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Let X and Y be independent and geometrically distributed with the same parameter p. Compute $E(X|X+Y=k)$ for all k=2,3,...

I tried to calculate $$\sum_{i=z}^P P(X)P(Y_z-x)/P(Z)$$ and then want to calculate $P(X+Y=k)$ but i am getting stuck in the first part. Can some one please give me a hint?

Raveesh
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Strange ansatz… what's P(X) if X is a random variable? That makes no sense…

It's an easy calculation for an expectation:

$$E[X|X+Y=k] = \sum_{j=0}^\infty jP(X=j|X+Y=k) = \sum_{j=0}^\infty j\frac{P(X=j,X+Y=k)}{P(X+Y=k)}$$

Now $P(X=j,X+Y=k) = P(X=j,Y=k-j) = P(X=j)P(Y=k-j)$

The same you can do for $P(X+Y=k)$

Gono
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  • Thanks Gono!! I am substituting the geometric distribution $ P(X=j)= p(1-p)^j$ and the same for $ P(Y=k-j)= p(1-p)^k-j$, what should i substitute for the denominator, my objective is to come with an answer for the expectation in terms of p – Raveesh Nov 10 '16 at 12:02
  • Obviously $P(X+Y=k) = \sum_{j=0}^k P(X+Y=k,X=j) = \sum_{j=0}^k P(Y=k-j,X=j)$ and then use geometric sum formular… – Gono Nov 10 '16 at 12:08
  • Sorry for the trouble, did that, got $p^2(1-p)^k(k+(k+1)+....) which seems suspiciously incorrect, Can you please check!! – Raveesh Nov 10 '16 at 12:49