# Properties of Poisson Distribution

• catsonmars
The resulting summation should be quite familiar, and you can use that to finish the problem.In summary, the Poisson distribution can be used to calculate the expectation value <n> by taking the summation of (λ^n/n!)*e^-λ, which can be simplified using a combination of factoring and recognizing familiar summation patterns.

## Homework Statement

Use the Poisson distribution W=(λ^n/n!)*e^-λ to calculate <n>

<n>=ƩW*n

## The Attempt at a Solution

Since W = (λ^n/n!)*e^-λ I wind up with <n>=[(λ^n/n!)*e^-λ]*n
But I really don't know where to go from here. Should I do a Taylor Series. I've tried crossing out the top n and ended up with

[(λ^n/(n-1)!)*e^-λ] but this doesn't seem to help. If anyone can point me in the right direction or a general problem solving strategy that would be great. I'd like to demonstrate more work but I don't know what to do.

Evaluate the generating function
$$Z(x)=\sum_{n=0}^{\infty} W(n) \exp(-n x).$$
Then you can get expectation values by taking derivatives of this function wrt. $x$ :-).

catsonmars said:

## Homework Statement

Use the Poisson distribution W=(λ^n/n!)*e^-λ to calculate <n>

<n>=ƩW*n

## The Attempt at a Solution

Since W = (λ^n/n!)*e^-λ I wind up with <n>=[(λ^n/n!)*e^-λ]*n
But I really don't know where to go from here. Should I do a Taylor Series. I've tried crossing out the top n and ended up with

[(λ^n/(n-1)!)*e^-λ] but this doesn't seem to help. If anyone can point me in the right direction or a general problem solving strategy that would be great. I'd like to demonstrate more work but I don't know what to do.
You need the summation.
$$\langle n \rangle = \sum_{n=0}^\infty \frac{\lambda^n}{n!} e^{-\lambda} n = \sum_{n=1}^\infty \frac{\lambda^n}{(n-1)!} e^{-\lambda}$$ Note that the lower limit changed from n=0 to n=1, since the n=0 term is 0. You just need to recognize that ##e^{-\lambda}## is a constant, so you can factor it out of the summation. Similarly, try pulling one factor of ##\lambda## out so that the exponent of ##\lambda## inside the summation matches up with the factorial.

## 1. What is a Poisson distribution?

A Poisson distribution is a statistical distribution that calculates the probability of a certain number of events occurring in a fixed time interval, given the average rate of events and assuming that the events occur independently and at a constant rate.

## 2. What are the properties of a Poisson distribution?

The properties of a Poisson distribution include that it is discrete, meaning it only considers whole numbers as outcomes, and that its mean and variance are equal. It also follows the law of rare events, where the probability of multiple events occurring at the same time decreases as the average rate of events increases.

## 3. What are some real-world applications of a Poisson distribution?

Poisson distributions are commonly used to model the number of occurrences of rare events, such as accidents, failures, or defects. They are also used in insurance risk analysis, queueing theory, and in the study of radioactive decay.

## 4. How is a Poisson distribution different from a normal distribution?

A normal distribution is a continuous distribution that can take on any value, while a Poisson distribution is discrete and only takes on whole numbers. Additionally, a normal distribution has a symmetric bell-shaped curve, while a Poisson distribution has a skewed right shape.

## 5. How is a Poisson distribution related to the binomial distribution?

A Poisson distribution can be thought of as a special case of the binomial distribution, where the probability of success is very small and the number of trials is very large. In this case, the binomial distribution approximates a Poisson distribution. Additionally, the sum of a large number of independent binomial random variables can also be approximated by a Poisson distribution.