4.6 Poisson Distribution

Poisson Distribution

Poisson distribution probability calculations

The Poisson distribution models the number of events occurring in a fixed interval of time, distance, area, or volume. It's built around a single parameter, $\lambda$ (lambda), which represents the average number of events in that interval.

The probability mass function (PMF) gives you the probability of observing exactly $k$ events:

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

• $X$ is the random variable representing the number of events in the interval
• $k$ is the specific number of events you're calculating the probability for (must be a non-negative integer: 0, 1, 2, …)
• $e$ is the mathematical constant, approximately 2.71828

To calculate a Poisson probability:

1. Identify $\lambda$, the average number of events in the interval
2. Decide on $k$, the number of events you want the probability for
3. Plug $\lambda$ and $k$ into the PMF formula and simplify

Example: A call center receives an average of 4 calls per hour, so $\lambda = 4$. What's the probability of receiving exactly 6 calls in a given hour?

$P(X = 6) = \frac{e^{-4} \cdot 4^6}{6!} = \frac{0.01832 \cdot 4096}{720} \approx 0.1042$

There's roughly a 10.4% chance of getting exactly 6 calls in that hour.

Other situations where the Poisson distribution applies:

• Number of defects per 1000 meters of cable (fixed interval of distance)
• Number of typos per page in a manuscript (fixed interval of area)

The key idea is that you're counting how many times something happens within a defined window.

Poisson approximation of binomial distribution

The Poisson distribution can serve as a handy approximation for the binomial distribution when the following conditions are met:

• The number of trials $n$ is large (typically $n > 20$)
• The probability of success $p$ is small (typically $p < 0.05$)
• You set $\lambda = n \times p$

When these hold, a $\text{Binomial}(n, p)$ distribution is well-approximated by a $\text{Poisson}(\lambda)$ distribution. This is sometimes called the law of rare events: you have many opportunities for something to happen, but on any single trial, it's unlikely.

Why bother with the approximation? Binomial calculations with large $n$ can get unwieldy because of the $\binom{n}{k}$ term. The Poisson formula is simpler to compute since it doesn't involve that combination.

Example: A factory produces 500 items per day, and each item has a 0.002 probability of being defective. Here $n = 500$ and $p = 0.002$, so $\lambda = 500 \times 0.002 = 1$. Instead of using the binomial formula with $n = 500$, you can use a Poisson distribution with $\lambda = 1$ to estimate the probability of, say, 0 or 1 defective items.

Interpreting Poisson distribution parameters

A distinctive feature of the Poisson distribution is that its mean and variance are both equal to $\lambda$:

• Mean (expected value): $E(X) = \lambda$
• Variance: $Var(X) = \lambda$
• Standard deviation: $SD(X) = \sqrt{\lambda}$

The mean tells you how many events to expect on average per interval. The standard deviation tells you how much the actual count typically varies from that average. Because both the mean and variance equal $\lambda$, a larger average also means more spread in the data.

Example: If an average of 10 customers arrive at a store per hour ($\lambda = 10$):

• You'd expect about 10 customers in any given hour
• The standard deviation is $\sqrt{10} \approx 3.16$, so it's common for the actual count to be roughly 7 to 13 in a typical hour

If you ever see count data where the mean and variance are close to equal, that's a good signal that a Poisson model might be appropriate.

Advanced Concepts in Poisson Distribution

• Independent events: The Poisson distribution assumes events occur independently of each other. One event happening doesn't make the next more or less likely.
• Intensity function: In more complex applications, the rate parameter can vary over time or space rather than staying constant. This leads to what's called a non-homogeneous Poisson process.
• Point process: The Poisson distribution is often used to model point processes, where events occur randomly across a continuous space or time interval (e.g., locations of trees in a forest or arrival times at a hospital).