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Math Help - Discrete rv question

  1. #1
    Senior Member Danneedshelp's Avatar
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    Discrete rv question

    I am stuck on the below problem. I am not sure how to approach. What I have done so far is used the fact that V (Y ) = V (X ) and expanded the the variance formula to include E(Y^{2})-E(Y)^{2}=E(X^{2})-E(X)^{2}, but I have not had much luck with it. I also thought about settin gup a system, but I am not sure how I would do that.

    Q: Suppose Y is a discrete random variable having P(Y= -1)=1/4, P (Y = 0) = 1/4,
    P (Y = 1) = 1/4, and P (Y = 2) = 1/4. Consider the discrete random variable X having
    P (X = <br />
-1) = p_{1} , P (X = 0) = p_{2} , P (X = 1) = p_{3} , and P (X = 2) = p_{4} . Also, suppose
    p2 = 2p1 . Find p1 , p2 , p3 , and p4 such that E(Y ) = E(X ) and V (Y ) = V (X ).

    Any help would be much appreciated, thanks
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  2. #2
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    Quote Originally Posted by Danneedshelp View Post
    I am stuck on the below problem. I am not sure how to approach. What I have done so far is used the fact that V (Y ) = V (X ) and expanded the the variance formula to include E(Y^{2})-E(Y)^{2}=E(X^{2})-E(X)^{2}, but I have not had much luck with it. I also thought about settin gup a system, but I am not sure how I would do that.

    Q: Suppose Y is a discrete random variable having P(Y= -1)=1/4, P (Y = 0) = 1/4,
    P (Y = 1) = 1/4, and P (Y = 2) = 1/4. Consider the discrete random variable X having
    P (X = <br />
-1) = p_{1} , P (X = 0) = p_{2} , P (X = 1) = p_{3} , and P (X = 2) = p_{4} . Also, suppose
    p2 = 2p1 . Find p1 , p2 , p3 , and p4 such that E(Y ) = E(X ) and V (Y ) = V (X ).

    Any help would be much appreciated, thanks
    p_1 + p_2 + p_3 + p_4 = 1 \Rightarrow 3p_1 + p_3 + p_4 = 1 .... (1)

    You need two more equations:

    Use the definition of expected value to calculate E(X) and E(Y) and hence get a second equation using E(X) = E(Y).

    Use the definition of variance to calculate Var(X) and Var(Y) and hence get a third equation using Var(X) = Var(Y).

    If you need more help, please show all your working and state where you're stuck.
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