Poisson Distribution (Discrete)

Last modified: July 28, 2024

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Poisson Distribution (Discrete)

A discrete random variable X follows a Poisson distribution if the events occur independently and at a constant average rate. The Poisson distribution is denoted as $X \sim Poisson (λ)$ , where $λ$ is the average rate (or mean) of events occurring in a given interval.

Probability Mass Function (PMF)

The PMF of a Poisson distribution is given by:

$P (X = k) = \frac{e^{- λ} λ^{k}}{k!}$

where $k \in 0, 1, 2, \dots$ .

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Cumulative Distribution Function (CDF)

The CDF of a Poisson distribution is given by:

$F (k) = P (X \leq k) = \sum_{i = 0}^{k} \frac{e^{- λ} λ^{i}}{i!}$

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Expected Value and Variance

The expected value (mean) and variance of a Poisson distribution are both equal to the average rate parameter $λ$ :

$E [X] = Var (X) = λ$

Moment Generating Functions and Moments

The moment generating function (MGF) of a Poisson distribution is:

$M_{X} (t) = E [e^{t X}] = e^{λ (e^{t} - 1)}$

To find the n-th moment, we take the n-th derivative of the MGF with respect to t and then evaluate it at t=0:

$E [X^{n}] = \frac{d^{n} M_{X} (t)}{d t^{n}} |_{t = 0}$

First Moment (Mean):

$E [X] = λ$

Second Moment (Variance + Mean^2):

$E [X^{2}] = λ^{2} + λ$

Example: Hospital Emergency Room Visits

A hospital's emergency room experiences an average of 5 visits per hour due to accidents. We can model this using a Poisson distribution.

Given:

Average rate of success (visits per hour) $λ = 5$

I. What is the probability exactly 3 accidents occur in an hour?

For exactly 3 accidents:

$P (X = 3) = \frac{e^{- 5} \cdot 5^{3}}{3!} = 0.1404$

II. What is the probability of more than 7 accidents in an hour?

For more than 7 accidents:

$P (X > 7) = 1 - \sum_{k = 0}^{7} \frac{e^{- 5} \cdot 5^{k}}{k!} = 0.1339$

III. What is the probability of at most 2 accidents in an hour?

For at most 2 accidents:

$P (X \leq 2) = \sum_{k = 0}^{2} \frac{e^{- 5} \cdot 5^{k}}{k!} = 0.1247$

Poisson Approximation to the Binomial Distribution

If the number of trials in a binomial distribution is large (n) and the probability of success (p) is small, then the binomial distribution can be approximated by a Poisson distribution with parameter $λ = n p$ :

$Binomial (n, p) \approx Poisson (n p)$