Q.4.36 - A First Course in Probability

Question

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 . 

Step-by-Step Solution

Answer

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

Step 1:Given Information

The random variable $X$ denotes the number of games played.

Observe that $X\in \left\{2,3\right\}$

X can be written as $X=2+I$ $\left(I is the indicator random variable\right)$

               $P(I=1)=2 p(1-p)$

Step 2:Explanation

Variance of the derivative is :

$Var\left(X\right)=Var(2+I)=Var\left(I\right)=2p(1-p)·\left(p^2+(1-p{)}^2\right)$

             

Step 3:Final Answer

Root of the derivative is $\iff -2(2p-1{)}^3=0\iff p=\frac{1}{2}$

                  $\iff -2(2p-1{)}^3=0\iff p=\frac{1}{2}$

The variance is maximized when $P=\frac{1}{2}$