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.

Short 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-by-step solution

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.

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.

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

Explanation (Part b)

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

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