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Poisson Distribution: Definition, Formula, and Examples

Poisson Distribution

Definition: 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 space if these events occur with a known constant mean rate and independently of the time since the last event. For example, it can model the number of phone calls received by a call center in an hour or the number of cars passing through a toll booth in a day.

Formula:P(X=k)=e−λλkk!P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}

Where:

  • kk is the number of occurrences (usually an integer)
  • λ\lambda is the average number of events in the interval
  • ee is Euler's number (e≈2.71828)(e \approx 2.71828)
Examples:

Suppose a call center receives an average of 5 calls per hour. Let's calculate various probabilities:

  1. Probability of receiving exactly 3 calls in an hour:P(X=3)=e−5533!≈0.1404P(X = 3) = \frac{e^{-5}5^3}{3!} \approx 0.1404
  2. Probability of receiving at most 2 calls in an hour:P(X≤2)=P(X=0)+P(X=1)+P(X=2)≈0.1247P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) \approx 0.1247
  3. Probability of receiving more than 6 calls in an hour:P(X>6)=1−P(X≤6)≈0.2851P(X > 6) = 1 - P(X \leq 6) \approx 0.2851

Properties of Poisson Distribution

  • Mean: E(X)=λE(X) = \lambda
  • Variance: Var(X)=λVar(X) = \lambda
  • Standard Deviation: SD(X)=λSD(X) = \sqrt{\lambda}
  • Poisson as an approximation of the Binomial distribution: If the number of trials nn is large, the probability of success pp is small, and the product npnp is moderate, the Binomial distribution can be approximated by the Poisson distribution with λ=np\lambda = np.

How to calculate Poisson distribution in R?

R
# Set parameter
lambda <- 5  # average rate

# Calculate P(X = 3)
prob_equal_3 <- dpois(3, lambda)
print(paste("P(X = 3):", prob_equal_3))

# Calculate P(X <= 2)
prob_less_equal_2 <- ppois(2, lambda)
print(paste("P(X <= 2):", prob_less_equal_2))

# Calculate P(X > 6)
prob_greater_6 <- 1 - ppois(6, lambda)
print(paste("P(X > 6):", prob_greater_6))

# Calculate mean and variance
mean <- lambda
variance <- lambda
print(paste("Mean:", mean))
print(paste("Variance:", variance))

How to calculate Poisson distribution in Python?

Python
import scipy.stats as stats

# Set parameter
lambda_ = 5  # average rate

# Calculate P(X = 3)
prob_equal_3 = stats.poisson.pmf(3, lambda_)
print(f"P(X = 3): {prob_equal_3:.4f}")

# Calculate P(X <= 2)
prob_less_equal_2 = stats.poisson.cdf(2, lambda_)
print(f"P(X <= 2): {prob_less_equal_2:.4f}")

# Calculate P(X > 6)
prob_greater_6 = 1 - stats.poisson.cdf(6, lambda_)
print(f"P(X > 6): {prob_greater_6:.4f}")

# Calculate mean and variance
mean = lambda_
variance = lambda_
print(f"Mean: {mean}")
print(f"Variance: {variance}")

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