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Math Help - Expected Value and Variance

  1. #1
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    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...
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  2. #2
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    Re: Expected Value and Variance

    Calculate this one.

    SD(X) = \sqrt{E(X^2)- [E(X)]^2}
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  3. #3
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    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.
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  4. #4
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    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.
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  5. #5
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    Re: Expected Value and Variance

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

    Let us not forget our algebra.
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  6. #6
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    Re: Expected Value and Variance

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