Finding the conditional distribution

In summary, the conversation is about finding the conditional distribution of ##u|Y## based on the given information. The current progress includes the manipulation of the joint distribution ##f(u,y)## and the probability ##p(y)## to get ##f(u|y)##. The remaining step is to cancel out some terms and determine the final distribution.
  • #1
Bazzinga
45
0
Hey guys, I'm trying to find a conditional distribution based on the following information:

##Y|u Poisson(u \lambda)##, where ##u~Gamma( \phi)## and ##Y~NegBinomial(\frac{\lambda \phi}{1+ \lambda \phi}, \phi^{-1})##

I want to find the conditional distribution ##u|Y##

Here's what I've got so far:

##f(u|y)= \frac{f(u, y)}{p(y)} = \frac{p(y|u)}{p(y)} f(u)##
##=\frac{(u \lambda)^{y}e^{-u \lambda}}{y!} \frac{u^{ \alpha - 1}e^{-u/ \beta}}{ \Gamma ( \alpha) \beta^{ \alpha}} \frac{y! \Gamma ( \alpha)}{\Gamma (y+ \alpha)} ( \frac{1+\lambda \beta}{ \lambda \beta})^{y} (1+ \lambda \beta)^{ \alpha}##

where ##\beta = \phi## and ## \alpha = \phi^{-1}##
(I'm using ##\beta## and ##\alpha## right now because that's what it does in the notes)

I'm not sure where to go from here. I can cancel some terms out, but then I'm not sure what distribution I'm supposed to end up with. Could anyone give me a push in the right direction?
 
  • #3
Bazzinga said:
I can cancel some terms out
Then please do so and post what you get.
 

1. What is a conditional distribution?

A conditional distribution is a type of probability distribution that shows the probability of an event occurring given that another event has already occurred. It is used to analyze the relationship between two variables and understand how one variable affects the other.

2. How do you find the conditional distribution?

The conditional distribution can be found by dividing the joint distribution (the probability of both events occurring) by the marginal distribution (the probability of the first event occurring). This can be represented mathematically as P(A|B) = P(A and B) / P(B).

3. What is the purpose of finding the conditional distribution?

The purpose of finding the conditional distribution is to understand the relationship between two variables and how one variable affects the other. It can also help in making predictions and decisions based on the given information.

4. Can a conditional distribution be used to determine causation?

No, a conditional distribution alone cannot determine causation. It only shows the relationship between two variables, but it does not prove that one variable causes the other. Other factors must be considered to establish causation.

5. How is a conditional distribution different from a marginal distribution?

A marginal distribution shows the probability of a single event occurring, while a conditional distribution shows the probability of an event occurring given that another event has already occurred. Marginal distributions are used to analyze one variable, while conditional distributions are used to understand the relationship between two variables.

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