Q. 5.29 - A First Course in Probability

Question

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). 

Step-by-Step Solution

Answer

We have proved that  $Y~\mathrm{Uni}f(0,1)$

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).

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.

Step:3 Explanation

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

$Y=\frac{X-a}{b-a}$

Since $X\in (a,b)$, we have that $Y\in (0,1)$. Also, for $y\in (0,1)$ we have that $P(Y\le y)=P\left(\frac{X-a}{b-a}\le y\right)=P(X\le a+(b-a\left)y\right)=\frac{a+(b-a)y-a}{b-a}=y$ so we have proved that $Y~Unif(0,1)$.

Step:4 Result

The result is $Y=\frac{X-a}{b-a}$