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Chapter 4: Q.4.8 (page 170)

Find Var(X) ifP(X=a)=p=1−P(X=b)

Short Answer

Expert verified

We have found to beVar(X)=(a-b)2(1-p)p

Step by step solution

01

Given information

Given in the question is,

P(X=a)=p=1-P(X=b)

02

Substitution

The random variable Xcan consider only two values, awith probability pand bwith the probability 1-p.

Using the meaning of the mean, we have that

E(X)=a×p+b×(1-p)=:μ

03

Calculation

Currently, operating the definition of the variance, we have that

Var(X)=E(X-μ)2=(a-μ)2·p+(b-μ)2·(1-p)

=(a-a×p-b×(1-p))2p+(b-a×p+b×(1-p))2(1-p)

Substitute the given expression,

=(a-b)2(1-p)2p+(b-a)2p2(1-p)

role="math" localid="1646669147074" =(a-b)2(1-p)p[1-p+p]

We get,

=(a-b)2(1-p)p.

04

Final answer

The solution of theVar(X)is found to be(a-b)2(1-p)p.

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Most popular questions from this chapter

Five men and 5 women are ranked according to their scores on an examination. Assume that no two scores are alike and all 10! possible rankings are equally likely. LetXdenote the highest ranking achieved by a woman. (For instance,X=1 if the top-ranked person is female.) FindP{X=i},i=1,2,3,…,8,9,10

A man claims to have extrasensory perception. As a test, a fair coin is flipped 10times and the man is asked to predict the outcome in advance. He gets 7out of 10 correct. What is the probability that he would have done at least this well if he did not have ESP?

Let X be a binomial random variable with parameters (n, p). What value of p maximizes P{X = k}, k = 0, 1, ... , n? This is an example of a statistical method used to estimate p when a binomial (n, p) random variable is observed to equal k. If we assume that n is known, then we estimate p by choosing that value of p that maximizes P{X = k}. This is known as the method of maximum likelihood estimation.

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Show also that for n large,

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in the sense that the ratio Var(X) ton/4approaches 1as n approaches q. Compare this formula with the limiting form of Var(Y) when P{Y =i}=1/n,i=1,...,n.
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