Question

On a given day, your golf score takes values from the range 101 to 110, with probability 0.1, independent of other days. Determined to improve your score, you decide to play on three different days and declare as your score the minimum X of the scores X1, X2, and X3 on the different days. (a) Calculate the PMF of X. (b) By how much has your expected score improved as a result of playing on three days?

Answer 1

The PMF of X is calculated by considering the probabilities of the individual scores and multiplying them together. The expected score is obtained by multiplying each score by its respective probability and summing the products.

Step-by-step explanation:

(a) To calculate the PMF of X, we need to find the probability of each possible value of X (minimum score) by considering the probabilities of the individual scores X1, X2, and X3. Since the scores on each day are independent, the probability of getting a score of X on each day is 0.1.

To find the PMF of X:

1. If X = 101, the probability is P(X = 101) = P(X1 = 101) * P(X2 > 101) * P(X3 > 101) = 0.1 * 0.9 * 0.9 = 0.081.
2. If X = 102, the probability is P(X = 102) = P(X1 = 102) * P(X2 > 102) * P(X3 > 102) = 0.1 * 0.1 * 0.9 = 0.009.
3. Continuing this pattern, we find the probabilities for X = 103, 104, 105, 106, 107, 108, 109, 110 to be 0.001, 0.0001, 0.00001, 0.000001, 0.0000001, 0.00000001, 0.000000001, 0.0000000001, respectively.

(b) To calculate the expected score, we need to multiply each score by its respective probability and sum the products. The expected score is E(X) = 101 * 0.081 + 102 * 0.009 + 103 * 0.001 + 104 * 0.0001 + 105 * 0.00001 + 106 * 0.000001 + 107 * 0.0000001 + 108 * 0.00000001 + 109 * 0.000000001 + 110 * 0.0000000001.