# Expected Value and Variance

• September 15th 2011, 08:13 PM
koudai8
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 Quote: 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:

Quote:

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...
• September 15th 2011, 08:18 PM
pickslides
Re: Expected Value and Variance
Calculate this one.

$SD(X) = \sqrt{E(X^2)- [E(X)]^2}$
• September 15th 2011, 08:19 PM
TKHunny
Re: Expected Value and Variance
Calculate the first two moments.

If X is the profit on a ticket random variable...

$E[X] = \sum x \cdot p(x) = \frac{1}{17,000}\cdot\left[(-999\cdot 1) + (-349\cdot 4) + (-99\cdot 13) + ...\right]$

Do that, then change to x^2 and find the second moment. The variance is a just a hop, skip, and a jump away.
• September 15th 2011, 08:40 PM
koudai8
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.
• September 15th 2011, 08:50 PM
TKHunny
Re: Expected Value and Variance
For simplicity of presentation, I used the Distributive Property of Multiplication over Addition.

Let us not forget our algebra. (Wink)
• September 15th 2011, 09:04 PM
koudai8
Re: Expected Value and Variance
Bah I'm losing my mind. Thank you so much.