1. Posterior Distributions

I have a couple of questions that I am having trouble with. Both require finding a posterior distribution.

a) Consider a random variable X with pdf f_X(x|theta) = (3*theta)*(x^2) *(exp(-theta*(x^3)), 0 < x < infinity. Assume theta has a prior distribution which is gamma with alpha = beta = 4. Find the posterior dist'n for theta.

b) A pattern is assumed to follow a geometric dist'n, p_X(k|theta) =
((1 - theta)^(k-1))*theta, k = 0, 1, 2, ..., n. Assume theta has a prior dist'n that is uniform on [0, 1]. Find the posterior dist'n for theta.

Thanks for any help!

2. This part is vague.... gamma with alpha = beta = 4
That can be written two different ways, the beta above the theta in the exponent of e or below the theta.

The point here is to multiply these two densities, that will give you their joint density. Then you integrate out the theta to give you the marginal of x. Finally divide the joint density by this marginal to give you the density of theta given x.

3. Also, check if these are conjugate priors. (You should look for a list of conjugate priors somewhere.) If they are, then the distribution of the posterior is the same as the distribution of the prior, just with updated parameter values. This means that you don't have to actually integrate, and makes the problem a lot easier.