# Probability Mass Functions

#### creative22

##### New Member
I am having some trouble on understanding some problems on my practice exam.

The prompt is: Let X1 and X2 have (the same) probability mass function
f(x) := (exp[-n]) * n/x! for x=(0,1,2,3....) where n is a positive constant. Assume that
{X1 = a} and {X2 = b} are independent events for any nonnegative integers a and b.

The question: For any nonnegative integer b, show that:
P(X1 + X2 =b) = the sum from x=0 to b is P(X1 + X2 = b intersect X2 = x) =
the sum from x=0 to b is P(X1 =b-x)P(X2 =x).

I am looking for a little guidance or insight on this problem. I'm stuck on how to get started. Thanks.

#### BGM

##### TS Contributor
General principle - Law of total probability.

Suppose $$A$$ be a event, and $$\{B_1, B_2, ..., B_N\}$$ be a sequence of events that forms a partition over the sample space ($$N$$ can be infinite), i.e. $$B_i \cap B_j = \varnothing ~~ \forall i \neq j$$ and $$\bigcup_{i=1}^N B_i = \Omega$$, where $$\Omega$$ is the sample space.

Then note that $$A \subseteq \Omega \Rightarrow A = A \cap \Omega$$

and the following crucial facts:

$$A \cap \left(\bigcup_{i=1}^N B_i\right) = \bigcup_{i=1}^N (A \cap B_i)$$

$$(A \cap B_i) \cap (A \cap B_j) = A \cap (B_i \cap B_j) = A \cap \varnothing = \varnothing ~~ \forall i \neq j$$

So the event $$A$$ is expressed as a union of the mutually exclusive events $$\{A \cap B_1, A \cap B_2, ..., A \cap B_N\}$$

and therefore, $$P(A) = \sum_{i=1}^N P(A \cap B_i)$$

In particular, the question is just a standard convolution formula.