Suppose a family has saved enough for a 10 day vacation (the only one they will be able to take for 10 years)and has a utility function U = V^1/2 (where V is the number of healthy vacation days they experience).Suppose they are not a particularly healthy family and the probability that someone will have a vacation ruining illness (V = 0)is 30%.What is the expected value of V? A)10 B)7 C)3 D)0

Question

Quizplus · Accepted Answer

Using the formula for expected value: E(V) = P(V = 0) * 0 + P(V = 1) * 1 + P(V = 2) * 2 + ... + P(V = 10) * 10

We know that P(V = 0) = 0.3, so we can simplify the formula to:
E(V) = 0.3 * 0 + P(V = 1) * 1 + P(V = 2) * 2 + ... + P(V = 10) * 10

We can assume that the family is equally likely to have any number of healthy vacation days between 1 and 10, so P(V = n) = 0.7 * (1/10) = 0.07 for all n from 1 to 10.

Substituting into the formula, we get:
E(V) = 0.3 * 0 + 0.07 * 1 + 0.07 * 2 + ... + 0.07 * 10
E(V) = 0.7 * (1 + 2 + ... + 10)
E(V) = 0.7 * 55
E(V) = 38.5

Therefore, the expected value of V is 38.5. However, since V represents the number of healthy vacation days, it needs to be an integer value. The closest integer to 38.5 is 38, which corresponds to 7 healthy vacation days (since the family is going on a 10 day vacation). Therefore, the best choice is B) 7.

Suppose a Family Has Saved Enough for a 10 Day