Vtyjkyuk

Let $X$ and $Y$ be independent random variables such that $Xsim operatornamePoisson(lambda_1 = 5)$ and
$Ysim operatornamePoisson(lambda_2 = 15)$. Let $Z = X + Y$ . Compute $operatornameCorr(X, Z)$.

Answer:

$operatornameVar(X) = 5$

$operatornameVar(Y) = 15$

$operatornameCov(Z) = operatornameCov(X, X+Y) = operatornameCov(X, X) = operatornameVar(X) = 5$

$operatornameVar(Z) = operatornameVar(X + Y) = V(X) + V(Y) = 5 + 15 = 20$

$$operatornameCorr(X, Z) = fracoperatornameCov(X, Z)sqrtoperatornameVar(X) sqrtoperatornameVar(Z) = frac5sqrt5 cdot 20 = 0.5$$

I'm just wondering if this statement is true for all covariance values or just if independent.

$operatornameCov(Z) = operatornameCov(X, X+Y) = operatornameCov(X, X) = operatornameVar(X) = 5$

Couldn't find it on wiki

$$Cov(X,Z) = Cov(X,X+Y)$$
Let $E(X) = mu_X$ and $E(Y) = mu_Y$ then
$$Cov(X,X+Y) = E(X-mu_X)(X-mu_X + Y - mu_Y) $$
which is
$$Cov(X,X+Y) = E((X-mu_X)^2) + E(X-mu_X)(Y-mu_Y) $$
hence
$$Cov(X,X+Y)= Var X + E(XY) - mu_XEY -mu_YEX +mu_Xmu_Y $$
Due to independence of $X$ and $Y$, we have
$$Cov(X,X+Y) = Var X + E(X)E(Y) - mu_Xmu_Y -mu_Ymu_X +mu_Xmu_Y$$
i.e.
$$Cov(X,X+Y) = Var X + mu_Xmu_Y- mu_Xmu_Y -mu_Ymu_X +mu_Xmu_Y = Var X$$

Lemma. $$textcov(X, Y) = 0$$ if $X$ and $Y$ are independent.

Proof. $$textcov(X, Y) = mathopmathbbE[XY] - mathopmathbbE[X]mathopmathbbE[Y] $$

which is 0 as the expected value of a product of random variables is the product of the expected values of those variables.

Original question

Now back to the original question:

$$beginalign
textcov(X, Z) &= textcov(X,X-Y)\
&= textcov(X,X)-textcov(X,Y)\
&= textvar(X,X) - 0\
&= 5
endalign$$