# Discrete rv question

• Oct 7th 2009, 08:26 PM
Danneedshelp
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 \$\displaystyle V (Y ) = V (X )\$ and expanded the the variance formula to include \$\displaystyle 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 \$\displaystyle P(Y= -1)=1/4\$, \$\displaystyle P (Y = 0) = 1/4\$,
\$\displaystyle P (Y = 1) = 1/4\$, and \$\displaystyle P (Y = 2) = 1/4\$. Consider the discrete random variable X having
\$\displaystyle P (X =
-1) = p_{1}\$ , \$\displaystyle P (X = 0) = p_{2}\$ , \$\displaystyle P (X = 1) = p_{3}\$ , and \$\displaystyle P (X = 2) = p_{4}\$ . Also, suppose
p2 = 2p1 . Find p1 , p2 , p3 , and p4 such that \$\displaystyle E(Y ) = E(X )\$ and \$\displaystyle V (Y ) = V (X )\$.

Any help would be much appreciated, thanks
• Oct 7th 2009, 09:25 PM
mr fantastic
Quote:

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 \$\displaystyle V (Y ) = V (X )\$ and expanded the the variance formula to include \$\displaystyle 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 \$\displaystyle P(Y= -1)=1/4\$, \$\displaystyle P (Y = 0) = 1/4\$,
\$\displaystyle P (Y = 1) = 1/4\$, and \$\displaystyle P (Y = 2) = 1/4\$. Consider the discrete random variable X having
\$\displaystyle P (X =
-1) = p_{1}\$ , \$\displaystyle P (X = 0) = p_{2}\$ , \$\displaystyle P (X = 1) = p_{3}\$ , and \$\displaystyle P (X = 2) = p_{4}\$ . Also, suppose
p2 = 2p1 . Find p1 , p2 , p3 , and p4 such that \$\displaystyle E(Y ) = E(X )\$ and \$\displaystyle V (Y ) = V (X )\$.

Any help would be much appreciated, thanks

\$\displaystyle 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.