# Relation between Gamma and Poisson

• lily_w
In summary, the conversation discusses a classic proof involving integration by parts and induction for the statement Pr(X>t)=Pr(Y ≤r−1), where X follows a gamma distribution and Y follows a Poisson distribution. The conversation also mentions the base case of r=1 and how to show the statement holds for r=k+1. Finally, it suggests writing down the probability density functions for X and Y and showing they are equal when r=1.
lily_w
I'm having trouble doing a classic proof (integration par part and induction on r) for this :

Pr(X>t)=Pr(Y ≤r−1), where X follows a gamma Γ(α = r, β = 1/λ) and Y a Poisson P (λt).

I don't really understand what induction on r really means.

I tried just showing the basic density equation for both (Gamma and Poisson) but i don't know how to make them simplify into the same thing.

First, you want to show that the statement Pr(X>t) = Pr(Y≤r-1) is true when r=1. This is the base case. Next, you assume the statement holds when r=k and show that the statement is true for r=k+1. That's what you need to do for a proof by induction.

When r=1, what are the probability density functions for X and Y? Write down expressions for Pr(X>t) and Pr(Y≤0) and show they are equal.

## What is the relation between Gamma and Poisson?

The Gamma distribution and the Poisson distribution are closely related to each other. The Gamma distribution is a continuous probability distribution that is used to model the waiting time between events. The Poisson distribution, on the other hand, is a discrete probability distribution that is used to model the number of events that occur in a certain time period. The relationship between these two distributions is that the Gamma distribution can be used to approximate the Poisson distribution under certain conditions.

## What are the conditions for approximating the Poisson distribution with the Gamma distribution?

The Poisson distribution can be approximated by the Gamma distribution when the mean of the Poisson distribution is large and the shape parameter of the Gamma distribution is small. In other words, when the number of events is large and the probability of an event occurring is small, the Gamma distribution can be used to approximate the Poisson distribution.

## How do you calculate the mean and variance of a Gamma distribution?

The mean and variance of a Gamma distribution can be calculated using the shape and scale parameters. The mean of a Gamma distribution is equal to the shape parameter divided by the scale parameter. The variance is equal to the shape parameter divided by the square of the scale parameter.

## What is the relationship between the mean and variance of a Poisson distribution?

The mean and variance of a Poisson distribution are equal. This means that the variance of a Poisson distribution is equal to the square of its mean. This relationship is important when using the Gamma distribution to approximate the Poisson distribution, as the mean and variance of the Gamma distribution can be adjusted to match the mean and variance of the Poisson distribution.

## How is the Gamma distribution used in relation to the Poisson distribution?

The Gamma distribution is used to approximate the Poisson distribution when the mean of the Poisson distribution is large and the shape parameter of the Gamma distribution is small. This approximation can be useful in situations where the Poisson distribution is difficult to work with, as the Gamma distribution is a continuous distribution and therefore can be easier to use in calculations.

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