Question

3. Let X and 6 be two random variables. Let X given 0 have a Bernoulli distribution with parameter θ, this is, X | θ ~ Bernoulli (9), and let θ have a beta distribution with parameters a and b, this is 9 Beta(a, b), where a and b are known positive constants (a) Find the joint distribution of (X,6 (b) Find the marginal distribution of X.

Answer 1

$(a)~The~joint~distribution~of~(X,\theta)~is\\ f(x,\theta)=P(X=x|\theta)f(\theta)=\theta^x(1-\theta)^{1-x}\times \frac{1}{B(a,b)}\theta^{a-1}(1-\theta)^{b-1}\\ = \frac{1}{B(a,b)}\theta^{x&plus;a-1}(1-\theta)^{b-x},~x=0,1,~0<\theta<1\\ =0~otherwise\\ (b)~The~marginal~pmf~of~X~is\\ P(X=x)=\frac{1}{B(a,b)}\int_0^1\theta^{x&plus;a-1}(1-\theta)^{b-x}d\theta\\ =\frac{1}{B(a,b)}\times B(x&plus;a,b-x&plus;1)\\ =\frac{\Gamma(a&plus;b)}{\Gamma(a)\Gamma(b)}\times \frac{\Gamma(x&plus;a)\Gamma(b-x&plus;1)}{\Gamma(a&plus;b&plus;1)}\\ =\frac{b}{a&plus;b}~when~x=0\\ =\frac{a}{a&plus;b}~when~x=1\\ =0~otherwise\\ Hence~X\sim Bernoulli(a/(a&plus;b))$