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 ?