dpois(x = 13, lambda = 15)[1] 0.09560681Poisson distribution
Poisson distribution
A Poisson process calculates the number of times an event occurs in a period of time, or in a particular area, or over some distance, or within any other kind of measurement, and the process has particular characteristics:
The experiment counts the number of occurrences of an event over some other measurement,
The mean is the same for each interval,
The count of events in each interval is independent of the other intervals, and
The intervals don’t overlap.
The probability of the event occurring is proportional to the period of time.
Probability formula for Poisson
\[ P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!} \]
P(X=k) is the probability that the random variable X takes the value k (i.e., the probability of having exactly k events occur).
λ (lambda) is the average rate of events per time period or area. It is also known as the rate parameter of the Poisson distribution.
e is the base of the natural logarithm, approximately equal to 2.71828.
k is the number of events we are interested in (a non-negative integer value).
Example
Poisson distribution is useful for finding the probability that a specific number of events will occur in a given period of time.
The mean number of cars passing through the intersection in any particular hour is λ = 15, and if we want to know the probability that x = 13 cars will pass through it in the next hour.
\[ P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!} \]
\[ P(X =13) = \frac{2.71828^{-15} \cdot 15^{13} } {13!} \]
So there's an approximately 9.56 % chance that 13 cars will pass through the intersection in the next hour.
The Poisson formula can also be used to calculate cumulative probabilities. For instance, if we want to know the probability that at most 7 cars pass through the intersection in an hour
\[ P(X <= 7) = P(X=0)+P(X=1)+(P=2)+(P=3)+(P=4)+(P=5)+(P=6)+(P=7) \]
\[ P(X <=7) = \frac{2.71828^{-15} \cdot 15^{0} } {0!} + \frac{2.71828^{-15} \cdot 15^{1} } {1!} +\frac{2.71828^{-15} \cdot 15^{2} } {2!} +\frac{2.71828^{-15} \cdot 15^{3} } {3!} +\frac{2.71828^{-15} \cdot 15^{4} } {4!} +\frac{2.71828^{-15} \cdot 15^{5} } {5!} +\frac{2.71828^{-15} \cdot 15^{6} } {6!} +\frac{2.71828^{-15} \cdot 15^{7} } {7!} \]
ppois(q = 7, lambda = 15, lower.tail = TRUE)[1] 0.01800219So there's an approximately 1.8 % chance that 7 or fewer cars pass through the intersection in an hour.