Question

In the section on combining Poisson variables, my book states:
P(X+Y) = P(X) + P(Y)
I'm confused by this, because if X and Y are independent, then doesn't the probability of X AND Y happening equal to the product of their respective probabilities, not the sum?
Perhaps X and Y is different from X+Y? If so, then how?
Thanks!

Answer 1

I'm a bit confused by the notation. Usually, for probabilities from random variables, you see notations like \(P(X=x)\), or like \(P(X \le x)\) for cumulative properties.

As for independence, the rule I know is:
\(P(X=x, Y=y)=P(X=x)\cdot P(Y=y)\).

The notation \(P(X=x, Y=y)\) is the same as \(P(X=x \cap Y=y)\).

The random variables X, Y and "X+Y" are different random variables though. In fact: for independent Poisson random variables, you have

If \(X_1\sim\text{POI}(\theta_1)\) and \(X_2\sim\text{POI}(\theta_2)\), then \(X_1+X_2\sim\text{POI}(\theta_1+\theta_2)\)
which can easily shown by moment generating functions.

Answer 2

So, maybe to make it more "concrete" If you had
\(X\sim\text{POI}(4)\) and \(Y\sim\text{POI}(7)\) , then \(X+Y\sim\text{POI}(11)\) , so what changes is the parameter.