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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鈮�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鈭�U>a]P(U>a)=P(U>s)P(U>a)=1-P(U鈮�s)1-P(U鈮�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鈭�U<a]P(U<a)=P(U<s)P(<>a)=s-01-0a-01-0=sa

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Most popular questions from this chapter

Consider a directory of classified advertisements that consists of m pages, where m is very large. Suppose that the number of advertisements per page varies and that your only method of finding out how many advertisements there are on a specified page is to count them. In addition, suppose that there are too many pages for it to be feasible to make a complete count of the total number of advertisements and that your objective is to choose a directory advertisement in such a way that each of them has an equal chance of being selected.

(a) If you randomly choose a page and then randomly choose an advertisement from that page, would that satisfy your objective? Why or why not? Let n(i) denote the number of advertisements on page i, i = 1, ... , m, and suppose that whereas these quantities are unknown, we can assume that they are all less than or equal to some specified value n. Consider the following algorithm for choosing an advertisement.

Step 1. Choose a page at random. Suppose it is page X. Determine n(X) by counting the number of advertisements on page X.

Step 2. 鈥淎ccept鈥� page X with probability n(X)/n. If page X is accepted, go to step 3. Otherwise, return to step 1.

Step 3. Randomly choose one of the advertisements on page X. Call each pass of the algorithm through step 1 an iteration. For instance, if the first randomly chosen page is rejected and the second accepted, then we would have needed 2 iterations of the algorithm to obtain an advertisement.

(b) What is the probability that a single iteration of the algorithm results in the acceptance of an advertisement on page i?

(c) What is the probability that a single iteration of the algorithm results in the acceptance of an advertisement?

(d) What is the probability that the algorithm goes through k iterations, accepting the jth advertisement on page i on the final iteration?

(e) What is the probability that the jth advertisement on page i is the advertisement obtained from the algorithm?

(f) What is the expected number of iterations taken by the algorithm?

Derive the distribution of the range of a sample of size 2 from a distribution having density function f(x)=2x,0<x<1.

ThejointprobabilitydensityfunctionofXandYisgivenbyf(x,y)=e鈭�(x+y)0鈥�x<q,0鈥�y<qFind(a)PX<Yand(b)PX<a

If X1 and X2 are independent exponential random variables, each having parameter 位, find the joint density function of Y1=X1+X2 and Y2=eX1 .

In Problem 6.3, calculate the conditional probability mass function of Y1given that

(a) localid="1647528969986" Y2=1;

(b) localid="1647528979412" Y2=0.
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