2026-06-08

Poisson Distribution

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time (or volume or distance or bites of cheese). We can use the Poisson distribution to determine the likelihood of improbable events.

Condition for Poisson Distribution

  • The events occur randomly and independently

Poisson Formula

\[\color{#00213D}{P(X=x) = \frac{\lambda^x e^{-\lambda}}{x!}} \]

X = Our random variable
\(x\) = The value we want to know the probability for
e = Euler’s number ≈ 2.71828
\(\lambda\) = The mean/variance probability of the poisson

Because we are dividing by the factorial of \(x\), which grows very quickly, as we move farther away from \(\lambda\) the probability moves towards zero but never fully reaches zero.This means that the probability of an unlikely event is low but not impossible.

Specific Example: Lightning Strikes

Lightning strikes a particular location 8.7 times an hour. We want to know the distribution of probability for lightning strikes at this location. With this information we can look at probabilities like:

  • The likelyhood that lightning will strike 20 in 1 hour
  • The likelyhood that lightning will strike 2 in 1 hour
  • The likelyhood that lightning will strike 200 in 10 hour
  • The likelyhood that lightning will strike 2 in 10 hour 

prob = c()
lambda = c()
hours = c() 
Number_of_Occurances = c()

i = 8.7
  for (j in 1:10) {
    for (k in 0:125) {
      lambda =c(lambda, i)
      hours =c(hours, j)
      Number_of_Occurances = c(Number_of_Occurances, k)
      
      pb = (((j*i)^k) * (exp(-(j*i)))) / (factorial(k))
      prob = c(prob, pb)
    }
  }

poisson = tibble(
  Hours = hours,
  Lambda = lambda,
  x = Number_of_Occurances,
  Probability = prob
)

ggplot(
  data = poisson, 
  aes(x = x, y = Probability)) + geom_point(aes(color=Hours)
)

General Example

In this example we want to look for the general trends of what a Poisson distribution looks like. This time rather than \(\lambda\) being fixed we are going to show what it looks like when we measure for many different \(\lambda\). As we did in the lightning example we also adjust time and \(x\)(the number of instances we want the probability for).

The graph makes a few things more clear:

  • For short time periods and small values of \(x\) the distribution is more right skewed.
  • As time and \(x\) increases the graph becomes less right skewed but the overall max probability decreases.

This is due to that factorial in the denominator of the function

3D Plot of General Example

It may be easier to understand this graph if we look at it in 3D perspective. \(\lambda\) is now represented by color and change in time, \(x\), and probability are each one of the three axis.

Finding the Probability that lightning Strikes Three Times or Less

In order to accomplish this we need to find the sum of the probabilities for all \(x\) such that \(x\) is three or less.

\[\color{#00213D}{P(X\leq 3) = \Sigma^{3}_{x=1} \frac{\lambda^x e^{-\lambda}}{x!}} \]

Since in our example lightning strikes 8.7 times per hour, \(\lambda\) = 8.7.

\[\color{#00213D}{P(X\leq 3) = \frac{8.7^0 e^{-8.7}}{0!} + \frac{8.7^1 e^{-8.7}}{1!} + \frac{8.7^2 e^{-8.7}}{2!} + \frac{8.7^3 e^{-8.7}}{3!}}\]

Thank You