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Chapter 4: Q. 4.5 (page 173)

Suppose that P{X=0}=1−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−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

⇒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⇒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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