Consider two bets on two independent flips of a fair coin. If a coin lands on heads, the payoff of the bet is 5 and if the coin lands on tails, the payoff is 0. Each of the flips is independent of the other. Let X_i represent the results of the i^th where i =1,2

a) What is the expected value and variance of X_2. Show all work

b)What is the expected value and variance of a new variable, Y= X₁ + X₂ which sums the total payoff over the two independent bets? Show all work.

c) Now Consider a different scenario. Instead of the two flips being independent assume that the second coin flip is dependent on the first, If the first coin lands on tails, the second one will also land on tails. The first coin is still a fair coin, equally likely to land on either heads or tails. what is the expected value and variance of X₂ under the new assumption?

d) will the variance of Y under this new set of assumptions be higher, lower, or equal to the value you calculated in [art b) above? Note that you do not have to calculate the variance of Y to answer this question, but you must make an argument. As a hint, recall the definition of the variance of a sum of random variables