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Iterative expectation of continuous and discrete distributions

  1. Sep 25, 2010 #1
    1. The problem statement, all variables and given/known data
    Suppose X ~ uniform (0,1) and the conditional distribution of Y given X = x is binomial (n, p=x), i.e. P(Y=y|X=x) = nCy x[tex]^{y}[/tex] (1-x)[tex]^{n-y}[/tex] for y = 0, 1,..., n.

    2. Relevant equations
    FInd E(y) and the distribution of Y.

    3. The attempt at a solution
    f(x) = [tex]\frac{1}{b-a}[/tex] = [tex]\frac{1}{1-0}[/tex] =1

    E[Y] = E [E[Y|X=x]
    = [tex]\int[/tex] E[Y|X=x] f(x) dx where the integral is from o to 1
    = [tex]\int[/tex] [[tex]\Sigma[/tex] y f(y|x)] f(x) dx
    = [tex]\int[/tex] [[tex]\Sigma[/tex] y nCy x[tex]^{y}[/tex] (1-x)[tex]^{n-y}[/tex]] f(x) dx

    ...but I do not know anymore what to do next...please help.
  2. jcsd
  3. Sep 25, 2010 #2


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    Homework Helper

    You know the conditional distribution of Y given X. Use that to find E[Y | X]. The answer is a function of X - find its expectation with respect to X to get E[E[Y |X]] = E[Y]
  4. Sep 29, 2010 #3
    Thank you so much for your very good idea. Because of that, I already got the E[Y].

    Can you still help me in finding the distribution of Y?

    I am confused about this one I made:

    P[Y] = [tex]\int^{0}_{1}[/tex] [tex]\left[nCy x^{y} (1-x)^{n-y} dx\right][/tex]

    I understand that is a a beta function if we ignore the constant. But can you help me find the final distribution of Y?
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