Suppose $X_1,X_2,X_3$ are iid with distribution $\mathbb{N(\theta, \sigma^2)}$ and $Y_1 = X_1 + X_2$ and $Y_2 = X_2 + X_3$.
I need to find the joint distribution of $Y_1, Y_2$.
Here is my attempt:
Since the summation of the Gaussian Random variables is also gaussian, so $Y_1,Y_2 \sim N(2\theta,2\sigma^2)$.
If $Y_1, Y_2$ are independent then the joint pdf should be the product of their individual pdfs. But since they both depend on $X_2$, they are not iid. How to find the pdf in such a case?
Thank You!
Best Answer
$(Y_1,Y_2)$ has a two dimensional normal distribution with mean vector $(2\theta,2\theta)$ and variance co-variance matrix $A$ given by the following calulation:
$EY_1Y_2= E(X_1+X_2)(X_2+X_3)=EX_1EX_2+EX_1X_3+EX_2^{2}+EX_2EX_3=\theta^{2}+\theta^{2}+EX_2^{2}+\theta^{2}$ Now $EX_2^{2}=Var (X_2)+(EX_2)^{2}=\sigma ^{2}+\theta^{2}$ Hence, $EY_1Y_2=\sigma^{2}+4\theta^{2}$ Also, $(EY_1)(EY_2)=(2\theta) (2\theta)=4\theta^{2}$ Hence $Cov (Y_1,Y_2)= \sigma^{2}+4\theta^{2}-4\theta^{2}=\sigma^{2}$. Of course, $Var Y_1)=Var(Y_2)=2\sigma^{2}$.
For the joint density function refer to: https://en.wikipedia.org/wiki/Multivariate_normal_distribution