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Uniform and Triangular Distributions
The uniform distribution is the easiest continuous random distribution to understand, and it is actually a continuous distribution with equal probability. Its PDF is
f ( x ) = 1 b − a , x ∈ ( a , b ) f(x)=\frac{1}{b-a}, x\in (a,b) f(x)=b−a1,x∈(a,b)
Its samples are at ( a , b ) (a,b)(a,b ) In the interval, almost a rectangle is formed, so it is called rectangular distribution.
Correspondingly, PDF is a triangular distribution, which is a triangular distribution. For lower bound aaa , the upper limit isbbb , the mode of which isccThe triangular distribution of c , its PDF is
f ( x ) = { 2 ( x − a ) ( b − a ) ( c − a ) a ⩽ x ⩽ c 2 ( b − x ) ( b − a ) ( b − c ) c < x ⩽ b f(x)=\left\{\begin{aligned} \frac{2(x-a)}{(b-a)(c-a)}\quad a\leqslant x\leqslant c\\ \frac{2(b-x)}{(b-a)(b-c)}\quad c<x\leqslant b \end{aligned}\right. f(x)=⎩ ⎨ ⎧(b−a)(c−a)2(x−a)a⩽x⩽c(b−a)(b−c)2(b−x)c<x⩽b
In scipy.stats, uniformit is the uniform distribution class; triangit is the triangular distribution class. uniformThe input parameters are the left endpoint and length; triangcompared with the uniform distribution, there is one more mode position, and the value range is [ 0 , 1 ] [0,1][0,1 ] as shown below.
from scipy.stats import uniform, triang
fig = plt.figure()
ax = fig.add_subplot(121)
ax.hist(uniform(1,5).rvs(10000), bins=50)
ax = fig.add_subplot(122)
ax.hist(triang(0.5, 1, 5).rvs(10000), bins=50)
plt.show()
The result is
Uniformly distributed addition
Two uniformly distributed random variables X 1 , X 2 X_1, X_2X1,X2, where the samples can be added or subtracted to obtain a triangular distribution, and the lower limit and upper limit of X1 are a 1 , b 1 a_1,b_1a1,b1, the lower limit and upper limit of X2 are respectively a 2 , b 2 a_2,b_2a2,b2, from the characteristics of this triangular distribution, we can see that the lower limit and upper limit of the triangular distribution are a 1 + a 2 , b 1 + b 2 a_1+a_2, b_1+b_2 respectivelya1+a2,b1+b2, whose mode is a 1 + b 1 + a 2 + b 2 2 \frac{a_1+b_1+a_2+b_2}{2}2a1+b1+a2+b2。
a1, d1 = 1,5
a2, d2 = 3,6
x1 = uniform(a1,d1).rvs(10000)
x2 = uniform(a2,d2).rvs(10000)
plt.hist(x1+x2, density=True, bins=100)
st = a1+a2
D = d1 + d2
rv = triang(0.5, st, D)
xs = np.linspace(st, st+D, 200)
plt.plot(xs, rv.pdf(xs))
plt.show()
The result is
log uniform distribution
scipy.statsloguniformThe distribution class is also provided in , whose probability density function is
f ( x , a , b ) = 1 x log ( a b ) f(x,a,b)=\frac{1}{x\log(\frac{a}{b})} f(x,a,b)=xlog(ba)1
It is conceivable that its probability density curve is an inverse function. If the logarithm of the random variable is taken, it can be transformed into the interval [ log a , log ba ] [\log a, \log\frac{b}{a}][loga,logab] Uniform distribution, next test
import numpy as np
from scipy.stats import loguniform
import matplotlib.pyplot as plt
a, b = 1, 5
rs = loguniform(a,b).rvs(10000)
plt.hist(np.log10(rs), density=True, bins=100)
rv = uniform(np.log10(a), np.log10(b/a/a))
xs = np.linspace(rv.ppf(0.01), rv.ppf(0.99), 100)
plt.plot(xs, rv.pdf(xs))
plt.show()
