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Chapter 4: Q.4.8 (page 170)

Let Xbe a random variable having expected value μand variance σ2. Find the expected value and variance ofY=X-μσ.

Short Answer

Expert verified

Mean is 0, and variance is 1.

Step by step solution

01

Given information

Let X be a random variable having expected value μ and variance σ2.

02

Calculation

Utilizing the linearity of expectation, we have that

EY-μσ=1σ(E(Y)-E(μ))=1σ(μ-μ)=0

And using the effects of the Variance (adding a constant accomplishes not change the variance and multiplying by a constant increase the variance by the square) we have that

VarY-μσ=1σ2Var(Y)=1σ2×σ2=1

03

Final answer

The expected value and variance are0,1.

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Most popular questions from this chapter

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?

A box contains 5red and 5blue marbles. Two marbles are withdrawn randomly. If they are the same color, then you win \(1.10; if they are different colors, then you win -\)1.00. (That is, you lose $1.00.) Calculate

(a) the expected value of the amount you win;

(b) the variance of the amount you win.

Three dice are rolled. By assuming that each of the 63=216 possible outcomes is equally likely, find the probabilities attached to the possible values that X can take on, where X is the sum of the 3 dice.

A fair coin is flipped 10times. Find the probability that there is a string of 4consecutive heads by

(a) using the formula derived in the text;

(b) using the recursive equations derived in the text.

(c) Compare your answer with that given by the Poisson approximation.

Here is another way to obtain a set of recursive equations for determining Pn, the probability that there is a string of kconsecutive heads in a sequence of nflips of a fair coin that comes up heads with probability p:

(a) Argue that for k<n, there will be a string of kconsecutive heads if either

1. there is a string of kconsecutive heads within the first n-1flips, or

2. there is no string of kconsecutive heads within the first n-k-1flips, flip n-kis a tail, and flips n-k+1,…,nare all heads.

(b) Using the preceding, relate PntoPn-1. Starting with Pk=pk, the recursion can be used to obtain Pk+1, thenPk+2, and so on, up to Pn.
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