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Chapter 4: Q.4.21 (page 174)

Suppose that

P{X=a}=p,P{X=b}=1-p

(a) show that X-ba-bis a Bernoulli random variable

(b) Find Var(X).

Short Answer

Expert verified

In the given information the answer os part (a) isX-ab-a~01p1-pwhich show that is Bernoulli random variable.

(b) isVar(X)=(b-a)2p(1-p)

Step by step solution

01

 Step 1 :Given Information (Part-a)

With the probability p,random variable Xassumes 伪.Hence, with the same probability, random variable X-ab-aassumes0

with the probability localid="1646975947088" 1-p,random variable Xassumes bHence, with the same probability ,random variable X-ab-aassumes1

02

:Explanation (Part-a)

X-ab-a~01p1-p

which shows that it is a Bernoulli random variable

03

:Final Answer (Part-a)

The answer is X-ab-a~01p1-p

which shows thatX-ab-ais Bernoulli random variable

04

:Given Information(Part-b)

We know thatVarX-ab-a=p(1-p).

05

Step 5:Explanation (Part-b)

VarX-ab-a=1(b-a)2Var(X-a)=1(b-a)2Var(X)

so we get thatVar(X)=(b-a)2p(1-p)

Var(X)=(b-a)2p(1-p)

Var(X)=(b-a)2p(1-p)

06

:Final Answer (Part-b)

Var(X)=(b-a)2p(1-p)

The answer isVar(X)=(b-a)2p(1-p)Var(X)=(b-a)2p(1-p)

Var(X)=(b-a)2p(1-p)Var(X)=(b-a)2p(1-p)

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