Question

) Suppose you flip a coin. If it comes up heads, you win $20; if it comes up tails, you lose $20. a) Compute the expected value and variance of this lottery. b) Now consider a modification of this lottery: You flip two fair coins. If both coins come up heads, you win $20. If one coin comes up heads and the other comes up tails, you neither win nor lose – your payoff is $0. If both coins come up tails, you lose $20. Verify that this lottery has the same expected value but a smaller variance than the lottery with a single coin flip. (Hint: The probability that two fair coins both come up heads is 0.25, and the probability that two fair coins come up tails is 0.25.) Why does the second lottery have a smaller variance?

Answer 1

if a coin is flipped once,

the outcome will be H,T

outcome on the coin	head	tail
probability	1/2	1/2
payoff = X	+20	-20

expected value of the lottery = E(X) =

variance =

now if the coin is flipped twice,

the outcomes will be ; HH,HT,TH,TT

outcome of the coin	2 heads	1 head and 1 tail	2 tails
probability	1/4	1/2	1/4
payoff = X	+20	+0	-20

expected value = E(X) =

we see that the expected value in both the cases is same i.e. zero ( verified )

variance measures the variability from the expected value. The variance in the second case is less than the variance in the first case, this is because the probability of events +20 and -20 has decreased. the decrease in the probability of extreme values has indeed decreased the probability that the random variable will take values that are farther away from the mean value i.e. 0, hence reducing the variability.