Exponential Distribution

The exponential distribution is a continuous probability distribution that models the time between events in a Poisson process.

We remember that the Poisson process describes the number of events that occurred during some period. On the other hand, exponential distribution describes the time between the occurrence of two successive events (the distance between two adjacent non-zero points in the graph above).
Poisson distribution is parameterized by the mu parameter, which describes the average number of accidents by the time unit. The exponential distribution is parametrized by the parameter scale which determines the average time between two accidents.

Note

There is a clear relationship between the Poisson process parameter mu and the exponential distribution parameter scale:
mu = 1 \ scale for one unit of time

Example

Suppose the average time between customer arrivals at a store is 5 minutes. What is the probability that the next customer arrives within 3 minutes?

We have a Poisson process where an event is the customer's arrival. The average time between two arrivals is 5 minutes. As a result, we can use exponential distribution to calculate the corresponding probability:

from scipy.stats import expon

# Parameters of the exponential distribution

mean_waiting_time = 5

# Calculate that new customer will arrive in less than 3 minutes

probability = expon.cdf(3, scale=mean_waiting_time) - expon.cdf(0, scale=mean_waiting_time)

# Print the result

print(f'The probability that the next customer arrives within 3 minutes is: {probability:.4f}')


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from scipy.stats import expon

# Parameters of the exponential distribution
mean_waiting_time = 5

# Calculate that new customer will arrive in less than 3 minutes
probability = expon.cdf(3, scale=mean_waiting_time) - expon.cdf(0, scale=mean_waiting_time)

# Print the result
print(f'The probability that the next customer arrives within 3 minutes is: {probability:.4f}')

We also use .cdf() method of the scipy.stats.expon class with a specified scale parameter to calculate the corresponding probability on the interval [0,3].

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