Let X and Y be independent random variables each having a geometric density with parameter p. Find E[Y | X + Y = z] where z is a nonnegative integer.

Medenovgj

Medenovgj

Answered question

2022-09-25

Let X and Y be independent random variables each having a geometric density with parameter p. Find E[Y | X + Y = z] where z is a nonnegative integer.

Answer & Explanation

Wischarm1q

Wischarm1q

Beginner2022-09-26Added 7 answers

We have that
P ( Y = y | X + Y = z ) = 1 z + 1
which means that given X+Y=z, Y is Uniformly distributed over 0,1,..., z.
Therefore, the conditional expected value of Y is
E ( Y | X + Y = z ) = z 2
Result:
Since Y is conditionally uniformly distributed given X+Y=z, we have that the conditional expected value is equal to z 2 .

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