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Chapter 5: Q. 5.27 (page 216)

If xis uniformly distributed over (a,b), what random variable, having a linear relation with x, is uniformly distributed over (0,1)?

Short Answer

Expert verified

Random variable having linear relation withxisY=X-ab-a

Step by step solution

01

Random variable

Given that X~Unif(a,b).

Approach Random variable

Y=X-ab-a

02

Explanation

SinceX∈(a,b).

Y∈(0,1).

For y∈(0,1),

P(Y≤y)=PX-ab-a≤y

=P(X≤a+(b-a)y)

=a+(b-a)y-ab-a

=y

Proved that Y~Unif(0,1).

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