Probability Mass Functions

#1
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
#2
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.