# Thread: Expected Value and Variance

1. ## Expected Value and Variance

Hi, I have a problem on Expected Value and Variance, and having spent hours but still couldn't figure out

One state lottery has 200 prizes of $1 100 prizes of$5
40 prizes of $25 13 prizes of$100
4 prizes of $350 1 prize of$1000
Assuming that 17,000 lottery tickets are issued and sold for $1 1. what is the lottery's expected profit per ticket For this problem, I solved it two ways. First by summing the [products of all the numbers of tickets with their payouts], which is 5400 then (17000-5400)/17000 to get$0.6824
Second way by summing the [product of the payouts of tickets with their probabilities] = $0.6824 Having explained what I did for the first problem, here is the "real" problem: 2. What is the lottery's standard deviation of profit per ticket? Since I found the answer to the expected value indirectly, I'm a blind goose in a hailstorm on number 2... 2. ## Re: Expected Value and Variance Calculate this one.$\displaystyle SD(X) = \sqrt{E(X^2)- [E(X)]^2}$3. ## Re: Expected Value and Variance Calculate the first two moments. If X is the profit on a ticket random variable...$\displaystyle E[X] = \sum x \cdot p(x) = \frac{1}{17,000}\cdot\left[(-999\cdot 1) + (-349\cdot 4) + (-99\cdot 13) + ...\right]\$

Do that, then change to x^2 and find the second moment. The variance is a just a hop, skip, and a jump away.

4. ## Re: Expected Value and Variance

Thanks for the reply. Though I do have 1 more questions: Shouldn't the 1/17000 go inside the bracket or am I missing something.

5. ## Re: Expected Value and Variance

For simplicity of presentation, I used the Distributive Property of Multiplication over Addition.

Let us not forget our algebra.

6. ## Re: Expected Value and Variance

Bah I'm losing my mind. Thank you so much.