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Chapter 4: Q.4.6 (page 170)

Let Xbe such thatP{X=1}=p=1-P{X=-1}

Find c≠1such that EcX=1.

Short Answer

Expert verified

The two possible values of c are p and 1p-1.

Step by step solution

01

Given information

Given in the question that,

P{X=1}=p=1-P{X=-1}.

02

Calculation

Observe that random variable Xassumes only two values with positive probability. We have that

P(X=1)=p

P(X=-1)=1-p

Using the theorem about the expectation of a function of a random variable,

We have that

1=EcX=c×p+1c(1-p)

Multiply both sides with c, we end up with an equation

pc2-c+(1-p)=0.

03

Solution

Two solutions are,

c1,2=1±1-4p(1-p)2p

Observe that the term under the root is equal to (2p-1)2, so we have

c1,2=1±|2p-1|2p

Therefore, we get that we have two answers

c1=1,c2=1-pp.

04

Final answer

There exist two possible values of c. They are p and 1p-1.

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