Assume that X is uniformly distributed on (0, 1) and that the conditional distribution of Y given X=x is a binomial distribution with parameters (n,x). Then we say that Y has a binomial distribution with fixed size n and random probability parameter.I have to find the marginal distribution of Y. So I have to use that: P ( Y = y ) = ∑ x = y ∞ P ( Y = y | X = x ) P ( X = x )But when we have that X is uniformly distributed I think the sum becomes a integral so we get: P ( Y = y ) = ∫ 0 1 ( n y ) x y ( 1 − x ) n − y d xBut what to do next? I think maybe take the binomial coefficient out from the integral, but how then using this to find the marginal distribution of Y? Hope anyone can help me? Do I also have to use that Y has a binomial distribution with fixed size n and random probability parameter?

Question

Assume that X is uniformly distributed on (0, 1) and that the conditional distribution of Y given X=x is a binomial distribution with parameters (n,x). Then we say that Y has a binomial distribution with fixed size n and random probability parameter.I have to find the marginal distribution of Y. So I have to use that:  P  (  Y  =  y  )  =      ∑          x      =      y              ∞        P  (  Y  =  y      |    X  =  x  )  P  (  X  =  x  )But when we have that X is uniformly distributed I think the sum becomes a integral so we get:  P  (  Y  =  y  )  =      ∫    0    1                      (                    n      y                      )                  x    y    (  1  −  x      )          n      −      y        d  xBut what to do next? I think maybe take the binomial coefficient out from the integral, but how then using this to find the marginal distribution of Y? Hope anyone can help me? Do I also have to use that Y has a binomial distribution with fixed size n and random probability parameter?

Jaida Sanders · Accepted Answer

Your integral is correct. One can show that      ∫    0    1        x    y    (  1  −  x      )          n      −      y          d  x  =            y      !      (      n      −      y      )      !              (      n      +      1      )      !        ,see the Beta function or order statistics of i.i.d. uniform random variables. Multiplying by                     (                    n      y                      )              =            n      !              y      !      (      n      −      y      )      !       and canceling gives a simple expression for P(Y=y).

Assume that X is uniformly distributed on (0, 1) and that the conditional distribution of Y given X=

Answered question

Answer & Explanation

New Questions in Pre-Algebra