/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q. 4.5 Suppose that P{X=0}=1鈭扨{X=1}. ... [FREE SOLUTION] | 91影视

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Chapter 4: Q. 4.5 (page 173)

Suppose that P{X=0}=1鈭�P{X=1}. If E[X]=3Var(X), find P{X=0}.

Short Answer

Expert verified

Observe two cases. If EX=0, the answer is p=1. Otherwise, answer13.

Step by step solution

01

Given information

Given in the question is a function,

P{X=0}=1鈭�P{X=1}. If E[X]=3Var(X).

02

Computation

Define P(X=0)=p.

Then we have that P(X=1)=1-p.

From the given relation between the mean and variance, we have that

localid="1646920607051" E(X)=3Var(X)=3EX2-3(EX)2=3EX-3(EX)2

鈬�3(EX)2=2EX.

03

Calculation

The last inequality in the first row holds, since Xonly assumes values 0and l, so X~X2.

If localid="1646920627387" E(X)=0, we have that p=1. So, suppose that localid="1646920639456" E(X)>0. Then, we have that

localid="1646920651370" E(X)=23

Hence,

localid="1646920661162" 23=E(X)=1-p鈬�p=13.

04

Final answer

We have observe two cases.

If E(X)=0, the answer is p=1. Otherwise, answer is1/3.

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Most popular questions from this chapter

An urn initially contains one red and one blue ball. At each stage, a ball is randomly chosen and then replaced along with another of the same color. Let X denote the selection number of the 铿乺st chosen ball that is blue. For instance, if the 铿乺st selection is red and the second blue, then X is equal to 2.

  • (a) Find P{X>i},i鈮�1
  • (b) Show that with probabilityrole="math" 1, a blue ball is eventually chosen. (That is, show thatP{X<鈭�}=1.)
  • (c) FindE[X].

Aand Bwill take the same 10-question examination. Each question will be answered correctly by Awith probability.7, independently of her results on other questions. Each question will be answered correctly by B with probability .4, independently both of her results on the other questions and on the performance of A.

(a) Find the expected number of questions that are answered correctly by both A and B.

(b) Find the variance of the number of questions that are answered correctly by either A or B

Suppose that the number of events that occur in a specified time is a Poisson random variable with parameter 位. If each event is counted with probability p, independently of every other event, show that the number of events that are counted is a Poisson random variable with parameter 位p. Also, give an intuitive argument as to why this should be so. As an application of the preceding result, suppose that the number of distinct uranium deposits in a given area is a Poisson random variable with parameter 位=10. If, in a fixed period of time, each deposit is discovered independently with probability 150, find the probability that

(a) exactly ,

(b) at least 1, and

(c) at most 1deposit is discovered during that time.

Suppose that the random variable Xis equal to the number of hits obtained by a certain baseball player in his next 3at-bats. If P{X=1}=3,P{X=2}=2andP{X=0}=3P{X=3}, find E[X].

Suppose that the number of accidents occurring on a highway each day is a Poisson random variable with parameter 位 = 3.

(a) Find the probability that 3 or more accidents occur today.

(b) Repeat part (a) under the assumption that at least 1 accident occurs today.
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