$E(X_1X_2)=\frac{7}{3}$, $E(X_1)=\frac{3}{2}$.Find the joint distribution of $X_1,X_2$.

Question

$X_1$ and $X_2$ are identically distributed (not necessarily independent) random variables taking values in $ \left\{ 1,2 \right\}$. Given, $E(X_1X_2)=\frac{7}{3}$, $E(X_1)=\frac{3}{2}$.Find the joint distribution of $X_1,X_2$.

Clearly , if we assume that $P[X_1=1]=a=1-P[X_1=2]$, then we get $a=\frac{1}{2}$ from $E(X_1)=\frac{3}{2}$ As $X_2$ is identically distributed as $X_1$ , we also get the distribution of $X_2$. But how to get the joint pmf? They are not independent. Help!

Answer 1

Define $$P(X_1,X_2=1,1)=a$$ $$P(X_1,X_2=1,2)=P(X_1,X_2=2,1)=b$$ $$P(X_1,X_2=2,2)=c$$ Then the following system of equations holds: $$a+2b+c=1$$ $$a+b+2(b+c)=\frac32=E(X_1)$$ $$a+2(2b)+4c=\frac73=E(X_1X_2)$$ Solving gives $a,b,c=\frac13,\frac16,\frac13$, which immediately leads to the joint pmf.