Q.6.20 - A First Course in Probability

Question

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. 

Step-by-Step Solution

Answer

a. The conditional distribution is $P(U>s, U >a)=\frac{1-s}{1-a}, a<s<1$

b. The conditional distribution is $P(U<s, U<a)=\frac{s}{a},0<s<a$

Step 1: 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.

Step 2: Explanation (Part a)

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

The cumulative distribution of U is 

$$\begin{gathered} P(U\le u)=F\left(u\right) \\ =\frac{u-0}{1-0} \\ =u \end{gathered}$$

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

$$\begin{gathered} P(U>s, U>a)=\frac{P [U>s\cap U>a]}{P(U>a)} \\ =\frac{P(U>s)}{P(U>a)} \\ =\frac{1-P(U\le s)}{1-P(U\le a)} \\ =\frac{1-s}{1-a} \end{gathered}$$

Step 3: Explanation (Part b)

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

$$\begin{gathered} P(U<s, U<a)=\frac{P [U<s\cap U<a]}{P(U<a)} \\ =\frac{P(U<s)}{P(<>a)} \\ =\frac{{\displaystyle \frac{s-0}{1-0}}}{{\displaystyle \frac{a-0}{1-0}}} \\ =\frac{s}{a} \end{gathered}$$