Is it possible to generate two random variables $X$ and $Y$ that are both $Poisson(5)$ with $Corr(X,Y)=-1$? Why?
I was thinking about generating $3$ independent Poisson random variables $Z_1,Z_2, and Z_3$ with parameter $5$. Then let $X\sim Z_1+Z_2$ and $Y\sim Z_1+Z_3$ are correlated Poisson random variables with parameter 5. Am I right? Then what should I do? Thank you.
The answer below proves the key fact that if the correlation coefficient is $-1$, then there exist constants $\alpha$ and $\beta$, with $\alpha$ negative, such that $X=\alpha Y+\beta$ with probability $1$. Essentially we are proving the Cauchy-Schwarz Inequality, with particular attention to the equality case.
Let $U=X-\mu_X$ and $V=Y-\mu_Y$. Note that $U$ and $V$ have mean $0$. Also, the variance of $U$ is the same as the variance of $X$, and the same holds for $V$ and $Y$. Consider $E((U+tV)^2)$, where $t$ is a variable. Note that $$E((U+tV)^2)\ge 0. \tag{1}$$ for all $t$.
By the linearity of expectation, we have
$$E((U+tV)^2)=E(U^2)+2tE(UV)+t^2E(V^2).$$ Since this quadratic is always $\ge 0$, its discriminant is $\le 0$. It follows that $4(E(UV))^2-4E(U^2)E)V^2)\le 0$. This implies that $\frac{|E(UV)|}{\sqrt{E(U^2)}\sqrt{E(V^2)}}\le 1$.
But $\frac{E(UV)}{\sqrt{E(U^2)}\sqrt{E(V^2)}}$ is the correlation coefficient of $X$ and $Y$. We have proved the familiar fact that the correlation coefficient is between $-1$ and $1$.
The inequalities we have used are strict unless the quadratic $t^2E(V^2)+2tE(UV)+E(U^2)$ has minimum value of $0$. Thus the only way we can have correlation coefficient $\pm 1$ is if there is a $t_0$ such that $E((U+t_0 V)^2)=0$. Thus the only way we can have correlation coefficient $\pm 1$ is if there is a $t_0$ such that with probability $1$, we have $U+t_0V=0$.
Putting this in terms of $X$ and $Y$, the only way we can have correlation coefficient $\pm 1$ is if for some $t_0$, with probability $1$, we have $$X-\mu_X+t_0(Y-\mu_Y)=0.$$ Equivalently, the correlation coefficient is $\pm 1$ is the exist constants $\alpha$ and $\beta$ such that with probability $1$, we have $$X=\alpha Y+\beta.$$ The case correlation coefficient $-1$ corresponds to negative values of $\alpha$.