The Kraft Foods Company presents Willard Waterman as the great Gilded Sleeve. The Sleeve is brought to you by the Kraft Foods Company. Kraft, makers of the one and only Miracle Whip salad dressing. And one and only is right, friends, because there is no other salad dressing like Miracle Whip. No other salad dressing has that distinctive flavor, that lively teasing flavor that's peppy, yet not a bit too sharp. It's a flavor millions prefer. Enjoy it on your salads. Tomorrow, get a jar of the one and only Miracle Whip. With spring here, there's a lot of house cleaning going on in Summerfield. The great Gilded Sleeve has refinished his parlor furniture. Mr. Pee-Vee has sprayed his screens. Chief of Police Gates has whitewashed the jail. Even the Jolly Boys Club over Floyd's barbershop hasn't escaped. And while the paint is drying up there, the gang is meeting at Judge Hooker's house. Oh, it's all right. Home sweet home, Commissioner. Right off the needle point on the judge's wall. Home sweet home. I love that picture. My mother in Salinas has one over the mantle. Play something else, Floyd. The Chief's getting sentimental. Okay, let me see. That's fine, Floyd. Oh, it certainly is, Floyd. Oh, my goodness. What's the matter with the judge? Perhaps it'd sound better, Floyd, if you'd pull out all the stock. Well, I ain't a bad ol' organ, Judge. Thank you. It hasn't been played for quite a while. There are a few cracks in the bellows. Uh, if the air's escaping, Floyd, try pumping your feet a little faster. Uh-uh. You're gonna sing any songs. I better save my legs. Hey, what's that? It's my doorbell, Floyd. Yeah, I thought we was raided. Yeah, that must be Pee-Vee. Hi, Pee-Vee. Hello, Mr. Pee-Vee. Pee-Vee? We were just about to sing a song. There is. You're covering in the time. Yeah, my goodness. Well, now, Pee-Vee, I have a lot of good music. Yeah, why don't we try one of these antiques here on the organ? Yeah, let's see what the judge has here. Uh, your bird in the gilded cage. Cubanola glide. Alma, where do you live? How about this one? I picked a lemon in your garden of love. That was a hit song in 1906. I didn't know they had lemons in 1906. Yes, they did. I used to go with one. Say, fellas, here's a dandy. My wife's gone to the country hurrah hurrah, 1909. Let's take this one. We all know it. Yeah, that has a solo line for me. All together now, gang. Hit it. By the line, by the line. It was nothing, Commissioner. Well, if I do say so myself, the acoustics are excellent here in my park. Yeah, music always sounds better in a musty room. Lloyd? Huh? Well, I do the best I can. Since I don't have my housekeeper now, it's a chore to keep things spick and spank. I think it looks mighty nice, Judge. I've never seen your pussy willers over the fireplace look better. Thank you, Chief. Now, if you'll excuse me, I'll go brew some tea.  Forgive what I said, Judge.  That's all right. Any a crummy housekeeper? Well, he didn't have to tell himself, Lloyd. Judge's house wouldn't look bad at all if he'd get some of the junk out of here. He's had those old cattails behind that picture for 20 years. Well, cattails I can stand, but them peacock feathers in the umbrella stand ought to go. Well, the place could stand to spring cleaning. Things accumulate over the years. Yeah, I'll say. Last week when me and Lovey was cleaning, we threw away a truckload of stuff. Well, the judge doesn't have anybody around to help. Say, why don't we jolly boys clean up the judge's house? Us? Sure. He's going out of town tomorrow. We can surprise him. Wait a minute, Commissioner. I just got through renovating the jail. Took me a whole day to lamp black the cell bars. Oh, come on, fellas. Now what do you say? He'll be a jolly boy's good deed. I wish you'd sorted these good dudes last week when Mrs. Peavey and I were cleaning. We ought to do it for the judge. He's lived it so long, he doesn't realize how cluttered his house is. Remember our slogan now? All for one and one for all. Well, I'm game. I'll donate half a day from the barbershop.  It'll be worth it to get rid of all this underbrush he's collected. What do you say, Chief? Mm-hmm. If everybody's for it, I'll help. Well, Mrs. Peavey always likes a chance to tend to jolly store it. Yeah, okay. It's a deal. But don't say anything to the judge. Let's keep it a secret. Here comes the tea. What are you gentlemen huddling about? Oh, nothing, Judge. Yeah, we're just deciding on another song. Yeah, that's right. Yes, indeed. But how about Goodbye Pussy Willows Goodbye? That was a fine breakfast, Bertie. Thank you, Mr. Gilsey. Now, where's the scrub pail in the mop? Scrub pail in mop? That's what I asked for in my dream. What's the matter? I can spill your prunes? No, Leroy. The pail in mop, please, Bertie. Yes, sir. And some floor wax, furniture polish, and dusting claws. Yes, sir. And I want the vacuum cleaner, Bertie. Mr. Gilsey, what are you trying to do? Get Bertie's job? No, not exactly, Bertie. That's what he's trying to do, get Bertie's job. What's this all about? It's a secret. It's a secret. What are you up to? No, kitties. Don't get curious. Who's curious about a scrub bucket? Maybe the water commissioner's going to scrub up the office. No, nothing like that. I know. The mayor's transferring on to the street cleaning department. He is not. As a matter of fact, I'm just doing an old friend a favor. Oh? Last night at the Jolly Boys meeting, I had a great idea. We were at the judge's house, and I noticed it's in pretty bad shape, since he doesn't have his housekeeper. So it occurred to me that while he's out of town today, there'd be a nice gesture if the Jolly Boys pitched in and cleaned house for him. I thought of it. Yeah, we got that impression. Well, I think it's a wonderful idea, Anki. Yeah, I'm always thinking of nice things like that. All the Jolly Boys are meeting here this morning, and then, oops, here they come now. I'll get it! Yeah, I'll get it, Bertie. Hey, that's the judge. I see his car up front. Uh-oh, I have to get rid of him. The judge mustn't know what I'm doing. That is spoil everything. Coming! Yeah, remember, kitties, it's a secret. Mum's the word. Who's talking? Morning, Gilday. Judge, I thought you'd gone to the county seat. I just got by to bring your cigar lighter. You left it on the organ last night. Yeah, oh, thanks, Judge. Well, see you later. Have a nice trip. Gilday, are you trying to rush me off? Yeah, Gilday, I'm expecting some other people. Stop pushing me! Who? Well, some friends of yours. We're going to do you a favor. Even if you aren't supposed to know about it. It's my idea. Oh, a surprise for me? Yeah, a little idea I cooked up with the Jolly Boys. Well, I won't ask what it is. I'll be on my way. You wait a minute, Judge. Aren't you curious? What? We're doing this for you, remember? You might show a little interest. I thought I wasn't supposed to know about it. You're not. Either way, Judge, you still keep your house key under the welcome map. Well, yes. How about those antiques? Will that finish? Take a furniture polish? Of course, Gilday. Why do you ask? You had that as part of my big idea. You what about your rugs, Horace?  Or vacuum them? Gilday, what are you up to? You mean you haven't guessed by now? Judge, while you're away, I'm cleaning your house. You? Well, like I say, I'm pulling the Jolly Boys in to help me. I don't want to take all the credit myself. Why, Gilday, old friend, what a splendid thing to do for me. That you, fella, shouldn't give your time. You're glad to do it, Judge. Now, be on your way before the others come. Oh, yes, I better go. Goodbye. You're right. And remember, Judge, this is our secret. You're right. Unky, I thought you weren't going to tell the judge. Well, Marjorie, you whittled it out of me. Yes, yes. Don't tell anybody. It's a secret. Mom's the word. Big deal. That boy. Well, Unky, you were pretty obvious with the judge. Well, my dear. Excuse me? Yes, Bertie. I got it. Well, thank you, Bertie. I'll get it. Yeah, I'll get it, Bertie. What if I wasn't a lobbyist? You're doing something for somebody. You deserve the credit. No use hiding my life under a mop bucket. Hello. Hi, Commiss. This is Floyd. Oh, Floyd, I'm waiting for you. We're due at the judges. I can't leave the barbershop today, Commiss. You can't? I got nine. Farmer Nelson just brought... Oh, my goodness. Oh, Floyd, can't you put them off? I had a buck ahead.  Sorry I can't be with you, Commiss. Yo, Bert. Have a good time. Yes, yes. Goodbye, Commiss. Goodbye. Look at that, Floyd. Down at his barbershop making money. What a sneaky way to get out of cleaning house. Oh, Mr. Gillespie, what can I do for you this morning? What can you do? Phoebe, don't look so innocent. Last night, you promised to help me clean the judge's house. Uh-oh, yes, yes. Well, I'm afraid you'll have to count me out, Mr. Gillespie. Phoebe, not you, too. That's the way it looks. Well, I thought Mrs. Phoebe wanted to run the store. Yes, but this is her day to run their garden club. Phoebe, which is more important, the garden club or cleaning the judge's house. I'd hate to put that to vote before the garden club. Yeah, all right, Phoebe. If that's the final word. That's the final word, Mr. Gillespie. I heard it this morning before I left home. Well, here comes the chief. He'll help me. Good morning, Phoebe. Oh, chief. Hello, commissioner. I was just by your house, and they told me I'd find you at the drugstore. Well, chief, are you ready? Ha, ha, you bet I am. I'm letting a deputy prepare lunch for the prisoners, so I'm free. You're true blue chief. Of course, we'll have to do this job alone. Uh, we will. Phoebe and Floyd won't help us. Fair weather, friends. Well, it isn't as if we wouldn't like to, chief. We ran into complications. Oh, well, in that case, commissioner, perhaps we should clean the judge's house another time. No, I can't do that, chief. I told the judge we were going to clean his house today. How's that?  Commissioner, I thought this was to be our little secret. Well, uh, blabbermouth. Oh, Phoebe. On second thought, commissioner, I don't think I can help you today, either. You ask this, chief. Well, my deputy doesn't make a very good stew, the prisoners might complain. Yeah, for that's an excuse and you know it. Yes, I know it. You all right? Back out all of it. You'll be sorry you didn't help me clean the judge's house. Well, now I wouldn't care to add. Whose idea was this, anyway? Great guilders leave returns in a moment. Nothing is quite as eye-appealing on a dinner table as a colorful, tempting salad. For example, just imagine a shimmering lime gelatin, salad mold surrounded with lettuce cups, filled with plump, juicy red strawberries. Just be sure that salad tastes every bit as good as it looks. Serve the very best salad dressing you can buy. Delicious miracle whip. Miracle whip has a perfect, just right flavor, a flavor that's distinctive, because miracle whip is actually a different kind of salad dressing made from a secret craft recipe that combines the best qualities of old-fashioned boiled dressing and fine mayonnaise. No wonder miracle whip outsells the next 20 leading brands of salad dressing combined.  And say, here's something to look forward to. It's the big salad carnival that'll be in full swing. It's your grocer soon. There'll be salad makings galore to give you lots of new salad ideas. Plan now to visit your grocer's wonderful big salad carnival and stock up on all the famous craft salad dressings. Let's get back to the great Gilderslee. He had the brilliant idea of getting the Jolly Boys and cleaning Judge Hooker's house on the day the judge was out of town. It was to be a big surprise. But now, guess who's surprised? Yeah, I never dreamed the Jolly Boys wouldn't be able to help me. I have a feeling I put myself on a spot. Still, why should I go over to the judge's house and do all that work myself? Yeah, better lie down on the couch and think this over. Anyway, wouldn't be fair to the others if I went over alone. If I did all the work, I'd get all the credit. It was supposed to be a Jolly Boys project. Ouch! I thought you were going to the judge's house. Well, Marjorie, the plans have changed a little. Oh? P.D. has to stay at the pharmacy. And Floyd has a big rush on at the barbershop. And the chief says he has to be at the jail. Everybody has an excuse. What's your excuse? Well, I'll think of something everybody else has. Well, I don't see how you can refuse to go through with it now, Uncle. Well, why not? Well, it was supposed to be a secret. But you had to tell the judge. Now you can't back out. Guess not. Blabbermouth. What? Nothing. Boob. Stoop. Pigeon. Uncle, you can't just sit there calling yourself names. You've got to keep your promise. Well, if you don't, the judge will be disappointed in you. It might even break up your friendship. Well, that might teach me a lesson. Oh, I'd hate to see anything come between you and Judge Hooker. Oh, I don't think it'll come to that. Maybe the judge thought I was kidding this morning. After all, Marjorie, it is a pretty silly idea. A bunch of grown men going over and cleaning a house. Sure, I'll bet the judge has forgotten all about it. Miss Gelfleave. Yes, Bertie? Here's a package somebody sent you. Me? What an interesting-looking box. Open it, Anki. Yeah, you'll be sending me a package. What is it? It's a box of cigars. My favorite brand. Ain't that something? Here's the card, Anki. Yes, let's see. Do my thoughtful friends with a thousand thanks for his hooker. Oh, he hasn't forgotten, Anki. You were the heck of a time to remember. No, I've got to clean his confounded house. I'll take the buckets of Leroy. You bring them up. Okay. Marjorie, come over and help me a claim. This is women's work. Yeah, Marjorie has the twins to look after my boy. Well, I can't like go home and look after the twins and send Marjorie over. They like me. I make them laugh when I wiggle my ears. They probably miss me right now. No, Leroy, I need you here. What about Bertie? Well, Bertie has your choice to do at home. Besides, she just cleaned our house. Can't ask her to clean another one. Hey, maybe the judge's key isn't under the mat. Yeah, no such luck. Maybe it's the wrong key. Maybe it'll jam the lock. Maybe it can't work it. Step in, Leroy. Now, we'll start. Let me see, where will we start? You know, not in the parlor. Wussy Willows, Cattails, who saved a hard job for the last. Awe, I'm not too interested in doing this. You are? I can't clean house. I'm just a little kid. Leroy, I wasn't going to mention this until we'd finished the job. But I plan to give you 50 cents for helping. 50 cents? Oh, boy, I'm the best little house cleaner you ever saw. That's the spirit, my boy. Let's begin with the judge's study. See, stacks of them. And this is only one room. You're right, Leroy, 75 cents. Scrubbing kitchen floor. You water commissioner with housemaid's knee. Yes, Leroy. Maybe I shouldn't have hauled the ashes out through the kitchen. For some reason, we don't seem to be making much progress. You take all those papers outside. Yeah, and I found this can of paint in the garage, young. Maybe I could touch up some of the scratches on the pantry door. Well, it wouldn't hurt. Oh, boy, I like your paint. Look it on. Yes, Leroy. That isn't the way to handle a brush. Let me show you the stroke. Yeah, but, gosh. Painting is ticklish business. Yeah, don't be afraid of it. You have to get some paint on the brush, like this. All over the floor. Oh, brother. Now you've got to scrub the floor again. I have a better idea. This is Mr. Gilder's lead. Uh-oh. Perfectly, Leroy. Sure. It's smoking. But when I finish this rug, I'm through. I'm through. Oh, I'm pushed. The potter rug looks nice, Bertie. Thank you, sir. Them old cap tails sure did shed. Good riddance. I left them up on the ash can. I'll put the peacock feathers and stuff. Good boy. How does this table look, Bertie? Well, it looks as though it could have a little more help, old grease there. I said I was going to get. Gosh. House looks pretty good. Yeah, the old judge will like what we did.  Thank you, Mr. Gilder's lead, but anybody can look at me until I don't need candy. Well, I thought you'd like that. Of course, I could use the five dollars. Yeah, just as soon as we get home, Bertie, you deserve it. Thank you, sir. Well, heck, don't I deserve more than 75 cents? You bet, my boy. You deserve a dollar. Hey, King. And I deserve a medal. Yes, sir. Anybody home? Uh-huh, Mr. Munson. Hi, Leroy. Kamish. Oh, Bertie. Good evening, Delba. Lloyd. And Pee-Vee. No, no, no, Mr. Gilder's name. Thought we'd not buy him. See if we could be of any help.  Hey, the place looks different. Don't it, Pee? My, yeah. Well, it's sure. We've worked here all day. Yeah, Bertie. Fluff up the judge's Chicago World's Fair Pillow. Yes, sir. When the judge comes home, he's going to think he's in the wrong house. Yeah, could be, Bertie. Yes, sir. Mr. Gill sleep throughout the cattails, the pussy-willers, and the peacock feathers. And now the judge is going to think he's in the wrong house. Yes. When the judge walks in his study and sees all them newspapers and magazines gone, he's going to think he's in the wrong house. Yeah, that's about right, Bertie. Mr. Gill, Steve, you know what the judge is going to think when he sees all that stuff now? Yes, Bertie. That's right. I think Bertie's got something there. He cost me a little money. It was a lot of work, but by George, it was worth it. I'll sleep like a baby tonight. It makes me tired just to look at all you've done. Well, now the judge's place wasn't really dirty. He just had it all cluttered up. Hey, it's the judge. My. What a shiny kitchen I have. You come on in, judge, and take a look. Well, B.V. Floyd. Hi, judge. And gilded. How can I thank you, old friend? How beautiful my parlor is. Yeah, I'm glad you, judge. What do you got there? Let's see.  Got tails here. Peacock feathers in the umbrella stand and the milkweed pods atop the organ. Is he kidding? Judge, where'd you find all those things? Out back. Apparently you forgot to bring them in after you cleaned. Good judge. Now, will you gentlemen help me carry in my papers and periodicals? They're out for the ash can.  Great gilded sleeve will be right back. Now that spring is here and summer is coming on, you'll probably be serving potato salad often again. And for the best potato salad you ever tasted, be sure you make it with Miracle Whip salad dressing. Miracle Whip will give your potato salad a wonderful, lively, teasing flavor a just exactly right taste your folks like millions will go for. Miracle Whip tastes so good it's become the most popular salad dressing ever created. Get a jar soon, won't you? Make your salads taste better than ever with America's favorite salad dressing, the one and only Miracle Whip. This is Gilded Sleeve again. Just thought I'd mention that all the folks in Summerfield are working very hard in the civil defense program. Hope you and your community are doing the same. This is most important to each one of us. I suggest you get your copy of the official civil defense instructions either from your local civil defense organization or by writing the superintendent of documents in Washington, D.C. and enclosing five cents in coin or stamp. Thanks for being with us. See you next week. Good night all. And white, the music by Robert Armbruster. Included in the cast are Walter Tetley, Mary Lee Robb, Lillian Randolph, Arthur Q. Bryan, Ken Christie, Earl Ross, and Dick Legrand. This is John Easton saying good night for the Kraft Foods Company, makers of the famous line of Kraft quality food products. Be sure to listen in next Wednesday and every Wednesday for the further adventures of the Great Gilded Sleeve. Here are the winners in the fifth and final week of Parquet Marjoram's Great Name the Twins Contest. For entries postmarked before midnight, April 14th, glamorous new Ford Victorias go to Mrs. Clarence Morris, Russellville, Arkansas, Marion Atherton, Temple City, California, Rodney George Anderson, Newington, Connecticut, Mrs. Henry Millican, Dallas, Texas. Winners of other prizes will be notified by mail. The makers of Parquet Marjoram hope you have enjoyed this exciting contest and they hope too that when you shop for Marjoram you'll get the marjoram that tastes so good because it's always fresh. Get P-A-R-K-A-Y, Parquet Marjoram, made by Kraft. Don't miss the Falcon each Sunday over the station. Check your newspaper for time of broadcast and listen next Sunday as the Falcon solves the case of the Flaming Club. Marjoram Arts plays You Bet Your Life with 1-


Today, we will start discussing random variables. Random variables are motivated by the following fact. So, when you have random experiment, the experiment may not be interested in the specific elementary outcome the results. So, the if you have a sample space big omega, the experimental may not really care about what specific little omega shows up. Here, she may be more interested in some numerical function of the elementary outcome. So, an example is that if you toss a coin 10 times, you may not be interested in what specific sequence of heads and tails resulted, but you may just want to know how many heads came up. So, that is a numerical function of your elementary outcome. The elementary outcome is a sequence of heads and tails, 10 long string of heads and tails and you simply counting the heads. And there are many more complicated examples, where your probability space and your elementary outcome may be something very complicated. So, if you are looking at something like weather as a random phenomenon, it is very difficult to even propose a proper sample space or you know talk about what the elementary outcome is, because it is a very complicated phenomenon. Whereas, you may just want to know what the temperature is, which is just a numerical function of the days elementary outcome. So, that is what motivates random variables. So, random variables are simply numerical functions of your outcomes. They are defined. So, it is a function defined on the sample space and it gives a real number in general. So, I will make this more formal. So, I just want to before I really formally tell you what a random variable is. I want to tell you what it is not, because it is a misnomer. So, a random variable is neither random nor is it a variable. This is something you have to keep in mind. So, it is not random. It is a deterministic function and it is not a variable. It is a function. So, it is a misnomer to call a random variable or random variable, because it is neither random nor is it a variable. So, this pictorially and I will keep drawing this picture a few times. So, this is your sample space and it is endowed with some sigma algebra and some probability measure. Omega p is a probability space given to you and a random variable is a map from omega to r. So, this is your map. We will call it x. We will always use capital letters to capital letters like x, x 1, x 2, y z, etcetera to denote random variables. So, normally for events and sets we use also capital letters, but we use letters like a, b, c. You may also use a, b, c for random variables, but we generally keep a, b, c, etcetera for sets and x, y, z for random variables to the extent possible and always capital. Random variables are always denoted by capital letters. So, what happens is, so you have this probability space and your goddess of chance picks a little omega. That is what happens in the random experiment. So, your little omega realizes and for every little omega there is a unique real number. This random variable identifies a unique real number. So, it is a mapping from omega to r and it is a deterministic mapping. There is nothing random about it. Once you fix your little omega, it is a unique real number. Every time for little omega is fixed, if for little omega 1 you may have some real number. Here for some other little omega 2 you may have some other real number. So, the only source of randomness in the random variable is the little omega itself. That is the only random thing in probability that little omega being selected by mother nature or whatever that is the only random thing. After little omega realizes, there is no randomness in the random variable. It is a deterministic mapping. You give an omega, it will give you x of omega. So, this will be x of that little omega and it will be some real number. So, capital X within brackets to denote the random variable which is the function itself and x of little omega will denote the specific value taken by the random variable when the elementary outcome is little omega. If you are talking about the specific, so if you want to say that the specific real number is the value x, you say x of omega is equal to little x. So, always capital letters for random variables and small letters for the value it takes. So, please retain this notation in your written work. So far any questions? So far I have said that random variable is a mapping from sample space to real line. So, one little subtlety here is that. So, just like not all subsets of omega are considered events, not all functions from omega to r are considered random variables. So, random variables are special certain nice functions called measurable functions. So, I will I have to define what a measurable function is. Let me. So, let me just define that first. Let me just consider a function x from omega to r is said to be f measurable if for every Borel said b the pre image x inverse of b which is. So, the pre image x inverse of b is f measurable here x inverse of b is equal to the set of all omega for which x of omega belongs to b. So, I will explain this in a minute. I will also define a random variable on the probability space omega f p is a measurable function is a f measurable function from omega to r. So, I will explain this definition now. So, I am defining.  Actually in this case I mean I have used the notation of a random variable, but this omega could omega f could jolly will be any measurable space. You do not even need a probability measure on it. It could be any measure. All you need is a set and some sigma algebra on it and any f measure any function f from omega to omega to r is said to be f measurable if the pre image of all Borel sets are f measurable. So, what is that mean? So, you take any. So, this real real line is endowed with the Borel sigma algebra remember the Borel sigma algebra it is generated by open sets on open intervals on r. So, you take any Borel set you want let us say that is your Borel set. This is your B I am drawing it like an interval, but it could be something very complicated. This is your Borel set. It could be some cantor set or some complicated set and you look at all those. So, this omega maps here. So, actually let me take my Borel set here. So, this omega maps here into B and there may be some other omega that maps here. You take those omegas. So, let me just draw a pre image of this. So, if you look at those omegas that map into that Borel set that will be some subset of your sample space. Now, that subset of your sample space need not be in your sigma algebra F need not be a F measurable set. But, a function is set to be F measurable if every Borel set has a pre image which is F measurable or in the case of a probability space the pre image of every Borel set must be a event correct everybody with me. So, that is what this is saying. So, X inverse B. So, this is notation X inverse B is the pre image of the Borel set B. So, you take a Borel set B and you are looking at all those omegas in the sample space for which X of omega is in B. X of omega is a real number. So, the X of omega is this one for example, and you look at all those omegas for which your X of omega is in that Borel set. And that will be a subset of the sample space and for every Borel set if it. So, happens that that pre image is F measurable then the function is set to be F measurable. And after all then we are saying that a random variable on probability space omega F P is just an F measurable function. So, here as I said in this for this definition you only need a measurable space you do not even need a probability measure on it. But a random variable is defined on a probability space F measurable function can be defined on any measurable space are there any questions on this definition. Random variable is a function from omega to R and we are saying that not all functions from omega to R are considered random variables. So, whether X is a random variable or not depends on what your sigma algebra is essentially right.  So, that is what makes a function a random variable it is a misnomer as I said it is a function right. So, it is a function it is not random either it is completely deterministic there is nothing random about it right. If it is a little omega it produces a deterministic X of omega right. The only randomness lies in the little omega itself right because little omega realizes randomly right. So, random variable is not random either is this definition yes. No see this could be this set omega may not be even a set of numbers it could be a set of numbers, but it could contain coin tosses it could contain diroles it could contain anything right and you all you have a some sample space consisting of elementary outcomes under sigma algebra on it. Borel set to the sigma algebra. So, every element in Borel set there should be a inverse mapping. No the inverse mapping. So, this is not the inverse mapping in the sense of an inverse function right this is notation for all those omegas for which X of omega is in B. I am not saying X is an invertible function for example, that is not what I am saying X inverse of B is the pre image of B the set of all omegas that map to B that is all. So, is this notation clear. So, we will use this a lot any other questions. So, if it is for example. So, you may have some. So, you can give examples of functions which are not random variables if you simply have let us say if you have non measurable sets here right. If you have subsets of omega which are not f measurable right even if you find one Borel set or some Borel set here whose pre image is that non f measurable set that function will not be a random variable right we can create many such examples. So, you know that you take any non Borel set. So, if you are looking at functions from R to R you take any non Borel set and put the map the function to 1 if it is in the non Borel set and 0 to if it is not right and then the. So, that is like an indicator of that set right. So, if you have indicators of non Borel set you will have a something that is not a random variable you will give you will give this example. So, not all functions are considered random variables that is all I want to say. So, why do you insist. So, first of all. So, why do you insist that you want a random variable to be this very special function right I mean why do you want this to be f measurable why cannot what is the problem with saying that all functions from omega to R are random variables. So, ultimately the probability measure is only defined for f measurable sets right. If you give me some arbitrary subset of omega it may not be an event and they I may not define a probability measure for arbitrary subsets of omega. So, I may have a situation. So, what I ultimately want to do is assign probability measure to Borel sets on R right. So, if you have a Borel set whose pre image is not f measurable you cannot assign a probability to Borel set on R right some. So, I would not avoid that problem right, but for a random variable because pre images of all Borel sets are in fact, f measurable I can assign a there is a well defined probability measure on R which is what is called the probability law of the random variable of random variable x. So, x is a random variable on probability space omega f p that I am not writing again. So, the probability law of x denoted p x is a function that maps Borel sets to 0 1 and is defined as p x of B is equal to p of x inverse of B alright which is. So, this is by definition. So, this is the probability of for which x of omega belongs to B right. So, I should write curly braces here. So, this probability law of x a random variable x takes as inputs Borel sets and for each Borel set will produce a number in 0 1 and this number is in fact, the probability of the Borel set the probability of the random variable lying in the Borel set. So, what are we saying here p x of B is defined as p of x inverse of B normal x inverse of B right is the pre image of the Borel set B. So, remember B is a Borel set on real real line this is a subset of this is a subset of the sample space it is in fact, an f measurable set why because it is a random variable x is a random variable. So, for every Borel set x inverse B is a f measurable set. So, it will have a probability measure assigned to it right. So, that is I am writing it out in full form here probability of those omegas for which x of omega is in B right and this p is your original measure on omega f. So, that is the definition of a probability law of random variable x. So, it produces. So, what is it in physical terms what does it mean it gives you the probability that the random variable takes value in your Borel set B you give me a Borel set it will give me a number between 0 and 1 which is the probability that my random variable takes values in the Borel set. So, in other words. So, to pictorially again this is a picture you may want to remember.  So, you have you take any Borel set let us say that is your Borel set it will have some pre image here right that is x inverse B right. Now, this is an f measurable set right. So, this is x inverse B and that is an f measurable set. So, the probability measure the original probability measure assigns a value in 0 1 to it. So, you can look at the probability law as a composition. So, it takes inputs as B it is a composition of x inverse and p right. So, often in shorthand notation people write p x is equal to p composition x inverse this is shorthand notation for what I just said right p x takes inputs as the Borel set. So, if you put a Borel set in here you put a Borel set in here x inverse B is f measurable set and what is the probability of it right this is more advance takes books use this somewhat cryptic, but shorthand notation it is a composition of your original measure p and your pre image under the random variable everybody with me. So, what can be shown now. So, this everybody understands probability law the probability that the random variable takes values in your Borel set. Now, what can be shown which you can easily do as homework is that R Borel p x is a probability space. So, for any random variable x R B R p x is a probability space what is that mean what is that mean. So, we already know B R B of R is sigma algebra on R right we only have to prove that p x which is the probability law is in fact a probability measure on this measurable space R B R correct. So, what you have to prove now you have to prove that p x of null set is 0 which is very trivial then you have to prove that p x of R is 1 right what is p x of R if you look at the entire real line it maps back to omega right because for every element omega you have to map some real number. So, the pre image of the entire real line will be the sample space itself and probability of sample space we knows what right and then finally, you have to prove countable additivity of p x for disjoint Borel sets right there again you will have to use that the disjoint Borel sets will have disjoint pre images and then use countable additivity of p on omega f is that clear you can do it it is a very it is just definition chasing. So, what does a random variable do it effectively pushes. So, we have a measure omega. So, we have a measure p on omega f it pushes that measure on to R B R on to the Borel sigma algebra on R right and p x is the probability measure induced by the random variable on the real line the probability loss nothing, but the probability measure induced by the random variable on the real line on the Borel sigma algebra on the real line I should say be precise right you have to prove it right you have to take prove this joint Borel sets. So, you have to take a countable you have to prove what it countable additivity right you take B 1 B 2 dot dot dot as disjoint Borel sets and you look we have to prove that p x is countable additive right it follows from the countable additivity of p. Maximum 4 is to 6, but here same every b have to have a pre image. So, there will not be 1 to 1 mapping.  So, in that case all functions from omega to r will be random variables why because your f is 2 power omega right. So, pre image you cannot have a pre image which is not f measurable right. So, only in cases where your f is not 2 power omega does is there a possibility that a function is not a random variable right. If your f is 2 power omega all functions of omega to r will be random variables right that ok. So, that is your probability law. So, in order to specify. So, once you have this probability space right. So, you have real line. So, you are interested in what this random variable measure. So, may be you are just interested in today's temperature or the number of heads in 10 tosses or some such thing then you are only interested in this probability space you can if you are not interested in anything else you can throw away omega f p right. You can just keep this probability space because you are only interested in what x values of x it takes. So, if I want to specify. So, the random variable x I can if I specify p x for all boral sets I can I know it is complete probabilistic description on the real line right. So, it appears as though in order to give you a complete probability description of a random variable I need to give you the probability law p x for every boral set right. It turns out that you do not need p x for all boral sets. It is enough to specify p x for certain nicer subsets right. Boral sets are can be very complicated right. So, that is what I will get to next. See I also when I introduced boral sigma algebra on the real line I mentioned you that you can there are multiple ways you can generate the boral sigma algebra on the real line. One of them is using open intervals the other is using semi infinite intervals right. So, you can look at the boral sigma algebra as being generated by sets of that form for x belonging to r right. Now, so you can. So, let me write this. So, this we know right. So, you know that sigma algebra generated by sets of this form sigma algebra generated by sets of this form is nothing but your boral sigma algebra on r right. You have to prove that this is the same boral sigma algebra as that generated by open intervals, but that is true you can prove it. Now, what I am trying to get at is this is that this probability law which is defined for all boral sets must also be defined for the generating class correct. So, note that minus infinity x. So, note that sets of the form minus infinity x x belongs to r are boral measurable boral sets right. Thus p x is well defined on sets of this form. Why? Because, they are the generating class of the boral sigma algebra you know p x is defined on all boral sets. So, it must obviously be defined on the generating class which is minus infinity x. Note that the sets of the form are boral sets. Thus p x of minus infinity x is well defined for all x in r right agreed. I am simply saying that minus infinity x being a very special boral set must have a probability law assigned to it right. So, this is well defined. So, if you look at this. So, another way of writing this is. So, if you want to write this I can write it as probability of omega such that what x of omega is less than or equal to x right. So, I should put in order to be precise I should put set like that right. Everybody with me? So, that is a boral set. So, p x is defined on that set right and that is nothing but the probability that omega is such that x of omega lies in this. So, remember this right. So, it is the set of all omega for which x of omega lies in minus infinity x which is nothing but the probability that x of omega is less than or equal to little x correct. So, you are essentially this specifies the probability law of the semi infinite intervals right which are nice boral sets after all. And this has a name what is it called? This is the cumulative distribution function of the random variable this is called c d f cumulative distribution function right. And it is usually denoted by f x of little x. So, capital F subscript whatever random variable you are talking about of whatever the argument is little x. It gives you the probability that x of omega is less than or equal to x. So, in short hand notation this is usually written as probability that x is less than or equal to x. This is bad notation. So, this is abuse of notation, but it is like a short hand. So, if I write this I really mean this I will keep writing this several times, but I actually mean this. And similarly, for something like this people write short hand probability that x is in b right. When I write this again I mean this probability of omega for which after all p is only defined on subsets of omega right. So, when I write this I mean this and similarly when I write that I mean that any questions. So, what we said so far is that a random variable being a measurable function from omega to r pushes this measure from omega f to the boral sigma algebra on r right. So, given any boral set I can talk about p x of b which is the probability that x lies in that boral set. Then I went ahead and said hey this boral sigma algebra is after all generated by this semi infinite intervals. So, they must have a probability law there because they are nice sets nice boral sets. So, it is nice to give that a name. So, I have called that the cumulative distribution function. It is well defined because these guys are boral sets right. So, if you give me the probability law for all boral sets I can easily figure out the cumulative distribution function. Because these are also boral sets right this minus infinity x is also boral set. So, what is not so obvious and requires a proof is that if you specify the CDF. In fact, you can specify the probability law for all boral sets see after all see in your probably in your under graduate treatment of probability that you may have studied. You are told that CDF tells me everything about the random variable right yes or no. If I give you CDF you know everything about the random variables distribution right. That is true it is a correct statement, but it is not obvious if you think about it. So, the way we have developed it in order to know everything about the random variable I need to know the measure of probability measure of all these boral sets. But, if I only give you the measure of this minus infinity x type of semi infinite intervals. It may not be it is not clear at all that you can find p x for all boral sets right, but it is true that you can do it. After all see given any boral set you can write it as countable unions and intersections complements of generating class right and then you can figure out what p x is from f x. See the way to formalize this is suit using an object called a pi system. So, I will tell you what this is very briefly. A collection G of subsets of omega is said to be a pi system if a b in G implies a intersection b is in G. So, a pi system see like sigma like algebra sigma algebra etcetera this is another notion this is another notion. So, collection of subsets of omega is called a pi system if it is closed under finite intersections only finite intersections no complements no countable unions intersections etcetera closed under finite intersections. So, it is much weaker than a sigma algebra it is in fact weaker than an algebra also right all algebras are clearly pi systems, but not the other way. So, why do we care about this pi systems. So, the reason we care is that there is a non trivial result in measure theory that says if you uniquely specify a measure on a pi system it gets uniquely specified on the sigma algebra generated by G it is this is a non trivial result. So, this is something I will state, but not prove theorem let G be a pi system on omega and let f is equal to sigma of G.  So, what this theorem which is a non trivial result from measure theory this we will just use it I will just tell you what it means.  Now, I am saying that suppose there are two probability measures defined on omega f. So, omega f p 1 is a probability space omega f p 2 some other probability space two different probability measures defined on the same measurable space omega f and I am saying that they agree on G. So, p 1 equals p 2 so p 1 of some set in G will be the same as p 2 was the set in the G. So, on this pi system they agree then they have to agree on the entire sigma algebra which means if you uniquely specify a measure on the pi system the measure is uniquely specified on the entire sigma algebra. So, why is this useful because pi systems are simple and sigma algebra are complicated. So, on the real line for example, so why did I go through the story you can show. So, let us pi of r equal to the set of all minus infinity x where x is in r is a pi system. So, you can see that if you have one set of the form minus infinity x another set of the form minus infinity y the intersection of two such sets will be also of this form. So, collections of sets subsets of r of this form are clearly a pi system on r fine and. So, I am saying now and sigma of pi of r is what is Borel sigma algebra is B r. So, now if I invoke this theorem what happens. So, if I tell you the C D F which is simply specifying the probability measure on the pi system pi r then it gets uniquely specified on the entire Borel sigma algebra correct. So, I can state the theorem the C D F uniquely specifies probability law P X any random variable. So, the argument so to summarize I will summarize what I have said so far. So, we started of saying that a random variable is a measurable function. So, which means pre-images of Borel sets are F measurable. So, for each Borel set B we can assign a probability measure which is called the probability law P X right. So, P X of B is the probability that the random variable takes values in the Borel set B.  So, in a sense what a random variable does is pushes this measure from omega F to R B R. And if you only care about that random variable you can forget about omega F and only look at R B R P X right. Now, then we further said that Borel sets are generated by this class semi infinite intervals. So, the probability of those sets are well defined and probability P X the probability law of the semi infinite intervals is what defines the CDF of a random variable. And finally, we said that sets of this kind form of pi system. And since specifying a probability measure on pi system is enough to specify it on the entire sigma algebra it is enough to just tell me if you tell me the CDF of a random variable I can uniquely specify the probability measure P X on the entire Borel sigma algebra. So, in short although P X is the complete description of the random variable in practice we will use F X the CDF because it captures all the information of P X. And it is more intuitive because it has to it deals with nice sets right. Whereas, this probability law has to be specified for all Borel sets which is a very complicated bunch of sets right. Whereas, these are very nice sets right. So, are there any questions at this point I think I am out of time. So, if there are no questions I will stop.