Laplace distribution

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Laplace The form of the probability density function of the distribution is ：

   commonly μ The values for 0, So the form is as follows ：

It is composed of two exponential functions , So it is also called double exponential function distribution （double exponential distribution）

Mean and variance

Solution of mean , if X The probability density function of is f(X), that X The mean of , After substitution, it can be found that the integral function inside is an odd function , therefore The mean for 0.

Variance basis , Because the latter item is 0, So we mainly ask for the previous item , According to the integral formula ∫udv=uv−vdu To solve the , The variance is .

Use pyplot Draw a probability distribution diagram

import matplotlib.pyplot as plt
import numpy as np
def laplace_function(x, lambda_):
    return (1/(2*lambda_)) * np.e**(-1*(np.abs(x)/lambda_))
x = np.linspace(-5,5,10000)
y1 = [laplace_function(x_,1) for x_ in x]
y2 = [laplace_function(x_,2) for x_ in x]
y3 = [laplace_function(x_,0.5) for x_ in x]

plt.plot(x, y1, color='r', label="lambda:1")
plt.plot(x, y2, color='g', label="lambda:2")
plt.plot(x, y3, color='b', label="lambda:0.5")

plt.title("Laplace distribution")
plt.legend()
plt.show()