Q. 5.29 Let X be a continuous random var... [FREE SOLUTION]

91影视

A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

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 $Y ~ Uni f (0, 1)$

Step by step solution

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 $X 鈭� (a, b)$ , we have that $Y 鈭� (0, 1)$ . Also, for $y 鈭� (0, 1)$ we have that $P (Y 鈮� y) = P (\frac{X - a}{b - a} 鈮� y) = P (X 鈮� a + (b - a) y) = \frac{a + (b - a) y - a}{b - a} = y$ so we have proved that $Y ~ Unif (0, 1)$ .

Step:4 Result

The result is $Y = \frac{X 鈭� a}{b 鈭� a}$

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91影视

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

Step by step solution

Step:1 Given Information

Step:2 Definition

Step:3 Explanation

Step:4 Result

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