Suppose a fair coin 10 is tossed times and you win W X 5 d

Suppose a fair coin 10 is tossed times and you win W = X - 5 dollars, where X is the number of heads flipped. Determine the mean and variance of the random variable using the rules for a linear combination of a random variable. Suppose the game is changed so that W = 2X - 10. Determine the mean and variance of W for this payoff scheme. Which of the two payoff schemes is more variable?

Solution

Here X be the random variable having a binomial distribution with parameter n=10, p=0.5

hence E(X)=5 and var(X)=2.5

a) W=X-5

hence, E(W)= E(X-5)=E(X)-5=0

var(W)=var(X-5)=Var(X)=2.5

b) W=2X-10

hence E(W)=2*E(X)-10=0

var(W)=2*2*var(X)=10

c) since the second pair has more variacne so in that case payoff scheme is more variable