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SciPy - Exponential Distribution



Exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. For example, customers arriving at a store, file requests on a server etc.

The probability density function (pdf) of exponential distribution is defined as:

Exponential Distribution

Where, β is the scale parameter which is the inverse of the rate parameter λ = 1/β.

An exponential distribution has mean β and variance β2.

The cumulative distribution function (cdf) evaluated at x, is the probability that the random variable (X) will take a value less than or equal to x. The cdf of exponential distribution is defined as:

Exponential Distribution

The SciPy.stats.expon contains all the methods required to generate or work with exponential distribution. The most frequently methods are mentioned below:

Syntax

scipy.stats.expon.pdf(x, loc=0, scale=1)
scipy.stats.expon.cdf(x, loc=0, scale=1)
scipy.stats.expon.ppf(q, loc=0, scale=1)
scipy.stats.expon.rvs(loc=0, scale=1, size=1)

Parameters

x Required. Specify float or array_like of floats representing random variable.
q Required. Specify float or array_like of floats representing probabilities.
loc Optional. Specify the location of the distribution. Default is 0.
scale Optional. Specify the scale parameter, β = 1/λ. Must be non-negative. Default is 1.0.
size Optional. Specify output shape.

expon.pdf()

The expon.pdf() function measures probability density function (pdf) of the distribution.

from scipy.stats import expon
import matplotlib.pyplot as plt
import numpy as np

#creating a array of values between
#-1 to 10 with a difference of 0.1
x = np.arange(-1, 10, 0.1)
   
y = expon.pdf(x, 0, 2)
   
plt.plot(x, y) 
plt.show()

The output of the above code will be:

Exponential Distribution

expon.cdf()

The expon.cdf() function returns cumulative distribution function (cdf) of the distribution.

from scipy.stats import expon
import matplotlib.pyplot as plt
import numpy as np

#creating a array of values between
#-1 to 10 with a difference of 0.1
x = np.arange(-1, 10, 0.1)
   
y = expon.cdf(x, 0, 2)
   
plt.plot(x, y) 
plt.show()

The output of the above code will be:

Exponential Distribution

expon.ppf()

The expon.ppf() function takes the probability value and returns cumulative value corresponding to probability value of the distribution.

from scipy.stats import expon
import matplotlib.pyplot as plt
import numpy as np

#creating a array of probability from
#0 to 1 with a difference of 0.01
x = np.arange(0, 1, 0.01)
   
y = expon.ppf(x, 0, 2)
   
plt.plot(x, y) 
plt.show()

The output of the above code will be:

Exponential Distribution

expon.rvs()

The expon.ppf() function generates an array containing specified number of random numbers of the given exponential distribution. In the example below, a histogram is plotted to visualize the result.

from scipy.stats import expon
import matplotlib.pyplot as plt
import numpy as np

#fixing the seed for reproducibility
#of the result
np.random.seed(10)

#creating a vector containing 10000
#exponentially distributed random numbers
y = expon.rvs(0, 1, 10000)

#creating bin
bin = np.arange(-1,10,0.1)  

plt.hist(y, bins=bin, edgecolor='blue') 
plt.show()

The output of the above code will be:

Exponential Distribution

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