Question

Let Y be the daily sales volume of Chill-

inProbability, an ice cream store, and X be the daily high temperature. On a day with no rain, the relationship between Y and X is Y = 5X + 10€, where є ~ N(0, 1) is a random error, and X and e are independent. On a rainy day, Y = 2X +56. Now suppose the distribution of daily high temperature on days with no rain is Uniform (15, 25) and the distribution of daily high temperature on rainy days is Unifrom (20, 30). Also suppose it rains on 20% of the days. Find E(Y) and Var(Y).

Answer 1

To find the expected value and variance of the daily sales volume for an ice cream store based on temperature and rain conditions, we calculate a weighted average of E(Y) for both scenarios and include the distribution of the stochastic error term for variance.

Step-by-step explanation:

The question asks us to find both the expected value (E(Y)) and variance (Var(Y)) of the daily sales volume for an ice cream store, where the sales depend on the daily high temperature and whether it's a rainy day or not. We have two separate linear relationships for rainy days and days without rain, and different uniform distributions for the temperature X on both types of days. Furthermore, we know that it rains on 20% of the days.

Expected Value Calculation

To calculate E(Y), we'll take a weighted average of the expectations for rainy and non-rainy days, considering the probability of rain.

Variance Calculation

The variance of Y, Var(Y), must also consider the two scenarios and incorporate the distribution of the error term ε.