Question

A fair coin is tossed n times. Each coin toss costs d dollars and the reward in obtaining X heads is aX2 +bX. Find the expected value of the net reward.

Answer 1

a) A fair coin is tossed n times. Each coin toss costs d dollars and the reward in obtaining X heads is aX2 +bX. Find the expect

Let a fair coin be tossed times and each coin toss costs "ff dollars.

The reward for say X heads is aX^2+ bX

The cost of n toss is nd

The net reward is:

R = Reward in obtaining heads — Cost

R =aX2 +bX-nd

Here, “X” is the number of heads and it follows binomial distribution with the probability of success p = 0.5

The mean of the binomial distribution is,

E(X)=n p

       =n (0.5)

E(X)=0.5n

The variance of the random variable X is,

Var(X) = np(1-P)

            =n(0.5)(1-0.5)

            =n(0.5)(0.5)

Var(X)=0.25n

The expected value of X^2 is,

Var(X)=Var(X)+(E(X))^2

            =0.25n+(0.5n)^2

Var(X) = 0.25n+0.25n^2

The expected for net reward is,

The expected value of X2 is,

E(x^2)=Var(X) +(E(X))^2

           =0.25n+(0.5n)^2

            = 0.25n+0.25n

The expected for net reward is,

E[ax2 +bX -nd] =AE[X^2] + bE[X] - nd

                          =a(0.25n + 0.25n2) + b(0.5n) — nd

E[ax2 +bX-nd] = 0.25an +0.25an2 + 0.5nb—nd