So, why is A and B related to standard deviation? So, that is, why this particular form? Why is A is 1 by 2 pi sigma square and B is 1 over 2 sigma square? So, you can know this just by comparing. So, let us, let me write it here. So, what one function here is A e power minus B x square and it is also 1 by, so this is the Gaussian function. When normal distribution is written, it is often written that if you look at any book, normal distribution is written such a way that x minus sometime it is even written x average whole square by 2 sigma square. So, if you compare this and this, it is clear that this is A. So, A is 1 over, sorry this is a sigma square here, 1 over 2 pi sigma square and B this is 1 by 2 sigma square and x average in this case is 0. So, this is the, if you compare this and this, just you get this particular thing. Now, x average can be anything, but A is always 1 by 2 sigma, 1 over 2, B is 1 over 2 sigma square. So, why this particular, how is sigma is coming here? Why is it suddenly sigma coming here? Why this particular form for A and B? We will understand this in this lecture. So, let us start with the simplest case. So, the simplest question is, so if you have this Gaussian function, so if you have a Gaussian function P of x is A e power minus B x square, how do we find A? So, this is the function of normal distribution. If you plot this function, you will get the normal distribution of bell shape curve. Now, from this, how do we find A? So, that is the question. So, let us think a bit. So, this is probability distribution P of x. What does this mean? This means that, the probability of finding, having a value of x is this. So, probability is this, that what is the total probability? Total probability is always 1. Why? Because, like if you have, you know that in the case of head and tail, half, half is the probability. So, half, half plus half, probability of finding head is half, probability of finding tail is half, half plus half is 1. This is something that you know. Total probability has to be 1. In the case of marks that we discussed in the last lecture, if you have a probability of finding 1 mark, probability of finding 2 marks, probability of finding 3 marks, probability of finding 4 marks, up to probability of finding 100 marks, the total probability has to be 1. Because, you will surely find a student with probability, you will find all, even if all students got 100 marks, probability of finding 100 is, the sum will be 1. Let us say, all students get 53 marks, all students get 60 marks, still the sum, total probability of finding all the marks is 1. So, the total probability has to be always 1. That is the definition of probability. So, when we defined probability the other day, yesterday, in the previous lecture, we said that P of x, when we wrote, we had written that, in the case of H we had, like in the case of P of H i, we had written that, this is, can be written as the, the, the function. So, you had, you had n of H i into sum over i, sum over i n of H i. So, sum over i, if you divide this. So, now, if you do, now sum over of P of H i, this has to be 1. So, total probability always has to be 1. So, knowing this, knowing the total probability has always to be 1, we have this function P of x is a e power minus b x square integral minus infinity to infinity P of x d x has to be 1, minus infinity to infinity P of x d x has to be 1. What does this mean? This means that, minus infinity to infinity P of x is this. So, a e power minus b x square d x has to be 1, a e power minus. So, this is, if this is the case, a is, a can take a outside. If a is outside, this would imply that, a is equal to 1 over minus infinity to infinity e power minus b x square d x. So, just by knowing the integral of this function, we get a. a is 1 over e power minus infinity to infinity e power minus b x square d x. This is a. So, we get a. Now, how do we, how do we calculate this integral? What is the value of this integral? So, let us, quickly find out the value of this integral. So, at this, to begin with, I will just tell you the answer that, minus infinity to infinity e power minus b x square d x is root of pi by b. How do we get it? Later, we will discuss, how do we get it? How do we do this integral? But, for the moment, just understand that, the answer of this integral is root of pi by b. If you have a function e power minus b x square and if you integrate minus infinity to infinity, the answer is e power minus b x square is root of pi by b. How, we will show this is the answer indeed. But, for the moment, just take my word. This is the answer. If this is the answer, a is 1 over root of pi by b. It is root of b by pi. This is a. So, our probability distribution, if this is a, the probability distribution p of x, basically is, can be, can be written as p of x is equal to root of b by pi into e power minus b x square. So, this is our p x and surely we can phi c that minus infinity to infinity p of x d x is 1. So, this is the Gaussian function and a, we found already a. So, this is the full form of the Gaussian. Probability distribution, normal distribution has this particular form. Well, so now, this b by root pi is written such a way that minus, the total integral is 1. Well, now, what do we want to find out? So, we want to, we have, we had this first person. What is standard deviation? So, for the moment, we will consider, to begin with, we will consider this particular form for the probability distribution p of x e power minus b x, b by root pi e power minus b x square. Now, then we will find out for this particular case, what is the standard average and standard deviation? So, we know the definition of average and standard deviation. What is the definition of average and standard deviation? So, the average, as always. So, in the, in the previous lectures, we had defined average, x average is defined as minus infinity to infinity x p of x d x. So, just remember, this is always the definition of x average. If you are given a p of x, the probability distribution, the way to find x average is this. You multiply with x and integrate. So, just remember, this is the definition of x average. If p of x, p of x has to be, the integral of p of x is 1. So, the process, we just did a few minutes ago to calculate integral of a, this find out a is called normalization. So, this process, we had here, finding of a, this process is called normalization. So, this is the p of x is a normalized function. If p of x is normalized function, this is the definition of x average. So, we have x average and definition of x square average is minus infinity to infinity x square p of x d x. So, this is the definition of x, x square average. And we know the standard deviation sigma is nothing but, x square average minus x average square, square root. So, this is the standard deviation. So, we have to find out x average. We have to find out x square average and then, we can find out standard deviation. So, we have a p of x here. And our p of x is root of b by pi e power minus b x square.  So, I can write it outside. Now, it turns out that, what is the answer of this? Well, without doing this real calculation, just by looking the function plotted, you might be able to know the answer. So, let us just see how the function is. So, this is 0, this is plus x and this is minus x. So, this is minus 1, minus 2, this is 1, this is 2. And this is some function which is symmetric on both sides. It is this way is the same, this way is the same. So, we have a function which is exactly symmetric on both sides, exactly same symmetric on both sides. That means, whatever the value of plus 1 is the same value at minus 1. So, the e power minus b x square, if you find out at plus 1 is same as the value at minus 1. So, the only thing, but differs is x.  So, you multiply with 0.1 and multiply with minus 0.1 and sum of them.  So, since it is symmetric and since it is from minus infinity to plus infinity, the answer of this is 0, because for every positive value there will be a negative value which is cancelling. So, if you take one term in this, if you expand this, if you integrate this integral, if you write it as a sum. So, let us write this quickly. So, integral minus infinity to infinity x e power minus b x square can be written as sum over i x i. So, let us call i is equal to minus infinity to infinity x i e power minus b x i square delta x. So, this is d x, if you wish. So, this can be written. So, now, let us have a look at this function when x i is. So, there are many values. So, let us call this minus 1, 1, minus 2, 2. So, if x i is. So, let us look at this value 2 and minus 2. So, at this particular 2, what do you get? 2 into e power minus b x square x 2 square which is 4 plus minus 2 e power minus b 4. So, this is 0. So, just by summing both sides, you are multiplying with plus 2, you are multiplying minus 2. So, for every plus value here, there is a minus value multiplying on the other side. So, the total of this integral is 0. So, this is one way of saying it. So, it turns out that the x average is 0. So, x average which is integral x root of b by pi e power minus b x square is 0. Now, there is a rigorous way of showing this which we will… This is actually not very difficult at all. So, let us… If you want, let us do that the rigorous way. So, what we want? We want root of b by pi e power minus b x square x dx. I can write this as root of b by pi del by del x of… So, let us look at this. So, what is del by del x of e power minus b x square? This is e power minus b x square into derivative of minus b x square. So, this is e power minus b x square minus 2 b x. So, just by knowing this, just by knowing that that e power minus b x square into minus 2 b x, this is into times… This is the derivative. We immediately see, we can immediately see that just from this, I can say that del by… Just by this, you can see that del by del x of e power minus b x square divided by 1 over 2 b. So, I take 1 over 2 b the other side is equal to x e power minus b x square. So, x e power minus b x square can be written as minus 1 over 2 b del by del x of e power minus b x square. So, you substitute this there and do the integral yourself. You will find the answer that it is indeed 0. So, I will not do this clearly. You can do this yourself just by substituting the fact that x e power minus b x square is del by del x of e power minus b x square minus 1 over 2 b. Because, this you know, if you do this, you will get this. So, let us do that. So, let us substitute this here. So, what we want? Del by del x 1 over 2 b with a minus sign integral e power minus b x square d x. So, this integral has to be written outside. So, this is 0. You can convince yourself that minus infinity to infinity, this is 0. Do it yourself and convince yourself that this is indeed 0. .. The next question is, how do we calculate x square average? So, we have p of x is root of b by pi e power minus b x square. So, for most of the time, I will write, sometime I might write root of b by pi as a for convenience. So, what we want to find out is that x square average, which is nothing but x square e power minus b x square minus infinity to infinity d x. So, we want to find out x square average now. So, we know, p of x is root of b by pi e power minus b x square and x square average is a into a is root of b by pi, which is a. I write it as a sometime minus infinity to infinity x square e power minus b x square d x. So, this is the definition of x square average. Now, we want to find out, what is x square? So, first let us do this particular integral. We know that a is root of b by pi. So, let us calculate, what is integral x square e power minus b x square d x. So, let us, let us do this calculation. So, what do you want to calculate? We want to calculate integral minus infinity to infinity x square e power minus b x square d x. To do this, we do a standard trick, which is, we can write this particular function x square e power minus b x square as minus del by del b of e power minus b x square. So, if you, if you find the derivative with respect to b and multiply with a minus sign, you indeed get this. If you can, if you know, del by del b of e power minus b x square is e power minus b x square times the derivative of this, which is x square and with a minus sign. So, this, just by using the fact that x square e power minus b x square is minus del by del b, derivative of this function e power minus b x square. We can, instead of x square e power minus b x square, we can substitute this back and write it as minus infinity to infinity with a minus sign here and del by del b of e power minus b x square d x. So, del by del b is independent of x, because b and x are completely different variables. So, I can take this outside. So, this is minus del by del b of minus infinity to infinity e power minus b x square d x. So, what did we do? We wrote integral x square e power minus b x square d x as minus del b of del by del b of e power minus b x square d x minus infinity to plus infinity. We know the answer to this. We just told some time ago that answer to this particular integral minus infinity to infinity e power minus b x square d x is root of pi by b. So, this is the root of pi by b. So, the answer, the full answer, that is, if that is root of pi by b integral minus infinity to infinity x square e power minus b x square d x, there is nothing but minus del by del b of root of pi by b. What is this? There is a root pi there. Then, del by del b of root 1 over root b. 1 over root b can be written as b power minus half. So, this is nothing but root pi into derivative of b power minus half is minus half is a minus sign. So, this is a minus half. So, this is a half and minus minus cancels. So, plus half b power minus 3 by 2. So, this is the answer of this. So, this can be, if you wish, can be written as root pi by 2 into b power 3 by 2. b power minus 3 by 2 can be written as 1 over b power 3 by 2 and there is a half and there is a pi. So, integral x square e power minus b x square d x, there is nothing but root of pi by 2 b power 3 by 2. Now, what do we want? We want x square average. x square average is defined as, as we said, you have to multiply this with root of b by pi, which is our a integral minus infinity to infinity e power minus b x square d x, x square d x. So, this part we just found. This as, so this is equal to root of b by pi, root of b by pi into, this just we found. This as root pi by 2 b power 3 by 2 by 2 b power 3 by 2. So, b power 3 by 2 is what? So, b into b power 3 by 2. So, let us write b power 3 by 2 here. b power 3 by 2 can be written as b into root b, if you wish. So, this can be written as b into root b. So, root pi root pi cancels, b root b and b power 3 by 2, they together produce. So, if you wish root pi and root pi cancels, this can be written as root b by 2 b power 3 by 2. Root b is b power half. So, if you write root b as b power half, this is 1 over 2 b. So, x square average is 1 over 2 b. So, what did we find? We found two things. We found that, e x average of a Gaussian function of this particular probable normal distribution, if you look at here. If you look at the slide, you will see that x square. So, what we just found that x square average is 1 over 2 b and we had found that x average is 0. So, we found two things. We had found x average 0, x square average is 1 over 2 b. And what do we have? We have p of x is root of pi by b e power minus b x square. So, if you, this is what written the slide roughly here, that x average is x p power p x d x, which is 0 and x square average is x square p of x d x is 1 over 2 b. So, and what is the standard deviation, if you know this two? If you know this two, the standard deviation is x square average minus x average squared. So, the standard deviation, here x square average is, this x average is 0. So, standard deviation is x square average root itself. So, the standard deviation is… So, the standard deviation sigma is x square average minus x average square root. So, this is 1 over 2 b minus 0. So, it is like 1 over 2 b. What does this imply? This implies that b is equal to 2 sigma square. Sorry, b is equal to 1 over 2 sigma square. b is equal to 2 sigma square inverse. So, that is I sigma square is equal to 1 over 2 b, b is equal to 1 over 2 sigma square. So, we get a relation between b and sigma. This gives you a relation between b and sigma. So, b is 1 over 2 sigma square. This is b. b is 1 over 2 sigma square. So, once you know the relation between b and sigma, we can look back our function. Our function was, what is the p of x? p of x was root of pi by b e power minus b x square and we just found that b is equal to 1 over 2 sigma square. So, I substitute this back here. Sorry, this is wrongly written. So, this is the distribution function p of x is root of b by pi e power minus p x square and we found that b is equal to 1 over 2 sigma square. So, if I substitute this for here, what do you get? You get p of x is equal to root of b is 1 over 2 sigma square. So, you get 1 over 2 pi sigma square e power minus x square by 2 sigma square. This is 2. x square by 2 sigma square. This is 2. This is 2. So, p of x is equal to 1 over root of 1 over 1 over 2 pi sigma square e power minus x square by 2 sigma square. So, this is what precisely written here. So, we had asked this question. Why this particular a and b? We found that a is. So, there is a square root here. Sorry, there is a square root here, which is forgotten here. So, the correct form as it should be, it is p of x is root of 1 by 2 pi sigma square into e power minus x square by 2 sigma square. So, this should be the correct form for. So, this is the precisely the reason why sigma square is put here, because if you just call it a e power minus b x square. And we found that the, if you calculate from the, you take this function and if you take this function and calculate the sigma, that is x square average minus x average square, what we get is 1 over 2 b. So, that means root of 1 over 2 b. So, what means, what is that? Sorry, sigma is root of x square average minus x average square and if you found this, we found that it is root of 1 over 2 b. So, that gives a relation between b and sigma. So, that gives p of x this particular form. So, this is the reason why p of x is this particular form, where sigma is coming here and here. So, we found x average and x square average. Now, this is the function we took was, if the function we took was a e power minus b x square, if you had taken the function p of x is equal to a e power minus b x minus x 0 whole square. Same answer you would have got except x average, you would have got x 0 and x square average minus x average square, you would have got 1 over 2 b. So, this you can see yourself by just finding that x average is x 0 in this case and the standard deviation is the variance x square average minus x average square is a variance and variance is 1 over 2 b. So, the full function, the full function the probability distribution function p of x can be written as 1 by root of 2 pi sigma square e power minus x minus x 0 by 2 sigma square whole square. Now, given thus this particular form just by looking at this particular function just by a graph of this function, what all can you know just by looking at the graph of this function, what all can you know. So, let me take some particular value sigma is equal to 10. So, sigma square is 100 and let me take x 0 equal to 150, just some two values and just plotting this particular function by taking these two values you get this, what we had plotted earlier we get this. So, what does this tell us, so have a look at the slide here, what does this tell us. So, if you look at the slide here, this tells us that, so this is exactly the what is plotted here. So, what do you know from this loop by just looking at this, the peak is at 150. So, the peak is the point where is average value. So, x 0 is 150, so x average is 150. So, x average is 150. So, what does this mean, x 0, so peak is at 150. So, just by looking at the peak we can know what is the average value. So, average value is easy to find just look at the peak, the position of the peak is 150 that will be the average of this distribution. Now, how do we find the standard deviation, how do we find the sigma. So, just by look at this slide here, so you look at this paper here. So, we know that p of x is 1 by p of x is 1 by 2 pi sigma square e power minus x minus x 0 whole square by 2 sigma square and x average is 150 which is x 0. So, just note couple of things, if you know x 0 and if you know the sigma you know everything about p of x. So, you have to know only two things, you have to know x 0 and you have to know sigma then you can know you know everything about p of x. So, I told you if I find x 0 by looking at the peak we can find x 0 and how do we find sigma, we want to find the sigma. So, it turns out that sigma is related to the width of the distribution. So, if you look at here, if you look at the slide the width this distance. So, this is this are related. So, how do how is this related? You can measure the width here, the width here, the width here, you can measure the width of the distribution at various places. Which of this width will give you sigma, will this distance give you sigma or will this distance give you sigma or will this distance give you sigma. So, this is the question which of this distance will give you sigma. So, let us look at this again the Gaussian function. So, Gaussian function is a e power minus b x square, the Gaussian function is a e power minus b x square or p of x if you wish. Now, this we know that this is nothing but 1 by 2 pi sigma square square root into e power minus x square by 2 sigma square. So, for when will it let us say. So, when will it sigma will be equal to. So, what we want? We want just by looking at the width by measuring this distance. We want to get the sigma, but where will you measure the distance that is our question. We measure it here or will you measure it here. So, what we want? We want this distance. So, this is if this is x, this distance is 2 x. So, we want 2 x to be sigma. That is what we want for a particular value of y. So, if we if you look at this what we want? We want for a particular value of y for a for a given we want this distance that is this is x and this is also x. So, the 2 x we want a sigma for a value of y. So, given a y there is a place of there is a particular value of y for which 2 x is sigma. What is the value of y for which 2 x is sigma? So, let us first put a sigma is equal to 2 x and then calculate. So, what we have? We have p of x is 1 over 2 pi sigma square square root e power minus x square by 2 sigma square and substitute sigma is equal to 2 x. So, what do we get? 1 over root of 2 pi. So, this what we want is just. So, this particular part let us look at this particular part. If you look at this what you get e power minus x square by 4 2 into 2 into 2 x square. So, 2 x square is 4 x square and again 2 into 2 into 2. So, what we get? This particular part.  So, we will get p of x is a e power minus a x square by 8 x square x square and x square goes. So, what you get? You get a e power minus 1 by 8. So, what is a? When x is equal to 0, if you put x is equal to 0, p of x is a. So, if you e power minus 1 by a if you multiply a with e power minus 1 by 8 at that particular value a is this as a constant is a maximum value which we know a is the peak. So, a we know just by looking at it because if you have a function this values a this is the peak when x is equal to 0 what is the value of y that is a. So, if you multiply a with e power minus 1 by 8 that will be the y where sigma is 2 x if the width is equal to sigma. So, if you look at the distribution if you look at this particular distribution and multiply this peak with e power minus 1 by 8. So, what is e power minus 1 by 8? So, it turns out that e power minus 1 by 8 is approximately 88 dot some percentage. So, it is approximately 88 percentage. So, if you multiply p when p of x is a e power minus 1 by 8 if what did we find we found that if p of x is a e power minus 1 by 8 sigma is equal to 2 x or in other words we found that if sigma is equal to 2 x p of x is a e power minus 1 by 8. So, if you look at that 8 if I look at 88 percent from this peak. So, you multiply e e power the peak by this particular number which turns out to be 88 percent because this is e 1 by 8 is around 0.125. So, 0. e power minus a e power minus 0.125. So, this is coming 0.88. So, this is something like 0.88 of a approximately. So, if you look at the 88 percent of this peak somewhere here if you measure the width at the 88 percent at this particular point you will get the standard deviation. So, the width so the peak at the peak the x value of the peak is average and the 2 x the width the width at e power minus 1 by 8 of the peak is the standard deviation. So, just by looking at it we can know standard deviation as well as the average. So, to summarize what did we learn. So, we learned that we learned that x average if we is. So, we what we found out we learn how to calculate x average we found out how to calculate x square average we found out how to calculate sigma. So, if you have a particular function of the if a Gaussian function we found that sigma is related to if you have a function of the form a e power minus b x square sigma is 1 over 2 b square root. And we found that the peak is x average and the width at e power minus 1 by 8 times of the peak maximum value is 2 x which is at the 2 x the width at this particular point will give you the standard deviation. So, we learned you many things about normal distribution. So, use this information use this ideas next time when you see a Gaussian distribution just look at where is the peak and the width at 88 percent of the peak. And also do yourself what is the width the full width at half of the maximum we found that the width at 88 percent of the maximum is sigma what is the width at half of the maximum you find out yourself. So, keeping keeping this things that we will learn in mind that integral of e power minus b x square and average and x square average this will be very useful for various things in the future. So, discussing this we will stop today's lecture. Bye.