# Difficult Expectation problem

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!

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I cannot read the standard deviation, there is a typo. But if the random variable $X$ is the daily temperature, we seem to be told that the cost is $100X+50$. Wildly expensive! The mean of this is $100E(X)+50$, and $E(X)=90$. The variance is $100^2\text{Var}(X)$, so the standard deviation is $100$ times the standard deviation of the temperature. –  André Nicolas May 1 '12 at 2:52
The standard deviation = 5, and the mean is 90. I think X is just a probability, not sure how to calculate, that is what I am stuck on! –  jay May 1 '12 at 2:54
$x$ is the temperature; you are told that $x$ is a random variable with mean 90 and standard deviation 5; you are being asked for the mean and standard deviation of $100x+50$. Can you do that? –  Gerry Myerson May 1 '12 at 2:59
Not sure, I will see if I can figure it out . –  jay May 1 '12 at 3:03
100*EX + 50 = 100*90 + 50 = $9050 for the average cost, and for the deviation 100*5 =$500 ... sorry for the confusion, I'm just not understand exactly what the problem is asking based on the information given, maybe it's more a semantics issue. Thanks everyone!!! –  jay May 1 '12 at 3:15