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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∈(a,b), we have that Y∈(0,1). Also, for y∈(0,1)we have that P(Y≤y)=PX-ab-a≤y=P(X≤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−ab−a

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Most popular questions from this chapter

Compute the hazard rate function of a Weibull random variable and show it is increasing when 𝛽≥1and decreasing when 𝛽≤1

Suppose that the cumulative distribution function of the random variable Xis given by

F(x)=1−e−x2x>0

Evaluate (a)P[X>2];(b)P[1<X<3); (c) the hazard rate function of Fi(d)E[X]; (e)Var⁡(X).

Hint: For parts (d)and (e), you might want to make use of the results of Theoretical Exercise 5.5.

Show thatE[Y]=∫0∞PY>ydy-∫0∞PY<-ydy.

Hint: Show that ∫0∞PY<-ydy=-∫-∞0xfy(x)dx

∫0∞PY>ydy=∫0∞xfy(x)dx

Prove Theorem 7.1 when g(x) is a decreasing function.

Your company must make a sealed bid for a construction project. If you succeed in winning the contract (by having the lowest bid), then you plan to pay another firm $100,000 to do the work. If you believe that the minimum bid (in thousands of dollars) of the other participating companies can be modeled as the value of a random variable that is uniformly distributed on (70, 140), how much should you bid to maximize your expected profit?
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