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Chapter 6: Q.6.20 (page 276)

Let U denote a random variable uniformly distributed over (0, 1). Compute the conditional distribution of U given that

(a) U > a;

(b) U < a; where 0 < a < 1.

Short Answer

Expert verified

a. The conditional distribution is P(U>s,U>a)=1-s1-a,a<s<1

b. The conditional distribution isP(U<s,U<a)=sa,0<s<a

Step by step solution

01

Content Introduction

A random variable is a variable with an unknown value or a function that gives values to each of the results of an experiment. It's possible for a random variable to be discrete or continuous.

02

Explanation (Part a)

Let the random variable U follow uniform distribution over (0 , 1).

The cumulative distribution of U is

P(U≤u)=F(u)=u-01-0=u

Find the distribution conditional of U given that U > a.

P(U>s,U>a)=P[U>s∩U>a]P(U>a)=P(U>s)P(U>a)=1-P(U≤s)1-P(U≤a)=1-s1-a

03

Explanation (Part b)

Find the conditional distribution of U given that U < a.

P(U<s,U<a)=P[U<s∩U<a]P(U<a)=P(U<s)P(<>a)=s-01-0a-01-0=sa

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