Poisson and Gamma Distributions

In summary, the Poisson distribution is discrete and the Gamma distribution is continuous. To compute f(x), for a Poisson distribution, you integrate out x from the continuous part and then look at the discrete part. To compute P(Y=k), for a Gamma distribution, you sum over k for fixed m and look at P(Y=k|X). There is no summation index or summation limits for the cdf of a Gamma distribution.
  • #1
Artusartos
247
0
Let [itex]Y|X[/itex] be a Poisson(X), and X be [itex]Gamma(\alpha, \beta)[/itex]. Find E(X|Y)...

Since Y|X is Poisson(X), we have [itex]f(Y|X)= \frac{m^x e^{-m}}{x!}[/itex]...

Since X is [itex]Gamma(\alpha, \beta)[/itex], we have [itex]f(x)= \frac{x^{\alpha-1} e^{-x/B}}{\Gamma(\alpha) \beta^{\alpha}}[/itex]...

Since [itex]f(Y|X) = \frac{f(x,y)}{f(x)}[/itex] ====> [itex]f(x,y) = \frac{f(Y|X)}{f(x)}[/itex]...

So now we have...

[itex]f(x,y) = \frac{(\frac{m^x e^{-m}}{x!})}{\frac{x^{\alpha-1} e^{-x/B}}{\Gamma(\alpha) \beta^{\alpha}}}[/itex]

So now I have to find f(y), but I'm sort of confused...because the poisson distribution is discrete while the gamma distribution is continuous...so do I need to handle them seperately?

Thanks in advance.
 
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  • #2
Artusartos said:
Let [itex]Y|X[/itex] be a Poisson(X), and X be [itex]Gamma(\alpha, \beta)[/itex]. Find E(X|Y)...

Since Y|X is Poisson(X), we have [itex]f(Y|X)= \frac{m^x e^{-m}}{x!}[/itex]...

Since X is [itex]Gamma(\alpha, \beta)[/itex], we have [itex]f(x)= \frac{x^{\alpha-1} e^{-x/B}}{\Gamma(\alpha) \beta^{\alpha}}[/itex]...

Since [itex]f(Y|X) = \frac{f(x,y)}{f(x)}[/itex] ====> [itex]f(x,y) = \frac{f(Y|X)}{f(x)}[/itex]...

So now we have...

[itex]f(x,y) = \frac{(\frac{m^x e^{-m}}{x!})}{\frac{x^{\alpha-1} e^{-x/B}}{\Gamma(\alpha) \beta^{\alpha}}}[/itex]

So now I have to find f(y), but I'm sort of confused...because the poisson distribution is discrete while the gamma distribution is continuous...so do I need to handle them seperately?

Thanks in advance.

What is P{Y = k} for k = 0,1,2,... ? What is P{x < X < x+dx|Y=k} for k = 0,1,2, ... ?

RGV
 
  • #3
Ray Vickson said:
What is P{Y = k} for k = 0,1,2,... ? What is P{x < X < x+dx|Y=k} for k = 0,1,2, ... ?

RGV

When you say P(Y=k), do you mean P(Y=k|X)? (since we are told that it is poisson when it is conditioned by X...) However, if you mean that it is not P(Y=k|X)...actually that was what I was trying to compute by integrating out x from the continuous part...and then looking at the discrete part (but I was confused, because I wasn't sure if I could handle them seperately).

P(Y=k) = [itex]\sum \frac{m^k e^{-m}}{k!}[/itex]

and...

[itex] P(x < X < x+dx|Y=k) = \frac{P(x < X < x+dx) ∩ P(Y=k)}{P(Y=k)}[/itex]
 
  • #4
Artusartos said:
When you say P(Y=k), do you mean P(Y=k|X)? (since we are told that it is poisson when it is conditioned by X...) However, if you mean that it is not P(Y=k|X)...actually that was what I was trying to compute by integrating out x from the continuous part...and then looking at the discrete part (but I was confused, because I wasn't sure if I could handle them seperately).

P(Y=k) = [itex]\sum \frac{m^k e^{-m}}{k!}[/itex]

and...

[itex] P(x < X < x+dx|Y=k) = \frac{P(x < X < x+dx) ∩ P(Y=k)}{P(Y=k)}[/itex]

Where do you get the formula
[tex] P(Y=k) = \sum \frac{m^k e^{-m}}{k!}?[/tex] What is m? what are you summing over?

The way I read the problem, we have
[tex] P\{ Y = k | X = x \} = \frac{x^k e^{-x}}{k!}.[/tex]
That is *exactly* what the problem says!

Now, of course, you need to find P{Y = k}, an unconditional probability.

RGV
 
  • #5
Ray Vickson said:
Where do you get the formula
[tex] P(Y=k) = \sum \frac{m^k e^{-m}}{k!}?[/tex] What is m? what are you summing over?

The way I read the problem, we have
[tex] P\{ Y = k | X = x \} = \frac{x^k e^{-x}}{k!}.[/tex]
That is *exactly* what the problem says!

Now, of course, you need to find P{Y = k}, an unconditional probability.

RGV

The reason that I wrote summation is because this is the cumulative distribution...not the probability mass function. So for continuous cases, the cdf is the integral of the pdf, and for discrete cases the cdf is the summation of the pmf...and what you're summing over depends on the question. Isn't that correct?
 
  • #6
Artusartos said:
The reason that I wrote summation is because this is the cumulative distribution...not the probability mass function. So for continuous cases, the cdf is the integral of the pdf, and for discrete cases the cdf is the summation of the pmf...and what you're summing over depends on the question. Isn't that correct?

OK, fine: you want to find the cdf of something. However, your expression is still lacking in two ways: (i) what is the summation index?; and (ii) what are the summation limits? Just writing Ʃ does not tell anyone whether you are summing over k for fixed m or over m for fixed k, and what the summation limits are.

Anyway, nothing in this question asks about the cdf (unless there are parts of the question you have not written in your first post).

At this point I am finished with this thread.

RGV
 
  • #7
Ray Vickson said:
OK, fine: you want to find the cdf of something. However, your expression is still lacking in two ways: (i) what is the summation index?; and (ii) what are the summation limits? Just writing Ʃ does not tell anyone whether you are summing over k for fixed m or over m for fixed k, and what the summation limits are.

Anyway, nothing in this question asks about the cdf (unless there are parts of the question you have not written in your first post).

At this point I am finished with this thread.

RGV

Actually, I wasn't trying to find the cdf at first. But when you asked me to find P{Y = k} and P{x < X < x+dx|Y=k}...P(anything) can be found using the cdf.

Anyways, I'll probably ask my professor about this...
 

FAQ: Poisson and Gamma Distributions

What is a Poisson distribution?

A Poisson distribution is a statistical distribution that represents the probability of a certain number of events occurring within a specific time period, given a known average rate of occurrence and assuming that the events are independent of each other. It is often used to model rare events or count data, such as the number of customers arriving at a store in a given hour.

What is a Gamma distribution?

A Gamma distribution is a continuous probability distribution that is often used to model positive, skewed data such as wait times, insurance claims, and income. It is characterized by two parameters, shape and scale, and can take on a variety of shapes depending on the values of these parameters.

How are Poisson and Gamma distributions related?

The Gamma distribution can be seen as an extension of the Poisson distribution. In fact, when the shape parameter of a Gamma distribution is equal to 1, it becomes equivalent to a Poisson distribution. Similarly, the sum of independent Poisson distributions with the same rate parameter is equivalent to a Gamma distribution.

What are some real-world applications of Poisson and Gamma distributions?

Poisson distributions are commonly used to model rare events in fields such as insurance, finance, and epidemiology. Gamma distributions are often used in finance for modeling stock returns and in healthcare for modeling waiting times for medical procedures. Both distributions are also used in quality control to monitor product defects and in traffic engineering to analyze traffic flow.

How are Poisson and Gamma distributions calculated and interpreted?

Poisson distributions can be calculated using the formula P(x) = (e^-λ * λ^x) / x!, where λ is the average rate of occurrence and x is the number of events. This distribution is interpreted as the probability of observing x events in a given time period. Gamma distributions are calculated using the Gamma function and can be interpreted as the probability of obtaining a certain value within a continuous range. They can also be used to calculate confidence intervals and make predictions about future events.

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