Hi !

I'm doing an exercise, where I'm pretty stuck

So here it is.

Recall that if Y follows a geometric distribution (parameter p), then , for .

Let a sequence of rv independent and identically distributed : Bernoulli (p)

For any integer , we define

And for any integer , we define

1.Prove that is a random variable. I'm lost on this one... But I still have to think a little more ><

2.This question asked to prove that follows a geometric distribution, which was easy (I explained it with words)

3.Show that for ,

So this is mainly where I'm stuck...

It is asked to show a close reasoning. I don't know what it would change...

My thought on this so far : this implies that . But this is intuition (certainty that is not proved yet ).

I've tried an induction on k, but I don't think it's the correct way.

That's all I've got

Thanks for any help (not necessarily solutions ) provided !

EDIT: For question 3. :

So by intuition and "reasoning", it is equivalent to proving that equals the probability that and the sum of the first k-1 equals r-1 (which is a binomial distribution)

But how to prove it clearly ?