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Let $X_{1}, X_{2}, \ldots, X_{n}$ be random variables denoting $n$ independent bids for an item that is for sale.Suppose each $X_{i}$ is uniformly distributed on the interval $[100,200] .$ If the seller sells to thehighest bidder, how much can he expect to earn on the sale? [Hint: Let $Y=\max \left(X_{1}, X_{2}, \ldots, X_{n}\right)$Use the results of Sect. 4.9 to find $E(Y) . ]$ .

Intro Stats / AP Statistics

Chapter 4

Joint Probability Distributions and Their Applications

Section 1

Jointly Distributed Random Variables

Probability Topics

The Normal Distribution

Missouri State University

Piedmont College

Cairn University

Boston College

Lectures

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Let $X(s)$ be a random var…

Yeah. No. So let X one X two up to X N. Um B The random variable is denoting an independent state for an item that is for sale. So you're going to suppose X I is uniformly distributed. I'm just sorry, is uniformly distributed on this in the war mm. So the probability density function of X. Y is given vices it is uniformly distributed. They're going to get out of X. I hear X to the one over B minus A. Why? We know this is B and this is a. So we have 200 0. So you have one with 100. Okay, so the conclusive and distribution function, which is the CTF? We have the CTF. Yeah. Oh X I giving us a big F of X. Y of X 342 X minus a over B minus a. So we are going to get extra in this 100 over 200 0. So you have 100 over here where X belongs to this interval? We have 100. I'm not too hungry. Mhm. And so we are going to let Y be the highest bid. So we are going to get Y to be equal to the max of X. I. Sorry, X one up to X. M two. Excellent. Up to X. And over here. So our meaning is to find for the expected value of X. Do you want to find for this thing over here? So now the communicative and distribution function of Y can be obtained us. So the CDF for 4 to Cdef. Oh, why can be obtained us if for to be equal to be of, why legs are not ego? Too small wife? So we have p. Oh, max X. I where X I Square eyes from one up to 10, there's been no equal to Y. Mhm. So from here you're going to get a P X. one. There is an equal to worry Have to P of X one there is that uh It was too Why? Um So we are doing this because earlier we know X. I are independent. So we know the X eyes are independent. Yeah. So if they are independent, it means we are going to get the product. So we're going to use this in steps. So I from one to end the product of this. So we have hawaii in this 100. All over 100. Yeah. Oh, okay. So every four, which is that related will be equal to? Why am I in this 100? Yes. or divided by a 100 all to the power and where why belongs to I'm great. 200. So on this. And so now we can talk with the probability density function of why? By differentiating the CTF with respect to Y. So less different sheets. So they're going to find the P. D. F. So we have a small F of small F. Y of Y. To be equal to a differential of the big F. Yeah. Okay. With respect to Y. So what are we differentiated or function? So we have uh why minus 100 Or divided by 100 to the power? Uh huh. Right. So differentiating days you are going to get in over 100 in you know why my name is Hendrik. Let's get up and nine is 1 why we have why belonging to this interval? So moving further. You're not going to calculate the expected value of Why? As I said earlier this is our main aim. So we have E. F. Y. To me. So this will be ego to the integral of 100 2 200 of. Why? Thanks the pdf which is way off. Why do I, mm So we have 100 2 200 then we bring it away. Thanks. Um We have an divided by 100 to the bar in. Thanks. Why am I in this? 100? Go to the park and -1 do you worry? Mhm. All right. Yeah so we can um bring an end over 100 to the park and out since we are differentiating with respect to why not in. So we have 100 to 200 over here. Then we are going to get wide notified by a y minus 100 or to the bar. I mean this one don't worry. So now we are going to replace them. Why minus 100 activity? So if we do that we are going to get the Y. To be equal to D. T. Mhm. So moving on we are going to get the expectation of Y. To be equal to end over 100 to department you see girl Is there were 200. We have our tea plus 100 dying. See to grandpa N minus one did see for him. Right so from here we're going to get And over a 100 it's a difficult in We have a 0 200 you're going to get T to the far end class and right See to the ball and -1 get T. So now we can different we can integrate two. Mhm. So integrating this we are going to get do we have a n Over 100 to the bar in over here. So in the integral honey we integrate that we are going to get T to the bar. Yeah the first one. Yeah what divided by entrust work plus 100 over here. They have T to the bar in over in On this interval is early 200. So we need to get and over 100 to the bar in. So we have 100 and to the power M. T. And crusty All divided by N Plus one. Bless 100 over here we have 100 to depart in. Oh divided right in you know Sarah over here. So you're going to get and and times 100 To the bar and plus one Are divided by a 100 to the pardon? Yeah all days multiplied by N. Plus and last one all divided by in Into N Plus one. So from here We are going to get 100 into bracket two N plus one what's right is right and rest one overhand. And so his um if there's a lot sells to the highest weather they expected and on the sale will be equal to so they expected still which is the white and Expected value of Y will be evil 200 Then is two and last one for any glassware. Mhm.

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