Here is a difficult expectation problem I've been wrestling with, have not even come close to a solution
Suppose the average daytime temp. in degrees F on any one day during the month of August in a certain city is a random variable whose distribution is normal with mean u=90 and standard deviation o=5. Furthermore, suppose that on a day when the average daytime temp. is x degrees the cost in dollars of operating the air conditioning system for a certain company in this city is given by the function g(x) = 100x + 50. In the month of August, what is the expected daily cost of operating the air conditioning system for this company? What is the standard deviation?
Attempt: First, we know August has 31 days. Second, average daytime temp I get from using the formula for normal distribution and get 1. (This part I don't think is correct). The function g(x) depends on this first part, so I can not get that. However, I know that when I find g(x), by dividing by total number of days in August I can get the mean and then solve for standard deviation. Please help!