Calculating Expected Value for Independent Random Variables

[jakey]

Hi guys, if X1,X2,...,Xn are independent and identically distributed random variables, how do you find E(max(X1,...,Xn))?

Do you need to do order statistics or anything of that sort here? I got my answer by letting Y=max(X1,...Xn) and I got the CDF and then pdf of Y. For the CDF of Y, I just noted that P(Y<=y) = P(X1<=y, X2<=y,..., Xn<=y). Is this right?

 

[statdad]

Hi guys, if X1,X2,...,Xn are independent and identically distributed random variables, how do you find E(max(X1,...,Xn))?

Do you need to do order statistics or anything of that sort here? I got my answer by letting Y=max(X1,...Xn) and I got the CDF and then pdf of Y. For the CDF of Y, I just noted that P(Y<=y) = P(X1<=y, X2<=y,..., Xn<=y). Is this right?

Yes: write
<br /> P(Y \le y) = P(X_1 \le y, \dots, X_n \le y) = F_Y^n(y)<br />

This allows you to get the density (assuming the quantities are continuous) and then you find the expected value the usual way.

 

[jakey]

Thanks statdad! :D