Probability Mass Functions of Binomial Variables

[Kalinka35]

Homework Statement

Let X and Y be independent binomial random variables with parameters n and p.
Find the PMF of X+Y.
Find the conditional PMF of X given that X+Y=m.

Homework Equations

The PMF of X is P(X=k)=(n C k)p^k(1-p)^n-k
The PMF for Y would be the same.

The Attempt at a Solution

I am really not sure how to go about solving this problem though I have been told that the first part can be done with no computation or calculations at all. Not sure at all on the second part.

 

[jbunniii]

Do you know anything in general about the PMF of the sum of two independent discrete random variables? Does "convolution" ring a bell?

If not, don't worry about it. Think about what a binomial distribution with parameters n and p means. You can think of it as the number of "heads" resulting from flipping a coin n times, where the probability of "head" is p. If you do that experiment TWICE, independently, and the results are X and Y, respectively, then X + Y is equivalent to simply flipping the coin 2n times, right? What's the PMF for that experiment?

 

[Kalinka35]

Hmm, not that I know of.
I haven't heard the term convolution before.
Could you provide me with a definition?

 

[jbunniii]

Hmm, not that I know of.
I haven't heard the term convolution before.
Could you provide me with a definition?

You need it to answer the question in general (for arbitrary PMFs), but don't worry about it in this case - for this particular example you can solve it another way (see my edited post above).

 

[Kalinka35]

Ah yes, I understand it now.
Thanks very much for your clear explanation.