This calculator helps you compute the probabilities of a Poisson distribution given the average rate (λ) and the number of occurrences (k). You can find the probability of achieving exactly k events in a fixed interval of time or space. The distribution chart shows the probability mass function (PMF) of the Poisson distribution.
Examples for λ≈5: Emails per hour: 5.2, Bus arrivals per hour: 3.0
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.
Where:
Suppose a call center receives an average of 5 calls per hour. Let's calculate various probabilities:
lambda <- 5 # average rate
# P(X = 3)
prob_equal_3 <- dpois(3, lambda)
print(paste("P(X = 3):", prob_equal_3))
# P(X <= 2)
prob_less_equal_2 <- ppois(2, lambda)
print(paste("P(X <= 2):", prob_less_equal_2))
# P(X > 6)
prob_greater_6 <- 1 - ppois(6, lambda)
print(paste("P(X > 6):", prob_greater_6))
# mean and variance
mean <- lambda
variance <- lambda
print(paste("Mean:", mean))
print(paste("Variance:", variance))import scipy.stats as stats
lambda_ = 5 # average rate
# P(X = 3)
prob_equal_3 = stats.poisson.pmf(3, lambda_)
print(f"P(X = 3): {prob_equal_3:.4f}")
# P(X <= 2)
prob_less_equal_2 = stats.poisson.cdf(2, lambda_)
print(f"P(X <= 2): {prob_less_equal_2:.4f}")
# P(X > 6)
prob_greater_6 = 1 - stats.poisson.cdf(6, lambda_)
print(f"P(X > 6): {prob_greater_6:.4f}")
# mean and variance
mean = lambda_
variance = lambda_
print(f"Mean: {mean}")
print(f"Variance: {variance}")This calculator helps you compute the probabilities of a Poisson distribution given the average rate (λ) and the number of occurrences (k). You can find the probability of achieving exactly k events in a fixed interval of time or space. The distribution chart shows the probability mass function (PMF) of the Poisson distribution.
Examples for λ≈5: Emails per hour: 5.2, Bus arrivals per hour: 3.0
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.
Where:
Suppose a call center receives an average of 5 calls per hour. Let's calculate various probabilities:
lambda <- 5 # average rate
# P(X = 3)
prob_equal_3 <- dpois(3, lambda)
print(paste("P(X = 3):", prob_equal_3))
# P(X <= 2)
prob_less_equal_2 <- ppois(2, lambda)
print(paste("P(X <= 2):", prob_less_equal_2))
# P(X > 6)
prob_greater_6 <- 1 - ppois(6, lambda)
print(paste("P(X > 6):", prob_greater_6))
# mean and variance
mean <- lambda
variance <- lambda
print(paste("Mean:", mean))
print(paste("Variance:", variance))import scipy.stats as stats
lambda_ = 5 # average rate
# P(X = 3)
prob_equal_3 = stats.poisson.pmf(3, lambda_)
print(f"P(X = 3): {prob_equal_3:.4f}")
# P(X <= 2)
prob_less_equal_2 = stats.poisson.cdf(2, lambda_)
print(f"P(X <= 2): {prob_less_equal_2:.4f}")
# P(X > 6)
prob_greater_6 = 1 - stats.poisson.cdf(6, lambda_)
print(f"P(X > 6): {prob_greater_6:.4f}")
# mean and variance
mean = lambda_
variance = lambda_
print(f"Mean: {mean}")
print(f"Variance: {variance}")