What is the PMF of a sum of two discrete random variables?

[Alupsaiu]

Hi,

I'm working a problem and I'm stuck on one part. Consider, X and Y, two independent discrete random variables who have the same geometric pmf. Show that for all n ≥ 2, the PMF

P(X=k|X+Y=n) is uniform.

Now, this equals: P(X=k)P(Y=n-k)/P(X+Y=n), which follows from the definition of conditional probability. Since the X and Y have the same geometric pmf the numerator is easy to calculate, but I'm stuck on what exactly P(X+Y=n) is. I know it's the joint PMF, but how can I relate it to the problem (i.e. to the fact that X and Y have the same geo PMF, that X and Y are independent etc). Any help is appreciated.

Thanks,
Alex

 

[Alupsaiu]

To be more specific, the problem is to show for any integer n≥2

 

[chiro]

Hey Alupsaiu and welcome to the forums.

What are finding difficult about the P(X + Y = n)?

You are given the realization of X (X = k), and you are given n, so based on that you should be able to get the realization of Y to figure out your probability.

That probability is just the probability that given n, it represents the probability that X = k and Y = n - k, In other words it is the same as saying P(X = k, Y = n - k).

 

[Alupsaiu]

Hey, thanks for the reply. I figured the problem out a while ago, I don't know exactly why I found it confusing, long day I suppose haha. Thanks for the help though!