I don't have the answer and I'd appreciate any feedback.
Consider the following game involving two players and a bag containing 3 discs: 2 blue and 1 red. The players take turns. At each turn the player puts $X into the kitty, removes a disc from the bag, looks at the colour and replaces it in the bag. If the disk is blue, the game continues with the other player's turn. If the disc is red, the game stops, and the player who picked up the red disc wins the money in the kitty.
Suppose . Let be the number of turns in a game (Y=1 if the red disc is chosen on the first turn). Let Z be the amount of money in the kitty when the game ends.
i. Evaluate . Write down the probability mass function of Y.
ii. Derive the moment generating function of X and the moment generating function of Y. For each give an interval around the origin for which the function is well-defined.
iii. Derive the moment generating function of . Express as a polynomial of t and hence find E(Z) and Var(Z).
iv. Evaluate the probability that the person who starts the game wins the game. Given the choice, would you choose to start or to go second in the game? Give reasons.
I'll do the answer in a separate post, this one is getting too long.