Consider a model for the mix of male and female children in families. Let Ki denote the family size (number of children), K_i = 1, c. Let F_i denote the number of female children, F_i = 0, ... , K_i. Suppose the density for the number of female children in a family with K_i children is binomial with constant success probability θ:

We are interested in analyzing the “probability,” θ. Suppose the (conjugate) prior over u is a beta distribution with parameters a and b:

Your sample of 25 observations is given here:

a. Compute the classical maximum likelihood estimate of θ.
b. Form the posterior density for θ given (K_i, F_i), i = 1, ... , 25 conditioned on a and b.
c. Using your sample of data, compute the posterior mean assuming a = b = 1.
d. Using your sample of data, compute the posterior mean assuming a = b = 2.
e. Using your sample of data, compute the posterior mean assuming a = 1 and b = 2.

Transcribed Image Text:

= (K₁) ³ (1 - 0) ³₁-³₂ p(F;| K₁, 0) =