What is up a welcome back? oops Do you like to do a little bit of make it so do we and we have new videos each week? And today is Tuesday, and you know what that means test cut Tuesday But I feel like a fraud this Tuesday. These are not test cuts today They were test cuts from last week. We took them to the craft show. Yeah, May 4 and already been test-cutted and Test sold and yeah and test and test so that's good one. The whole process has been tested Which one are you doing today? I Will do I don't know whichever one you don't want to do. I don't know. I don't mind either one All right, I'll do the pine cone We just tested how we pick and this is how we did it. I'm doing the Santa Belly It's like a classic Santa Belly, but with some layers And this one's just a wintery one I wanted something again people love to have them not just people love the Christmas ones But they also like to have one that they can keep up all winter January and you know all the way up to Valentine's Day when then they change it to the Valentine's Day sign No, mine's pretty much dead December 26th But it can go up now yeah Halloween I've seen that this year. That's new this year somebody Yeah, well as we have been so I heard that on the radio or something But then as we were driving home the last couple of nights Christmas lights are everywhere. Remember me putting it and you were like that might be for Ramadan or something I don't think so. I saw more Christmas lights. Not everybody. I haven't feeling a lot of that It's just straight maybe people are starting to celebrate Thanksgiving like it should be well I saw a tiktok that said this how you do it you start celebrating Christmas now like all the way through December You take a break for Thanksgiving, but you're back into it after Thanksgiving like people don't really celebrate Thanksgiving Well, I mean guess. Oh, you guys saw that on film. I just caught the fly Yeah, I just caught that fly live Throw them on the table. I'll teach him a lesson. Maybe he won't be back All right, let's just say hello your friends. Yeah, let's say hello to my hangers Hey hangers, let's see who's here. You know, we're getting excited. We got a talk to you about behind me We got big news Oh, there you go, okay, of course, I see Tyra and Judy or Trudy I apologize bands chat Kimberly Helen. What's up Helen Veronica Bob Amy tiny hands Baker Terry Patricia Juanita Anthony Edwards that famous guy from ER Hollow Queen Rick Pooley That's a cool name Off-road everything. That's Corey Tracy Tasha G Cat Miriam Richard Eric Christina Hill that means Eric's not far behind T-Town t-shirts Cheryl Christian, what's up Christian? I'm waiting for that second half. I'm chomping at the bit Steve Alan Cat Ingrid. What's up Ingrid? Tom Great design Tom Minnesota makes Chris I like how you put the dash. Thank you for your trick or name Scott Laurie Chavez And that's all I see on YouTube. I can't see the Let me say hello to our Facebook friends. Yeah, you're scrolling down Shannon Let's see who else Laurie Donna Debbie Dina. That's funny Donna Debbie Dina. Oh the triple D's Nancy, I think that's it right now for the chat. I already did it Let me use this Chessone Chessone Becky and making life fun with Teresa and Cindy Hey, got a lot of people here. Go ahead and hit that like button for me Go look at the last comment. You got a good question. Oh, what's that? Open the door The last comment. Oh, it's actually next to the last now. Oh Is that the new paint? Yeah, we have a new background here. We've added elements Foxy Hughes paints is officially launched today. Yeah, I we we celebrated with some wah-wah We just launched it this afternoon Lunchtime I was telling Tanner I was like, we should have gone to the store and we should have party hats on you should have graphics on the screen We should have horns out. This was a lot of hard work We're trying to do that while we are getting ready for one of the biggest shows of our of our year. Yeah, so We kind of Took it off the other day and I keep forgetting. This is soft open. It's soft open. Yeah, this is yeah, it's just you and us It's between us a little hangers here. We don't have any official announcement yet. Well, that's not true It's going out everywhere. I'm not sure everywhere. Yes, it's kind of we're gonna flood it So we worked really hard. We you know, we bottle these we label these we picked our own Mixed it ourselves. Well filled each pouch. We didn't mix the color ourselves We had to mix each gallon, I think the way it's shipped is that all those Yeah paint additives are on top so we have to mix it in before we can bottle it color and everything Yeah, the pigment Someone was pretty cool some of that I Could not guess what the color would be Yeah, he looked over I was doing one of them and I know he was talking to me He looked away and came back. He's like, whoa, look at that one It went from like a dark dark green to like like I'm like, I don't need like an aqua like a really light Oh, that's right. That's gonna be like this one or something. Yes. I think it was that one But it started off looking like darker than this one Yeah, and you'll see everything every all of the names Fox Fox related Fox related partly related to Garrett's favorite movie. Yep, but not all of them and Not rad. It's not rad Yeah, no other favorite movie. Yes, I'm sure most of you not Pee Wee's big adventure Yeah, he made me watch Pee Wee's big adventure the other night while we were painting Well, we watched Mama Mia and how many times that we watched Don't say it. Oh, I can't say the name. Yeah, it might come alive Jumanji, yeah Let me get let us get busy. We can chat and work at the same time. Yeah, we can Yes, Amy Amy. You're awesome She was the first one to place a paint order the first one to place a paint order and I saw in here Somebody asked Amy She said I meant to ask you why you're called Amy tiny hands Baker And Amy said, let me send you the video Yeah, it was a good video Caught me off guard. That's a date of funny. Yeah, that's why he hasn't forgotten it. That's why you're forever Amy tiny hands Baker Made him laugh totally made him laugh out loud Got me got me Up Tom says Pee Wee's big adventure great movie great movie, right? Tom like I was zero interested in watching that movie zero interested in watching it But Garrett said I hear a lot of She's doing a lot of laughing to somebody who hates Pee Wee Herman.   It's just like Joe dirt Which is awful, but still I Can't even believe you watch that movie. I was like I did I've seen it a few times and I think it's ridiculously funny He was big adventure classic Classic And it was a What's that guy? It was the same guy that did like the Jack and Sally Tim Burton, yeah, it was Tim Burton that's who did Pee Wee's big adventure. Oh really? Yeah, and then the guy is interesting the guy that scored Batman and Like Edward Scissorhand did Pee Wee's big adventure Tom Davis said boo got to go to the Alamo You got to go to the basement of the Alamo. Yeah Oh Suba is here she made it. Oh We might not have any handles here. I Took them all up front for a workshop and I Can see this has a little dried paint in the top because I went and found it in The paint room where our kids paint and it was sitting just like this fully open So guess what? It'll be fine for a couple of days with it all the way open That's how I found mine. It's got just a little tiny bit of dried stuff in the top. Good to go So that was a unplanned test Steve's is it like a guy movie then Pee Wee's big adventure is that is that what it is? How old Queen yes, I know you like all those scary movies like totally get it Gil I can't wait to tell Garrett when he comes back. I'm gonna have to read your comment to him He will love it Gil's comment says mr. Miyagi man who catch fly with chopsticks to accomplish anything I've caught in a wasp with So funny yes, he was very proud of himself Then I was afraid to let him go because he got real man Exactly All right, let's put these out here and I already got a bunch of extra rulers And do I have the Paper plates yes, I do I was listing up all the reasons.  You just pop the top and squeeze That's what we said earlier. We're like no more FIFA bottles. Look at that And I took these two I had to do a workshop by myself And Kim was doing a workshop at a show So I didn't have the FIFA bottles and I did a workshop with these pouches And it worked great and they traveled way easier Than the FIFA bottles you guys did I tell you that story? He's Leave Garrett to his own devices. He can get very creative. So he had to find a way quickly to Well, you took everything all the wagons all the wagons Because it was all the craft show yeah all the trailer. Yeah, I had all of our Regular wagons that we take to the craft show Bad thing about this thing is I've got to pop out any of the leftover pinecone pieces There's a few of them in here anyway So Garrett had to find a way he had to Create a makeshift wagon and I almost sent a picture on our put it on our social media because I thought it was hilarious Oh, there is I call it Genius, I don't know about that. That's probably a stretch because I came in and it was the furniture dolly That he had rigged up And what did you zip tie?  There was a shopping basket. I don't know where it came from It's just get a task. It that's what it was Just get a task in a shopping basket full of paints full of the pouches of paints along with a couple of handles and He was pulling that on the furniture dolly. I was like, please tell me Please tell me you did not walk up to this to this establishment Representing our business with the furniture dolly with rope strung to it with a shopping basket with our paint and kits in it I mean, I was gonna say it like that. I still didn't know you know the bushes Know what you said was no, I'll put the furniture dolly in the bushes Yeah, when I got up there, I can't let him go by himself It was heavy and I had both hands full and I had to carry my tea so I Just pulled the thing and then when I got close I put my tea in the basket and I Hooked it on in there But I ditched I ditched it It's like it's like when you go to an interview and you have to ride a bike there You ditch the ditch the bike in the bushes near it like a block away And you come back for it later We've all been there All the flies back didn't learn his lesson. All right. This piece is scored. I didn't actually I Cut it in half before and it was two pieces, but it didn't save me any Cut time and it didn't save any like Painting it was just harder to glue back together.  So I think we'll get some stain This is how Garrett travels with the stain cans If you can see it it has a It's got a stain condom on it It's got a stainy hat Jimmy hat It has a rubber glove over the top in case it leaks it shouldn't leak but in case it falls over Lessons learned it did tip over in the car That was a mess so on the way back I gave them all But now I've been doing it the last few shows and it works works out well Are you gonna stain Yes Got them. I just took that fly out with a roller I Know it's wants to know can you do a sleigh shaped Merry Christmas door hanger? Sleigh shape I think we should Let's see Cat says she likes the pouches Thanks cat. Are your paint pouches refillable? Yeah, I mean I would say they're refillable. I filled them the first time Again, I Think so I haven't had to haven't had to refill them yet. I mean yeah, I don't think you can really do a safety seal But you can refill them. Yeah Yes, the thing about the cap on the end that little white cap when I first put it on it's a lock cap, so It's like seal safety seals and then you kind of break that cap when you Open it the first time but the cap goes right back on me. It is Re-sealable Did you bring that white in did you bring in? Avenger ash ash Avenger. Yes, I did Surprise Yeah, I didn't put it where you would look for it So busy last week so little sleep Yes, you guys, I don't know if you guys have had to do this yet, but Well, we had a really great show. We were in Richmond here at the craftsman's classic Which is right here in our backyard. Yeah, how great is that? I was so lucky. It was so lucky that it was in our backyard Why do you say that? Because we came home every night and had to remake yeah Well tons of inventory. Well, yes, we had to paint more Friday night, but paint more Well, yeah, I cut during the day, but Friday night we came back and we busted out 34 more signs You sure did 34 signs Before the next morning and then Saturday night we came back and did what we only did 18 We're tired. Oh my gosh.  What are you showing me? Are you not showing me? Oh It's going to dry and you're gonna need a second coat. So I'm gonna need that When you're done the white when you're done So we so we did we did great I mean, it was one of the best shows we've ever done ever done. The problem was the next day We were like, oh my gosh Saturday, yeah, that was on Friday and then I was like, what am I gonna do Saturday Saturday's busier than Friday a vendor came in To see me and we're chatting and I was like, yeah What stinks is all the good stuff is gone. He's like, whoa. Yeah. Whoa. Don't ever say that. Yeah And we were like all the best all the best the best yeah, the best sellers are gone We made a few more of the best sellers while we were on Friday night and then Saturday night We were just too tired Saturday night to even do much of anything, right?  Yeah Yes, it wasn't 18 signs. It was 18 like Door corners. Oh, right. We did all those door corners. Yeah, those sold door corners People loving the door corners, right? I see you guys commenting in the top tier patron group about how well the door corners are doing and the same for us Yeah, they're good dinner and great and thanks Laurie for the little fairy light tip I bought a hundred pack of those fairy lights and I've been putting them. Yes I charge an extra five dollars for the fairy lights and they do not care. They need them with the lights So except for one lady one lady said did he's have a timer and I was like, no, but you can turn them on and off How long will they last and then yeah, they keep get asked me. I was like, I don't know Can't really answer that How long have you kept them on you know, lots of things that go into that right who made them And then she was like Well, I don't buy anything with Christmas decorations that don't have a timer now I was like, okay, so I didn't think she was gonna buy it and I turn around I guess I was working with some checked out somebody else. I looked up and here she is She was just getting one without lights. I was like, oh, okay. You're about the only one that got one without lights But they love them We only did the two we only had time to get two sets done the set of the merry Christmas and the Grinch hand It was all we really got painted. I haven't had time to design anymore Yes, I'm anxious to do that I'm gonna get on that kick as soon as this I guess Maybe this week. I really want to so during the We had a workshop last night and during the workshop We said that we're gonna have kids workshops and stuff.  Yeah, she does the whole This is how our business works and everything You get these kids unfinished if you want to have a host your own paint party at your house And then she goes last night she paused and she goes Is that it? She goes, oh, and we're doing kid classes or something. Wait, wait, wait So I thought I was getting ready to say these are done in five easy steps and I was like these I was like, wait, I was like, oh wait one more thing I forgot to share that you can do these class. Of course, you can Attend a workshop at our workshop location. We host workshops every Friday, Saturday, Sunday and Garrett's you said something about the holidays and I was like, right? Where we'll be hosting kids workshops I mean, we've talked about it, but we don't really have anything planned so I don't have any good I just rolled that right out there and then must have three To five ladies. I don't know a bunch of people asked me three I talked to three ladies about it. You did and then kind of projects. What's the age when? When will they be posted and I said, um, well, they should be posted now, but We've been so busy. I said that we'll have them up by Thanksgiving. Look for them by Thanksgiving So I bought us a couple of weeks All right But we now have to we haven't done kids workshops before kids have come in. Yeah, we've had kids at workshops Yes, but we've never done a kid workshop. Yes, right and after the last kid was at the workshop I was like, we are never doing kid Because this kid used every color paint every roller. Yes kids workshops will be done different They get a paint palette. Yeah, and we will do the paint for them Uh-huh I don't know. He was although at the one that we did At the craft show we had a kid come and he did one and he did great He was like seven or eight. He did great His stuff looked great. I mean, well, he wanted to know how you would rate it. Would you rate it? Um, would you rate it a 10? How do you say that? What did he say? Would you rate it? Um 10 out of 10 he goes He goes, do you like it or something and we were like, yeah, I did great He goes, would you rate it 10 out of 10? I was like, absolutely Um, what does that say up there? Does that matter? No, we're good. There's no card in the camera. All right, we're good I don't normally see that. Do I I wanted to make sure you guys could still be there during every last resin dragon studios is here What's up resin dragon Oh, wait a minute. I'm way back way back. Hold on. Let me catch. Let me go to the bottom I work my way back up. We'll start at the bottom now. You're here Halloween kin says kids events are fun Then again, I've worked with chill in childcare for 10 years. Well, then you've got it. No problem I've just drawn a blank Where'd you draw it? Minnesota maker Chris No, this is different, right? This is the other one, right? This is the other one There can be only one Uh, Sue bear says I'm gonna make up DIYs for the kids for christmas gift making pin boxes ornaments keychains great idea. Yeah It's a great one picture frames With mom or dad and grandparents, etc. That would be good. Oh, yeah, that's a good one Let's see I was thinking door frames door corners And doing like some cool door corners like I really want to finish that Super mario Laurie wanted to know how did we attach the lights after all with hot glue with star bond Laurie, I have a quick story to tell you about that. It's funny. So, um with the door corners I just destroying it. Yep. I just ran. I just wrapped it and I glued the back of the little box On to the bottom with a little star bond. That's all I did for those. They were perfectly happy with that But I wanted to do if remember last tuesday We did the deer and it said seasons greetings and I had all of the The snowflake cutouts and I wanted to put lights behind it. We did that at the very end remember and then After that melanie my friend melanie. She was at the show with us. She was like, you know what you need You need lights on the back of this one Uh-oh, what happened? I dropped the top and I'm not talking about my 5.0 He dropped the top And took out some stuff dropped top All right, so melanie's like you need to put lights behind this. I think it would look awesome I was like, oh, yeah, we're we actually tried to but we didn't finish putting the lights on the back She goes, well, I'm gonna do it So she goes back there and I said I hear that you can that the star bond will work if you want to try it So she had the star bond.  And he said How long how long does she go sit back there? I was like, it's quiet right now She's gonna do it So she sat back there and she when she got done it was beautiful It was beautiful. It had all these colored lights behind it And then I she turned it around and she'd take them all she'd she star bonded Them on and then put tape over it because the star bond Had to dry needed time to dry and kind of thicken up. So she put Green tape this green tape on the back like little pieces of it all over it The whole thing was covered in green tape and then she hung it up And she was like look at that. I was like, yeah, that looks really good And then she goes, well, let's leave it there for a minute And this was at the end of the day. She's like, let's leave it there and let it dry She goes, I'm going to go shopping. So she walked around.  Oh, look at this one It has lights and I'm sitting across the booth going, uh, oh I know that that still has the tape on the back of it And so I so I look away. I do something to help somebody else I don't really remember but next thing I know here she comes and she lays it on the table and I was like Um, oh The thing about this sign is and I had to turn it around and show her that I had all this tape on the back of it And we hadn't I said I told her what we had done in the process that we were waiting for the Glue to dry. We just haven't had a chance to take the tape off yet And she was the tape is not coming off I mean you star bonded tape to the sign I said if you would like I'll take the tape off real quick We'll see how it looks and she was adamant. No No, my mom and I were big crafters. Oh, no, you're good. You're fine. This is fine And I was like, but it's this tape little piece of tape all over it. I mean it was practically covered in it And I was like, no, he let let me take the tape off and she was like, no, no, no seriously It's fine.  So she did no, she did this person wanted to make money or do they want to be an artist? Because this is neither one she said she said that Um, she goes if it doesn't stick no problem. I'll just put some hot glue hot glue on there. I'll just finish my sign Yeah, that's essentially what she said. She was going to be fine with finishing it. So Laurie I still Oh my god, I got a new idea We're thinking about how to do that sign bar. What if we do like them half baked pizzas? Where you just got to take it home and finish it yourself And what does that mean? They gotta glue their own well come on. We paint it and stuff and then that's it They just buy the kit painted and they got to glue it together all the pieces. That's how we do it We don't have any assembly. We save on the assembly time. Here we go. We're done. Now. I just put it in a bag It's like a taken bake All right coat number two. Let me do this and then everybody will be walking up to the to the door and they'll be like Is that a digiorno? Oh my goodness moving on. Let's see Oh, I need that white. Where'd the white go? Everything is fall winter your busiest time of year 100 percent Everybody wants to do a craft for christmas or I don't really know So it's like we're like bears at this point. We're just trying to gather and get nice and fat for the winter Because as soon as craft show dies All the trout leave all the trout leave so We hunker down for the winter Yeah So this is our This is like our only season And well then you know, you know I mean the rest of the year we got to hunt all year round But like right now the the trouts are going upstream We got to get fat The stars are aligning Right this the craft show season really does carry us the fall season carries us through January February But well, you've been talking this year. We got to make sure that we get those Well, that's part of reason why we're trying to stay close to home with our Craft shows because wow, did we do a ton of advertising at the show for our workshops? We brought a tv Yes, so we were showing the workshops. We were showing signs that We didn't bring because we also thinned out our You want to go overhead? We'll talk about this on our top tier patron thing. Yeah Yeah, that's a top tier patron topic Sorry guys. We got a lot of lessons learned lately these tips.  Should I use this green right here? Yeah Yeah, I already took one out of inventory one of everything I already took one of everything out of inventory for us So technically those are ghosts that you're handling Be careful. We're not overhead Oh, you're comparing them to the side Which well, no, I'm asking which one should it go should I go Or should I live or great escape green Now I like the great escape green, but that's the one you said look cartoonish. Let me go overhead so I can see what you're Pontificating Yes, I'm gonna kind of put this together. I only have one coat on the black and one coat on the white so Probably can't tell too. You can't see that too much on the screen and then these trees Let me get this out of the way Bring this down these trees are going to be green So do I go with this? It's not that dark of a green. I need a darker green. I'm gonna We're going to continue to add colors to our palette. So Um, these are the first 35 colors We chose that kind of go with the colors we've been using we started with a slim 35 colors And I I already see there's some that I want to Update add and so I already next order will have We'll roll out new additional colors for now Here's like this color green. You can oh that maybe that's a good way to show you Or should it be this color green? I like the darker one So that is more It's not really olivey Um It's like a look this is like one of the issues we're having I guess this one. I named I named all the paints and then kim came through and she's like This name doesn't feel brown I was like, yes, it does and I gave her my reasoning and she's like, no Darker darker go darker. Okay Nice. Thank you guys Are you reading all the comments? No, because I think you need to go back and read a couple I told one caught my eye a minute ago And that's when I realized you weren't reading all the comments No, I well like look over here and we chat for a minute and they go by and then it's hard to go back Yeah You need help finding the one I'm talking about. I guess so It's the one that says I'm funny I just thought you should read it That's all I'm gonna holster this guy I'm coming in with my poppy colors Oh, hey, do you want me to paint that piece? Yeah, Dexter. I'm just saying I love the sign behind you guys and I get a kick out of Gary's jokes I don't know if everyone gets them, but they are fantastic You know what that's what I'm saying He sometimes I don't know I don't know I don't know He sometimes he so sometimes he's very quick about a joke He's he's very quick-witted and he'll roll something random out there far left field And then you can see people like if we're with another couple and or at a party People look at him, but they keep talking like they're not sure if they if they caught that But I you know, that's what he always says because baby always get my jokes really fast I do I get them and I laugh and I can see they don't get them and then I look at them like They didn't get you weird Stop they didn't get your joke So for those of you in the top tier patron group That's what I that was another moment today when you put your I said you need to post something out there that says announcing Foxy Hughes paints or something like that. He's like I am that's what I'm working on right now I'm about to make a post and I was like, oh, okay. Go ahead And then I look when I saw his post His post was a random quote From fantastic mr. Fox That's just as like random and which I got what it was but is anyone else going to get what it is People might ask like why is there a brand foxy? I said Why not a horse or a beagle a beetle Saying this existentially or whatever. What do you say? Existentially, you know All right, you want to go back overhead so we can kind of This is the the darker green. I thought it was an appropriate Quote well because you knew they were going to ask like why fox why a fox But we've already talked about the fox stuff. Anyway, people kind of knew it was coming, right? Well, we talked about the name prior. I think we have We've done a lot of teasing paints that were never here That's because we we'd set a deadline and let it pass Well, the funny thing was Garrett had a small the reason why we kind of Made it over that hump as Garrett had a small meltdown on We are renaming test cut Tuesdays Test cut Tuesdays are now known as Paul Garrett out Tuesdays No, they're not I promised somebody somebody commented Somebody made a comment about uh What is your paint coming out and I was like soon And I was like I'm bought I'm gonna try to bottle it this weekend or package it this weekend And they immediately repel it replied So I can buy it next week And I just Was like, yes Ah, he didn't you didn't tell me that I wasn't going to tell you that so I was like, yes I thought I felt like I made a commitment at that point. I committed To to like someone personally sometimes you just have to draw that line in the sand. Yes, and and That's it. We have a deadline need it make it happen. So we're I work well with deadlines You're up nearly Sunday night after the craft show Yes packaging all these craft show into the five and By 5 30 that by 6 30 because we had to pack up and everybody and somebody parked in front of the trailer And we had to wait forever to hook up the way we could leave Oh craft show was Not the first time and then I was fired up I was gonna give that person a piece of my mind And then she comes limping up and I was like no problem Yes So I had to go back inside to make the have them make an announcement to get the guy to move the trailer The the truck out from behind our trailer so that we could get our trailer out We could hook up to the trailer and when get so the trailer was gone I mean the van was gone And he was already hooking up the trailer when I got back and I said Did you say something? He was like no Just like what why wouldn't you say something? She was limping Kim I'm sure there are Handicap spots or something maybe You didn't have to park right in front of my trailer though the whole day though It was the whole day before I got there in the morning. I was like we're at a craft show. She's probably a crafter. She knows Yes, we were in the vendor lot. Yeah I was just assuming I was making assumptions All right back to what you were saying. Oh, we were just we were saying we filled the paints up All right, we filled the paints up that night. Yeah came back and we I labeled them I had all the labels right he had already prepared with the labels. Yeah So he quickly laid we had some extra hands here the kids were here And they helped us quickly label them all and Then put them then help us fill them We have our iota semi-automated filling machine because my My arms are all manually feeling jacked right now Feeling pretty jacked like I could probably I could probably rip apart like a phone book and I only say that now because I haven't seen a phone book in a while And I'm assuming they're thinner now because everybody's got cell phones. Do they even make a phone book? But I could not rip apart a 1986 phone book Anyway, what am I missing for guys if if I've missed or comment and you have Something you really want me to see you can just put it in here again Or you want me to do that um Put the weight behind the trees real quick Because I just need to put some gold on this jam right there and I'm done Are those the four eight or 16 ounce packets? Very good question. Yeah, those are all the eight ounce packets up here These are the the four ounces have a spout at the top The eight ounces and the 16 ounces will have the spout on the side for pouring Yes I'm holding against the black. I'm not trying to hold it against my chance here. You're showing Yeah, the old name Not the new name Your new clever names All right, I'm going to go throw white behind your Well, do you think it needs it should I? Yes, I think it means it. Yes. Wow. It really makes it pop I was going to go put some gold on that Oh, yeah, his belt definitely needs a little gold on it Well, good, that'll give me a chance here in just a second to answer some questions No, I wouldn't dare Could you guys hear him? He says I give you a chance to call me out while I'm gone all right All of the colors at once. Yes, we are going to create because I figure I mean if it were me That's what I would want.  I don't want some cool branded packaging I don't have all that yet, but It's coming and it'll get here But for now, yes, I think we could package them as like a discounted bundle price if you get all 35 colors in Maybe eight ounce or something. I don't know if I could bundle all the 16 ounces that would be big And I don't really have a 16 ounce filled to bring in here and show you But it's going to stand about this tall. It's a little bit. So here's here's the eight the 16 will stand about this tall so just Obviously double this That's eight and 16 Let's see What am I missing? We still get phone books in the small town They're super thin compared to the yellow pages And the white pages of the old days Really, I didn't even know that they printed. I haven't seen a phone book. I didn't even think they printed them anymore let's see Patreon Yes, join us over on patreon. That was all we were talking about. I see helen has explained patreon if you guys want to go back and see that Yeah, I guess I should say do not eat. They look fun though, right? I love these little pouches I was worried they wouldn't stand they stand just fine You can see all of these are standing up just fine. I was worried about being able to store it on Your shelf could I store them all on a shelf? And you can see I have them behind me. They store just fine So we've done a little testing even though We haven't had them We keep talking about them. We have been kind of working with them behind the scenes. So we've tested all the colors I'm I'm happy with all the colors that I have although there are some new colors that I want That we don't exactly have yet, but I wanted that anyway with my old paint colors. So this will be my opportunity to get exactly what I need I think I can even get custom mixed colors, but I don't think I need that quite yet Yes excited for a starter pack of the eight ounce. Yeah me too I love anything that comes in a bundle or a pack if it comes in a pack. I want it. I need the whole Oh You mean you have a whole set then I'm going to need the whole set Anybody know what here? This is just a funny thing And it's just I just laugh every time I think about it. So Do you guys know what cutco knives are? They're very nice knives, right and my cousin was like, hey Someone's going to call you about these cutco knives and I was like, oh, no No But then he showed me he came into a demo and showed me these really nice knives told me all about them And then he was like you can get this whole bundle And I already told you if it comes in a bundle.  She loves to cook She loves to bake. She came over and she goes Are those cutco knives and I was like, oh, yeah Oh, yeah, and then I'd had him a while and then she said I've never seen anyone with the whole block. I was like I was like, oh my god. I'm gonna get I'm gonna get in trouble Somebody's gonna tell Garrett that those things are super expensive and I'm gonna get in trouble So that's our secret. Don't you tell him when he comes back in here? Let me grab some glue real quick So because I can start gluing mine minus the little The white so what he's doing is The backer here you can see these green trees They actually sit on the second level so the white that you see behind them He has created a spray paint template so that he can lay it down And he spray paints white glitter.  He just spray painted that and it's still wet. I can smell it and everything So and it's got a little glitter. I don't know if we can show if I get really close You might be able to see the glitter on them that gives them a little something extra and There's my template right so those of you with a laser.  Ah, I got you Who this thing is wet and it's very hotant We're not driving right after this All right, you want to show them And then here Look at that.  We really learned so if you're not at taking that extra step To add glitter to your signs and you try to sell them at the craft show My pro tip of the day is it's not a lot. It's not a lot of glitter Right. It's just a quick. It's a dusting. Yes Just to give her a little bit of shimmer I don't drench it.  I went outside. I would hold it out here And I'd spray up here and let it walk through Hit it with a little bit of glitter clear coat Again, just the quick dust So it sticks And that's it I don't Don't drench the board Do you remember the girl with the let it snow Um the let it snow sign.  She came in to get a bow And somebody said something about the glitter on their sign and she was like Oh, you have glitter Where did you get glitter? I was trying to keep it on the down. You must have because I said Oh, yeah, Garrett's doing glitter. I said you could she goes Can I get what did she say she told me she brought me the sign she said I was told that I have to put glitter on this Uh, uh, that is not what she said. That is not what I told her. Anyway, I said, oh, yeah Garrett's doing glitter out. I said go find the guy with the hat on out there And and she goes and say what I said say Um, what did I tell her to to ask you wait a minute? I said I need glitter on this. No. No, I told her to say Um What could I what do what do I need to do to get glitter or no? What was it shoot? I forgot what I said darn it I can't think right now, but it was funny. Whatever it was. It was funny At least the girl had to do to get glitter in this tent. It's not exactly that it was Um, what do I need to do to get some glitter on this thing or something? Whatever it was. It was funny and she was like, all right She repeated it on her way out the door And I thought for she she was gonna go tell you but she came back and it was Fully glittered and it was beautiful and she was super excited about it. I got her and I glittered her up I mean she was right back. So you did it quick. Yeah Because it's just a dusting No way it dries quick too. No, yes very fast. It dries very fast There's welcome I can glue the rest of that on here Yeah, I'm gonna well we had the one cone In front of our trailer anthony edwards We had the one cone there which we borrowed from some other show and ended up in our stuff And we had it kind of blocking but I guess Garrett had it too far Too close to the actual hookup So there was enough room in the parking space for them to park right in front of the cone Now the first day because we we set up thursday night and left the trailer there And then had the cone sitting in the same spot and friday morning when we came in That our space was open like they left it open for us and I was like, wow Nobody parked in front of it. The cone did its job. That's so great But saturday morning when we came in no sunday morning when we came in They totally ignored it Ignored what? The cone anthony edwards, so he said you need to get Four cones or something and like basically block off the spot Yeah, I was like, well, we had one orange cone and that didn't work I don't think my orange cone was intimidating enough. Yeah one orange cone didn't do it. I knew what it is It needs angry eyes.  Have you seen that tiktok with this girl?  Yeah Oh, I should dress them up like mr. T And maybe give them a sign that says I pity the fool that parks here No All right, how do I get this I was just trying to think of like a good bouncer and I thought mr. T started off as a bouncer Oh, it stuck to my finger I don't think I ran on it I got it Courtney style says glitter is my arch nemesis. Does it rub off after? Rub off at all after it dries. Yes it does But uh, there is something called um glitter sealer And that's what I that's what I hit it with is the glitter sealer. Mm-hmm after we spray it lightly He'll hit it with the glitter sealer. Yeah Oh, here you go. Look these are little things See how these fell out of these pieces. Oh, they are just the inside pieces. You just put them back in I'm about to do it here again. Yeah That's fun back in All right, we're almost done you guys. Oh, it's five o'clock. We're right on time. What time is it? Beakers o'clock Beakers o'clock All right, there's the pine cone one. He's working on his I have to go get it. How many door hangers of the same sign? Same design. Do you take to the show? That totally depends on Oh, hold on. Sorry you guys Hold on phone call It's the Kim and Garrett line It could be the commissioner Gordon Whoever's calling Come online. Hey if if we lose signal that's because Kim is back there monkey and with the computer Let's see. What was I saying? Oh, so It depends on the size of the show We don't bring the same number of signs of every sign. So those beliefs are top seller this year by far Are those uh, believe Santa night scape the believe night scapes. Oh my goodness those things Just every time we it would mean we every time No matter how many I bring they're selling Merry Christmas lights is still a big seller and so of the beliefs for three day show at um at the Like at a convention center at a big convention center Wow. Oh man. Is this blue? No Where's the uh maker's fund? Here. I brought a bunch No, what are you doing? Well, um They just need to be behind the holes. Yeah, I see. Oh just that little strip there. Yeah, okay So I think how many beliefs did we end up selling 15? 15 beliefs. Yeah And then 15 Merry Christmas. So how many of those? What was next after that? I'm not drawing a blink right now. Oh holy nights. We sold 12 Merry Christmas lights. We sold 12 It's another big seller Grinch miss. Oh wait. Merry Grinch miss souls. It's top tier stuff Can't give it away. All our business all our business advice I'm sure Helen's gonna say something about it. Jason made it. He just made it Shoo Right on time to see the good stuff Jason Oh the glue they shared what kind of glue we're using I'm doing big foot can use the four sign and add a big foot words Believe. Oh, totally. Yes. I think that's a great one Uh, do you guys know the Excuse me the happy happy winter a snow globe that we did um during Christmas in July Sorry, I was about to cough again Uh, somebody did that at the workshop last night and did it in a really Like a different color combination and it looked really cool Like a lot of dark colors Did you Want to hit that? Okay I don't know if you can see it on film. I was like a little tiny bit right here isn't painted I got tired. Yeah, you're done right there. I'm over it Okay, it's black enough there you go You get the juice Lois says you have the lasers make a do not block sign There we go Oh, you need to put your belt buckle. Yeah, can you hit me with a little bit of the black while I find my buckle?  That's the tricky part Here it is Look at him looking great Santa's looking awesome. I think I should have done his beard in like modified tube sock or something What like an antique?  Some more contrast. Okay, put a little nicotine in his beard You know, Santa was a heavy smoker He always had his pipe Look at that. That looks great Sue bear says I like the pine cone one is an ornament. Yep. I get that. Yeah, that'd be a good ornament Oh, what did uh Jason? No, who said that? No, Juanita said my tube went out on my omtech have to wait until Friday.  Oh man It was saturday.  And I go out to the back I go out to the back and I hear it again.  It looked like a cartoon Uh, yeah, it was yes, like I don't know what's in it. We don't we didn't know what was in it You still don't know what's in it, right? I got it out today. It was a tiny piece of Plastic, I don't know where it came from Yeah, but well, I don't know how it could have been tiny and then I can't believe it was plastic based on the noise That I was hearing probably half the size of this right based on the noise right But I was scared to go reach around and turn it off because the thing was tingin and pingin and Making all kinds of that noise. It was very angry. It was so angry All right, this is what we got this is what we're working with yes Thanks for hanging out with us and chitchatting with us. This is yes. These were posted A week or so ago, but this was uh, we got the opportunity to do these on test cut Tuesdays for a lot of people then I guess if I That's a long beard Yeah, or he shortened that Right, absolutely somebody asked about my hoodie all bone sons That's my uh mom's Oh, yeah My stepdad I don't know He's a relation. That's his company They do uh, yeah, I said to him earlier I was like, are you gonna put your shirt on because I just had the t-shirt on today So I just didn't change into my baseball tee. Yes.  It's a drill emergency I did. Yep. We have to quickly fill some more paints and I was like Wait, you want to fill paints seven minutes till show time? Well, I didn't have any more Because we've had these paints back here and when the kids because we've done so much painting for the craft shows When they're empty up front instead of filling it up They come out here and get these pouches out of here and I didn't have a white pouch I didn't have a black pouch. I mean, how was I gonna do these signs without those? So we had that's why we were a little bit late because we were filling paint Yeah, we were filling paints still with the paints Yeah, I got up sunday or sunday and I was like time to make the paints Um flip the locations of the belt and the ho ho ho Oh, you think the belt should be lower Yeah, no gives it more of a gap between us kind of where his uh This is below his waist, I guess. Yeah, I think you can try it. He's wearing really really high belt. I don't know. It's sold It's sold. I made four sold for Uh, Helen said she wasn't sold on the pine cone technique, but she likes it. Oh really? Yeah Thanks, Allen made six sold six Santa's beard looks like kind of like the surfer known beard makes me chuckle kind of I believe it is the same beard. Is it the same beard? It might be. I just think I Squatted it down. Oh, that's right. We made it a little bit wider because it was too narrow made it squatty and wide Well, that's how we like our beards around here All right, you can say bye. All right, I am about out of time And today's tuesday, so you know what I gotta go do Get your inch. I gotta go get my enchiladas enchiladas.


Today, we will continue discussing CDF's. So, yesterday we said. So, we have a sample space omega a probability space omega f p and we said a random variable is a measurable function on this space, which is just saying that if you take any Borel set on the real line, the pre image of Borel set of that Borel set must be an event. That is what a random variable is. So, we also define. So, we define the probability law of X as. So, p X of B for any Borel set B, p X of B, we defined as probability that X of omega belongs to B. I should write again curly braces here. I will abuse notation and just write this as p of X in B. This is. So, the probability law define gives the probability that the random variable takes values in a specific Borel set B and is defined for all Borel sets. Now, we also said that the Borel sigma algebra is generated by the pi system of semi infinite intervals minus infinity X. So, the probability measure p X is also defined on the generating class of the Borel sigma algebra. So, we said the CDF, we gave it a name CDF. So, f X of X was simply the probability law of sets of the form that and. So, that was simply the probability of omega for which X of omega less than or equal to little x. Again, I will abuse notation by writing this as probability of X less than or equal to little x. So, this actually has this bad notation for sure, but when I write this I mean this and similarly when I write that I mean that. This is just shorthand you like. So, this is the cumulative distribution function. We stated a theorem last class, we did not prove it, but we said that if you specify the probability measure actually this is true for all finite measure spaces. So, if a finite measure is specified on a pi system it gets uniquely specified on the sigma algebra generated by the pi system. And since this minus infinity X is a pi system on R and it generates the Borel sigma algebra. If I specify the probability measure of these semi infinite intervals the probability measure of all Borel sets on R gets uniquely specified. So, that is something we stated without proof. The proof is a bit non-trivial bit technical we just need to know the result that is all. Any questions so far? That was just a recap of what I said yesterday. So, this we will. So, from now on we will just focus more on the CDF, because this probability law although it is the most complete specification it is you have to specify it for all complicated Borel sets. Borel sets can be pretty crazy whereas, f X of X is simply the probabilities of nice sets and specifying this is enough to specify the probability law. So, we will worry about this for the most part. So, we will now give four properties of cumulative distribution functions which hold regardless of what random variable you are talking about. Any cumulative distribution function satisfies certain properties which I will list now. So, as usual so omega f p is a probability space and x from omega to R is some random variable.  First property limit X going to minus infinity f X of X is equal to 0. Second property limit X going to infinity plus infinity f X of X is equal to 1. Property 3 property 3 is monotonicity if X is less than or equal to y, f X of X is less than or equal to f X of y.  So, these are four properties that are satisfied by all CDF's regardless of what random variable you are talking about. So, the first says that when X is going to minus infinity f x is going to 0 and when X is going to plus infinity it goes to 1 and everywhere in between it is non decreasing. So, it is something that starts off at 0 at minus infinity and then it ends up at 1 at infinity and in between it has to only increase it can be flat or it has to increase it cannot decrease. It does not have to it does not have to increase in a continuous way, but it is right continuous. Have you heard of right continuity? Well if you have not heard of it you just heard of it this is what right continuity means. So, continuity means that see continuity means that there are no jumps right you can give a more rigorous definition, but it just means that the function does not jump right. So, but CDF's can jump they do not have to be continuous functions of X CDF's can jump, but what they cannot do is if you approach from the right the limit and functional value have to be equal. So, what this notation see I remember I here I did not write epsilon arrow like that I did not write a right arrow right if I do that I just mean epsilon is going to 0 whichever way when I do that epsilon down arrow 0 I mean that epsilon is going to 0 from the positive side. So, some people write this some people write instead of that some people write epsilon going to 0 plus which means it is going to 0 from the positive side it is not going to 0 from the negative side or jumping around or any such thing right it is going to 0 from the positive side. So, what this is saying is that if you have a CDF right. So, heuristically speaking you have if you are trying to plot a CDF FX of X. So, it has to at minus infinity it starts of at 0 right and remember it is a probability right. So, it always has to be between 0 and 1 right and at infinity it ends up at 1 right approaches 1 and everywhere in between it has to increase right, but potentially with jump. So, it could do something like it could increase continuously at some in some place for some places it could jump in some places there is no problem with that right you could do something you could stay constant for some time no problem it cannot go down right you can again jump and then go to 1 for example right I mean this is just one possibility does not have to necessarily look like this, but what this what this right continuity property is saying is that when you have a jump in the function. So, you have a jump in these places right whenever it jumps this is hollow and that is solid what that means is that the left limit is not it may not need not be the functional value, but the right limit is always equal to the functional value right. So, when I approach like that right that is what means right. So, if this is my X FX of X plus epsilon is approaching the functional value. So, in all these places I have drawn it continuously it can even go like that right it does not have to be flat. So, what I mean is that here it is flat, but it can also have for example if this is a jump it can do something like that right. So, what I mean here again here that is hollow and that is solid meaning that when I when I when I approach X like that from the right the limit is equal to the functional value, but I approach from the left it need not be equal to the functional value right. So, CDFs need not be left continuous, but they are necessarily right continuous. So, if it is both left continuous and right continuous it is continuous right, but it need not be continuous. So, is this I mean so this is just some canonical picture of how it might look it can look more complicated also you will see some fairly bizarre CDFs, but this is just some representation of what you can expect from a typical CDF. So, we have to prove all of this, but before that are there any questions on right continuity monotonicity is very trivial right. So, when I so it is usual convention in plotting functions that. So, when there is a jump in the function you have to specify where the functional value is right. So, the functional value could be here or here or it could be somewhere else in general. See what CDF it cannot be somewhere else right cannot go above for example right, but wherever you put a solid dot is where the functional value is at a jump that is just standard plotting convention for functions. And if you put a hollow dot it means that that is not the functional value right I mean this means that the function if you approach the if you approach x like that the limit of the left limit is not the functional value when there is a solid hollow dot over there any other questions. So, CDFs are always right continuous. So, we have to prove all of this right. So, what shall we prove first? So, this one so you are looking at so if you are looking at have to be careful now. So, if you are looking at the probability limit x tending to minus infinity f x of x that is equal to limit x tends to minus infinity probability of omega such that x of omega less than or equal to x is not it. So, now I have to argue that so can I not so I have to use some kind of continuity of probabilities right, because I have to send the limit inside right correct. So, in particular you can take let us say any sequence let x n be real sequence such that x n goes to minus infinity as n goes to infinity. So, I am just going to make this into some kind of a countable limit. So, I can write that as so this guy I can write as this is equal to limit n going to infinity probability of omega such that x of omega less than or equal to little x n. So, I am still maintaining the good notation, but at some point you throw it away I will just write x x less than or equal to little x. So, now what do I have to do right. So, I am looking at so if you look at this picture again. So, you looking at some x n right such that x n is let us say x n is you can actually take it as sequence not only going to minus infinity, but going monotonically to minus infinity. So, let x n be any real any sequence such that x n tends to minus infinity monotonically. So, x n is something that is monotonically decreasing to minus infinity in whichever way you like any sequence that is going to minus infinity. So, then what happens is that the pre image of minus infinity x n will be a bigger set than the pre image of minus infinity x n plus 1 because x n plus 1 will be somewhere here. So, the pre images will be Russian dolls correct they will be nested decreasing sets. So, maybe I can write this. So, I can probably write this is this right. So, limit have to take the limit inside is not it. So, I have the following omega such that x of omega is less than or equal to x n and if I look at omega such that x of omega less than or equal to x n plus 1. Let us say I look at these two sets this is true right this this containment is true. So, if I do that this containment is true right. So, if I call these sets like so, if this I call a n and that call a n plus 1 these are nested decreasing sets correct. So, I am trying to invert continuity of probabilities. So, what I know I do know is that probability of. So, limit n tending to infinity probability of a n is equal to. So, these are nested decreasing sets right what is the continuity probability result for that probability of. So, limit of that is equal to probability of intersection n equal to 1 to infinity i equal to 1 to infinity a i is that correct correct no. So, what is that set. So, I what I have here is the probability limit n tending to infinity probability of a n right correct. So, this is nothing but a n in my notation and that has to be equal to the probability of this countable intersection and what is the what is this countable intersection now tell me it is null set correct. So, this countable intersection will be the null set right because x n is going to minus infinity right the set of all omegas that map to something that is arbitrarily going to minus infinity right that is the null set because everything maps to the real line right. So, beyond some x n your essentially this intersection you have to argue is null. So, you can show that this is in fact 0 because this guy is null I am using continuity of probability. So, property number 2 is also very similar except now you will have you are going to infinity. So, you will want to send a sequence x n which is going to plus infinity and you will realize that instead of null it will become. So, you will have unions instead of intersections right and then you will have this countable union that case will be the sample space itself right after all the pre image of the entire real line is the sample space. So, you have to use to you have to show this properly using continuity that is the only thing I want to emphasize you can know that you know this you can do the second property also is it the other way round is it not true that I am saying that x n see this x n is a sequence that is monotonically going to minus infinity right. So, x n plus 1 is less than or equal to x n correct. So, the pre image of that set has to be contained in the pre image of that set no see you look at all those omegas that map here right that is some f measurable set there, but this does omegas that map here must be a subset of that right. So, that nested containment works you have nested containments you use continuity of probabilities. So, you see how crucial it is right it is it keeps coming up all the time continuity. And so property number 3. So, property number 3 is absolutely trivial actually property number 3 is. So, you have the following right. So, x is less than or equal to y right. So, you have that the set of omegas for which x of omega is less than or equal to x, but if you just argued right is which way is the containment this way is not it omega for which x of omega less than or equal to y correct again right. So, you x is less than or equal to y. So, you call that x you call that y I just argued that this containment has to hold right any questions. So, now what happens these are all these are both f measurable sets. So, they have probabilities and you know that if a is a subset of b probability of a less than or equal to probability of b right. Therefore, you can put. So, you put p on both sides you have done right. So, if you put p on both sides I should not be doing this, but I should write it again, but this is what happens right correct. Because a is contain in b probability of a less than or equal to probability of b then you get that inequality which means. So, this is nothing, but f x of x which is less than or equal to f x of y for x less than or equal to y correct any questions see all along see these you know these properties already, but what I am emphasizing here is proving them from probabilities of probability measure and continuity and so on right. So, can we prove this continuity of not continuity this is right continuity of c d f. So, right continuity let epsilon in p any sequence. So, epsilon this is a non negative sequence b any sequence such that epsilon in down arrow 0 as n tends to infinity. So, I am epsilon is going to down to 0 right. So, if I show that this is true for every sequence epsilon in which is going down to 0 then I am done right that is what that limit means after all right. So, I consider a sequence like that then I say then consider the following limit n tends to infinity f x of x plus epsilon in right. I have to show that this is equal to f x of x for every sequence epsilon in going down to 0 if this is true then this is true right. So, why am I doing this I have to yeah this is something I did not discuss earlier. So, this epsilon is some continuous thing right this epsilon is some real parameter that is going down to 0, but why am I replacing it with a discrete sequence the sequence epsilon in right. So, the epsilon in this epsilon itself is going down to 0 on the real line, but I am now considering a sequence indexed by n right why am I doing that because I will invoke continuity of probability. I want to somewhere see I know how to handle countable intersection and countable unions right I cannot handle uncountable intersections and unions right. So, I am just that is what I am doing right. So, I am going from a uncountable this epsilon to a sequence. So, that I can get these countable intersections and unions very standard trick in measure theory in probability theory. So, that is simply and I have to prove that for every such sequence only then will that be true right. So, limit n tending to infinity probability of x less than or equal to x plus epsilon in. So, from now on I will use this notation. So, no omega such that x of omega now again I have to be careful. So, this is like these are nested decreasing sets as subsets of omega right and so that will be equal to. So, I am going to skip a step here. So, this will be equal to. So, how do I write this. So, this be omega such that intersection n equals i equals let us say 1 to infinity sorry my bad. So, intersection of sorry i equals 1 through infinity omega such that x of omega less than or equal to x plus epsilon n. So, the intersection should be outside the set obviously right that is the mistake I made agreed. So, I am just taking this is of course, if you want to write it down this should be omega such that x of omega less than or equal to epsilon n you have nested decreasing sets and the limit is outside. So, bring it in by continuity of probability right then I get that epsilon i correct everybody with me. So, I am just saying that this. So, this set. So, if you I mean you have to bring this limit in limit in right for the continuity of probability and that is what you get this is by the nested decreasing property and continuity of probability measures. Now, what is that set equal to the whole intersection remember epsilon i is going to 0 right. So, that whole set will be equal to the omegas for which x of omega less than or equal to x right. So, that will in fact be equal to f x of little x and this is true no matter how I choose epsilon n. As long as it is some monotonically decreasing sequence to 0 this holds right. Therefore, this limit holds. . See for the left limit this argument will not go through because you will be looking at. So, you will be approaching from the right is not it. So, you will have. So, you are looking at for the left limit you are looking at limit. So, you are looking at something like this f x of x minus epsilon n as epsilon n is going from positive or positive right. So, epsilon n is still positive. So, I can do that right, but that will be limit n tending to infinity probability that x of omega is less than or equal to x minus epsilon n is not it right I am making any mistake. Now, this will be what kind of sets of these are they decreasing or increasing sets are not they. So, I should take. So, this will be the probability of the union of i equals 1 through infinity omega such that x of omega is less than or equal to x minus epsilon i is not it agreed. Now, what is that equal to that set is equal to x less than x which is not your CDF right. So, this will simply be this will be equal to probability that capital S is less than little x not less than or equal to little x. So, the left limit is not the CDF right and it will be equal to the probability x less than or equal to x if and only if probability of x is 0. So, you already proved a problem that was going to be homework problem. So, if you have if you have a CDF that is continuous at x that means the probability assigned to the singleton real number x is has to be 0 if and only if. So, you see what I mean after all this minus that is the probability that x equals x right and you will have continuity of the CDF at x if and only if the probability of the singleton x assigned by the probability law p x is 0 right. So, that is a good question. So, we dissolve the homework already for you right. So, left continuity need not hold and in particular left continuity holds if right left continuity holds the probability of that point is 0 and if the probability of that point is something positive. You will definitely have a discontinuity right you have right continuity and there will be a discontinuity and the jump in the discontinuity will be equal to the probability that x is equal to x. That is actually what we have shown are there any other questions. So, any CDF of any random variable CDF of any random variable has to satisfy these 4 properties. So, what is interesting what is somewhat more remarkable is that there are no other further properties that have to be satisfied by a CDF. Meaning that if I give you any function at all which maps which a function that maps are to 0 to 1 right which satisfies these properties. Namely it is start of at 0 ends up at 1 monotonically non decreasing in between and right continuous. There exist some random variable whose CDF is that function you give me any function that I do not know that it is a CDF of any function any random variable. You give me a function that satisfies all these 4 properties then it is necessarily the CDF of a random variable that also can be shown. So, which means that this is all you need for a CDF there are no further demands that you need to make for it to be a valid CDF. Even if one of these does not hold it will not be a CDF there any other questions have few more minutes. So, I will now introduce very useful random variable called an indicator random variable. So, as usual you have omega f p. So, let a be an event define I a of omega is equal to 1 if omega is in a 0 if omega is not in a. So, I am just looking at. So, it is called indicator random variable because it indicates 1 when your realization is in a it is in the it says 0 otherwise. So, what I have pictorially I have my sample space with some f p on it and I am looking at some other event a and of course, my mother nature picks omega. So, it could fall here or it could fall here right if it falls here I say 1 for I a if it falls here I say 0 it only takes 2 values depending on where omega falls. Now, this is called the indicator random variable of the event a. Now, before I call it the indicator random variable have to prove something I have to prove that it is a random variable right. So, it is see certainly a function from omega to r in fact it is a function from omega it only takes 2 values on the real line. So, it is a function from omega to r, but I have to prove that it is a measurable function f measurable function from omega to r right. So, how do I do that? So, here is my real line right and so, I have 0 and 1. So, these are the only values that the function takes right. So, if you give me any Borel set you give me. So, now what do I have to show you give me any Borel set the pre-image is something f measurable right. So, you give me a Borel set let us say you give me a Borel set like that what is the pre-image of that here null set right, but if you give me a. So, let us see if you give me a Borel set that includes 1 the pre-image will be a which is f measurable right. Similarly, if you give me a Borel set that includes 0 it is pre-image will be a complement which is also f measurable because a is f measurable. Then finally, if I give you so if I give you some Borel set that includes both 0 and 1 the pre-image will be the entire sample space right which is also f measurable. So, you can just proven that the indicator function is in fact a random variable the indicator of a f measurable set is an f measurable function correct. So, this is a legitimate random variable. So, if you want to construct a function from omega to r which is not a random variable you can also do that right what you will do is you will start with some set a which is not f measurable right and then the indicator function will not be f measurable right. So, the for a which is not an event the indicator of that event will indicator of that set will be a function which is not a random variable right. So, the very simple example so what will the CDF of this we have established that this is a random variable the CDF of this will look like what at 0 it is. So, if I try to plot so I am going to call this f I a of x against x right if x is less than or equal to 0 the function will be at 0 right. So, we at 0 and between so if this is 0 and this is 1 so that is 1 right I beyond 1 it has to be 1 right that is also clear and then in between it will be the probability of a or a compliment yes correct here the height of this. So, here this is the probability that x is less than or equal to 0.5 say which is means probability x is equal to 0. So, this height will be the probability of a compliment and this jump will be the probability of a right. So, in all this the functional value will be over there right and here it should be hollow similarly here it should be solid and hollow over there. So, that is how the CDF looks yeah that is at 1. So, this has 2 discontinuities right. So, looking at this itself you can say that the probability of 0 is as I argued there right probability all these points have probability 0 and 0 has probability P a compliment and this jump is equal to probability of a any questions. So, tomorrow there is no class the next class is on Friday we will discuss some important topics we will discuss about sigma sigma algebras generated by a random variable and also we will classify random variables based on the measure they induce on the real line. But we saw that P x is the measure induced on the real line by the random variable x depending on the properties of the measure P x random variables are classified as well under several heads you already know about a couple of heads I guess, but there are more heads right. So, there are several types of random variables depending on the kind of measure they induced on the real line. So, that those topics will be left for Friday I will stop now.