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Math Help - Distribution of X

  1. #1
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    Distribution of X

    A game consists of drawing tickets with numbers on them from a box, independently with replacement. In order to play you have to stake $2 each time you draw a ticket. Your net gain is the number on the ticket you draw. Suppose there are 4 tickets in the box with numbers -2, -1, 0, 3 on them. If, for example the ticket shows $3 then you get your stake back, plus an additional $3.
    a) Let X stand for your net gain in one game. What is the distribution of X? Find E(X) and Var(X).
    b) If you play 100 times, what is your chance of winning $25 or more?

    I'm having a hard time knowing how to figure out the distribution, and therefore am having a hard time with the rest of the problem. It would be great if someone could help me!
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  2. #2
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    Quote Originally Posted by bart203 View Post
    A game consists of drawing tickets with numbers on them from a box, independently with replacement. In order to play you have to stake $2 each time you draw a ticket. Your net gain is the number on the ticket you draw. Suppose there are 4 tickets in the box with numbers -2, -1, 0, 3 on them. If, for example the ticket shows $3 then you get your stake back, plus an additional $3.
    a) Let X stand for your net gain in one game. What is the distribution of X? Find E(X) and Var(X).
    b) If you play 100 times, what is your chance of winning $25 or more?

    I'm having a hard time knowing how to figure out the distribution, and therefore am having a hard time with the rest of the problem. It would be great if someone could help me!
    Let X be the random variable number on ticket, that is, net gain.

    It should be plain that Pr(X = -2) = Pr(X = -1) = Pr(X = 0) = Pr(X = 3) = 1/4.

    Now use the usual definitions.
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    Hmm ok so E(x) = 0 and Var(x) = 3.5?

    But then how do you solve part b?
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    Quote Originally Posted by bart203 View Post
    Hmm ok so E(x) = 0 and Var(x) = 3.5?

    But then how do you solve part b?
    Let Y be the random variable net gain after 100 games.

    n = 100 is large, so it's reasonable to say that Y = X_1 + X_2 + .... + X_{100} ~ \text{Normal}\left( \mu = n E(X), \sigma^2 = n Var(X)\right).

    Calculate \Pr(Y \geq 25).
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