Question

Suppose you participate in a chess tournament in which you play n games. Since you are an average player, each game is equally likely to be a win, a loss, or a tie. You collect 2 points for each win, 1 point for each tie, and 0 points for each loss. The outcome of each game is independent of the outcome of every other game. Let X; be the number of points you earn for game i and let Y equal the total number of points earned over the n games. (Hint: Xi is an iid random variable.) Find the range Sxi and PMF Pxi(x) of X.

Answer 1

To find the range Sxi and PMF Pxi(x) of X, we need to determine the possible values of X and their corresponding probabilities. The possible values of X are 0, 1, and 2, and the probabilities can be calculated based on the number of games and the probabilities of winning, losing, or tying each game.

Step-by-step explanation:

To find the range Sxi and PMF Pxi(x) of X, we need to determine the possible values of X and their corresponding probabilities.

In this case, X represents the number of points earned for each game, and we know that a win earns 2 points, a tie earns 1 point, and a loss earns 0 points.

Therefore, the possible values of X are 0, 1, and 2.

• If X = 0, it means all games are losses. The probability of this is (0.43)^n, where n is the number of games.
• If X = 1, it means one game is a win and the rest are losses or ties. The probability of this is nC1 * (0.57)^1 * (0.43)^(n-1).
• If X = 2, it means all games are wins or ties. The probability of this is (0.57)^n.

Thus, the PMF Pxi(x) of X is:

• Pxi(0) = (0.43)^n
• Pxi(1) = nC1 * (0.57)^1 * (0.43)^(n-1)
• Pxi(2) = (0.57)^n