HW7

87993 assignment
1. Given that X is Binomial with (n,p) and Z is binomial with (n+m,p)
Conditional prob distribution of X given Z=z is given by
X=x means x = no of successes in n trials of X with probability for each trial = p
P(X/Z) = P(XZ)/P(Z) by conditional property distribution definition
Numerator = P(XZ) = P(X=x and Z=z) = P(X=x, Y = z-x)
=
We can see that except binomial coefficients all others would cancel out thus giving
P = P(X/Z) =
This is nothing but pdf of a hypergeometric distribution
With N = m+n here
Thus proved
2.
Marginal density function of x or y is got by integrating over the full domain the jt density function by other variable
Hence we have
a) PDF of x is
Similarly
Pdf of y is...