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Chapter 4: Q.4.36 (page 166)

Consider Problem 4.22 with i = 2. Find the variance of the number of games played, and show that this number is maximized when p = 1 2 .

Short Answer

Expert verified

In the given information the variance is maximized when p=1/2

Step by step solution

01

Step 1:Given Information

The random variable Xdenotes the number of games played.

Observe that X∈2,3

X can be written as X=2+IIistheindicatorrandomvariable

P(I=1)=2p(1-p)

02

Step 2:Explanation

Variance of the derivative is :

Var(X)=Var(2+I)=Var(I)=2p(1-p)·p2+(1-p)2

03

Step 3:Final Answer

Root of the derivative is ⇔-2(2p-1)3=0⇔p=12

⇔-2(2p-1)3=0⇔p=12

The variance is maximized whenP=12

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Most popular questions from this chapter

Suppose that the distribution function of X given by

F(b)=0    b<0b4    0≤b<112+b−14    1≤b<21112    2≤b<31    3≤b

(a) Find P{X=i},i=1,2,3.

(b) Find P12<X<32.

Suppose that Xtakes on one of the values0,1and2. If for some constantc,P{X=i}=cP{X=i-1},i=1,2, findE[X].

If X has distribution function F, what is the distribution function of eX?

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