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Chapter 5: Q. 5.29 (page 216)

Let X be a continuous random variable having cumulative distribution function F. Define the random variable Y by Y = F(X). Show that Y is uniformly distributed over (0, 1).

Short Answer

Expert verified

We have proved that Y~Unif(0,1)

Step by step solution

01

Step:1 Given Information

Consider X, a continuous random variable with the cumulative distribution function F. Y = F defines the random variable Y. (X). Demonstrate that Y is distributed evenly throughout the board (0, 1).

02

Step:2 Definition 

Each variable has its own probability distribution function (a mathematical characteristic that represents the possibilities of incidence of all viable consequences). Discrete and continuous variables are the 2 styles of random variables.

03

Step:3 Explanation

We are given that X~Unif(a,b). Consider random variable

Y=X-ab-a

Since X鈭�(a,b), we have that Y鈭�(0,1). Also, for y鈭�(0,1)we have that P(Y鈮�y)=PX-ab-a鈮�y=P(X鈮�a+(b-a)y)=a+(b-a)y-ab-a=yso we have proved that Y~Unif(0,1).

04

Step:4 Result

The result isY=X鈭�ab鈭�a

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