# Question: Consider the following game Player A flips a fair coin

Consider the following game. Player A flips a fair coin until a head appears. She pays player B 2n dollars, where n is the number of tosses required until a head appears. For example, if a head appears on the first trial, player A pays player B $2. If the game results in 4 tails followed by a head, player A pays player B 25 $32. Therefore, the payoff to player B is a random variable that takes on the values 2n for n = 1, 2, . . . and whose probability distribution is given by (1 2)n for n = 1, 2, . . . , that is, if X de-notes the payoff to player B,

The usual definition of a fair game between two players is for each player to have equal expectation for the amount to be won.

(a) How much should player B pay to player A so that this game will be fair?

(b) What is the variance of X?

(c) What is the probability of player B winning no more than $8 in one play of the game?

The usual definition of a fair game between two players is for each player to have equal expectation for the amount to be won.

(a) How much should player B pay to player A so that this game will be fair?

(b) What is the variance of X?

(c) What is the probability of player B winning no more than $8 in one play of the game?

## Answer to relevant Questions

The demand D for a product in a week is a random variable taking on the values of –1, 0, 1 with probabilities 1/8, 5/8, and C/8, respectively. A demand of – 1 implies that an item is returned. (a) Find C, E(D), and ...A joint random variable (X1, X2) is said to have a bivariate normal distribution if its joint density is given by for –∞ < s < ∞ and –∞ < t < ∞. (a) Show that E(X1) = μX1 and E(X2) = σX2. (b) Show that variance ...Let X be a discrete random variable, with probability Distribution And P(X = x2) = ¾. (a) Determine x1 and x2, such that E(X) = 0 and variance (X) = 10. (b) Sketch the CDF of X. Suppose F is IFR, with μ = 0.5. Find upper and lower bounds on R(t) for (a)t = 1/4 and (b) t = 1. Consider a system consisting of five components, labeled 1, 2, 3, 4, 5. The system is able to function satisfactorily as long as at least one of the following three combinations of components has every component in that ...Post your question