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Chapter 6: Q.6.25 (page 277)

Suppose that F(x) is a cumulative distribution function. Show that (a) Fn(x) and (b) 1 − [1 − F(x)] n are also cumulative distribution functions when n is a positive integer. Hint: Let X1, ... , Xn be independent random variables having the common distribution function F. Define random variables Y and Z in terms of the Xi so that P{Y … x} = Fn(x) and P{Z … x} = 1 − [1 − F(x)] n

Short Answer

Expert verified

Consider CDF's of random variables min (X1,......Xn) and max (X1,....X,n)

Step by step solution

01

Content Introduction

Let X1,....Xn be independent and equally distributed continuously random variables with common CDF F.

02

Content Explanation

Consider random variables Y=max(X1,....Xn)and Z=min(X1,...,Xn). Take any x∈ℝ.

We have that,

role="math" localid="1647442498771" FY(x)=P(Y≤x)=P(max(X1,.....Xn)≤x)=P(Xi≤x,∀i)=∏i=1nP(Xi≤x)=∏i=1nF(x)=Fn(x)

So we have CDF of Y is Fy(x)=Fn(x)

On the other hand, we have that

1-Fz(x)=P(Z≥x)=P(min(X1,....,Xn)≥x)=P(Xi≥x,∀i)=∏i=1nP(Xi≥x)=∏i=1n(1-F(x))

So we have CDF of Z isFz(x)=1-(1-F(x)n)

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