# What is the mean and variance of this Poisson random variable?

## What is the mean and variance of this Poisson random variable?

The Poisson distribution has a particularly simple mean, E ( X ) = λ , and variance, V ( X ) = λ .

### What is the relationship between Poisson and gamma distribution?

Poisson distribution is used to model the # of events in the future, Exponential distribution is used to predict the wait time until the very first event, and Gamma distribution is used to predict the wait time until the k-th event.

#### What is the variance of a gamma distribution?

The mean of the gamma distribution is αβ and the variance (square of the standard deviation) is αβ2.

**How do you find the mean of a gamma distribution?**

In the Solved Problems section, we calculate the mean and variance for the gamma distribution. In particular, we find out that if X∼Gamma(α,λ), then EX=αλ,Var(X)=αλ2….For any positive real number α:

- Γ(α)=∫∞0xα−1e−xdx;
- ∫∞0xα−1e−λxdx=Γ(α)λα,for λ>0;
- Γ(α+1)=αΓ(α);
- Γ(n)=(n−1)!, for n=1,2,3,⋯;
- Γ(12)=√π.

**How to calculate the mean and variance of a Poisson distribution?**

For a Poisson Distribution, the mean and the variance are equal. It means that E (X) = V (X) V (X) is the variance. A random variable is said to have a Poisson distribution with the parameter λ, where “λ” is considered as an expected value of the Poisson distribution. E (x) = μ = d (eλ (t-1))/dt, at t=1.

## When is a compound Poisson distribution a gamma distribution?

Compound Poisson Gamma distribution. If X has a gamma distribution, of which the exponential distribution is a special case, then the conditional distribution of Y | N is again a gamma distribution.

### When do you use a Poisson random variable?

A Poisson random variable “x” defines the number of successes in the experiment. This distribution occurs when there are events that do not occur as the outcomes of a definite number of outcomes. Poisson distribution is used under certain conditions.

#### What is the variance of a gamma random variable?

The variance of a gamma random variable is: σ 2 = V a r (X) = α θ 2