Is there a difference between posterior denisty and posterior distribution? Or are they 2 names for the same thing?
ok so does that mean that if iv got a posterior distribution of $\displaystyle x^{101}$$\displaystyle e^{-29x/2}$, thats fine is it? so if i need it to be a pdf do i need to change it? because if i integrate it over 0....infinity it doesnt sum to 1?
Is this where i would need a multiplier "k" in the posterior density?
Suppose that a random sample of size n = 100 from an exponential distribution with parameter is about to be collected. It is thought that a Ga(2, 2) prior distribution for is appropriate but, since it is believed that values of exceeding 6 are impossible, the prior distribution is truncated above 6.
Calculate the prior probability that exceeds 6 assuming the truncated prior distribution. What is the probability when the prior distribution is not truncated?
Suppose the data yield ¯x = 1/8. Calculate the posterior probability that
exceeds 6 assuming the truncated prior distribution. What is the probability
when the prior distribution is not truncated?
so i integrated my prior function over the range 0....6 and set it equal to 1 to find my constant "k" to make it a valid distribution.
i found "k" to be 4/(1-13exp{-12})
then from here i started getting confused as to how to calculate the prior probabilities. I guess thats when i started getting mixed up between densities and distributions?
Would you be able to walk me through what you did, I have attained answers but really not sure whether iv used the correct integrals or not because of my confusion! :-s
The probability distribution is the function that maps sets to probabilities. The density is the function that you integrate over. So, for example a distribution would be a function $\displaystyle \phi$ such that $\displaystyle \phi(A) = P(X \in A)$ whereas the density is the function $\displaystyle f$ such that $\displaystyle \int_A f(x) \ dx = P(X \in A)$.
If the question asks to find the posterior distribution, reporting the density should be enough to satisfy whoever is asking the question.