Q.6.25 - A First Course in Probability

Question

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 

Step-by-Step Solution

Answer

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

Step 1: Content Introduction

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

Step 2: Content Explanation

Consider random variables $Y= max (X_1,....X_n)$ and $Z=min (X_1,...,X_n)$. Take any $x\in \mathrm{ℝ}$.

We have that, 

$$\begin{gathered} F_Y\left(x\right)=P(Y\le x) \\ =P\left(max\right(X_1,.....X_n)\le x) \\ =P(X_i\le x,\forall i) \\ =\prod _{i=1}^{n}P(X_i\le x) \\ =\prod _{i=1}^{n}F\left(x\right)=F^n\left(x\right) \end{gathered}$$

So we have CDF of Y is $F_y\left(x\right)=F^n\left(x\right)$

On the other hand, we have that

$$\begin{gathered} 1-F_z\left(x\right)=P(Z\ge x)=P\left(min\right(X_1,....,X_n)\ge x) \\ =P(X_i\ge x,\forall i) \\ =\prod _{i=1}^{n}P(X_i\ge x) \\ =\prod _{i=1}^n(1-F(x\left)\right) \end{gathered}$$

So we have CDF of Z is $F_z\left(x\right)=1-(1-F{\left(x\right)}^n)$