Discrete rv question

**Danneedshelp** · October 7th 2009, 08:26 PM

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

**mr fantastic** · October 7th 2009, 09:25 PM

Originally Posted by Danneedshelp

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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