Distribution functions

Hi there,

I was wondering if someone could explain how to go about the following problem.

Let X and Y be independent random variables having distribution functions F_x and F_Y , respectively.

a) Define Z = max{X,Y} to be the larger of the two. Show that F_Z(z) = F_X(z)F_Y(z) for all z.

b) Define W = min{X,Y} to be the smaller of the two. Show that F_W(w) = 1 - [1-F_X(w)][1-F_Y(w)] for all w.

Thanks!

Quote:

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Originally Posted by supermario88

Hi there,

I was wondering if someone could explain how to go about the following problem.

Let X and Y be independent random variables having distribution functions F_x and F_Y , respectively.

a) Define Z = max{X,Y} to be the larger of the two. Show that F_Z(z) = F_X(z)F_Y(z) for all z.

b) Define W = min{X,Y} to be the smaller of the two. Show that F_W(w) = 1 - [1-F_X(w)][1-F_Y(w)] for all w.

Thanks!

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since they are independent,



and

Hi harish21,

Thanks very much for the reply. So I can finish letter b thanks to your hint. Would you mind explaining the intuition of letter a? Specifically, could you talk about why

P[Z≤z] = P[(X≤z)∩(Y≤z)]

I just want to develop a better understanding of the problem. Thank you.

Z is the largest of X and Y
so IF Z is less than or equal to a, then both X and Y must be less than or equal to a.
For the minimum, use the complement twice

I see. Thanks very much!

I work with order statistics all the time.
I can google my name and find some of my papers...
http://w3.math.sinica.edu.tw/bulleti...d322/32203.pdf

Understood. Thanks matheagle.