In [13]:
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Polygon
%matplotlib inline
A continuous random variable, often denoted as $X$, is a quantity whose values range over an interval of numbers.
Any random variable is associated with a probability density function (PDF), denoted $f(X)$ so that :
\begin{equation*} P(a \leq X \leq b) = \int_{a}^{b} f(X) dx \end{equation*}Since a probability is between 0 and 1, our PDF must satisfy following two restrictions:
1) The PDF must be greater than or equal to zero
\begin{equation*} f(X) \geq 0 \text{ for all } X \end{equation*}2) The total probability must be 1
\begin{equation*} \int_{-\infty}^{\infty} f(X) dx = 1 \end{equation*}Assume that we have a random variable $X$ with a probability density function:
\begin{equation*} f(X) = \frac{1}{9} e^{-X/9} \end{equation*}Q: How do we compute $P(10 \leq X \leq 20)$? What is the probability that $X$ is between 10 and 20?
A: We take the definite integral from 10 to 20.
In [28]:
def pdf(X):
return 1/9 * np.exp(-X/9)
X = np.linspace(0, 60)
y = pdf(X)
fig, ax = plt.subplots(figsize=(10, 5))
ax.plot(X, y)
ax.set_ylim(0)
ax.set_xlim(0)
a = 10
b = 20
ix = np.linspace(a, b)
iy = pdf(ix)
verts = [(a, 0)] + list(zip(ix, iy)) + [(b, 0)]
poly = Polygon(verts, facecolor='0.9', edgecolor='0.5')
ax.add_patch(poly)
ax.text(a+(b-a)/2, pdf(b)/2, r"$\int_{%s}^{%s} f(x)\mathrm{d}x$" % (a, b),
horizontalalignment='center', fontsize=14)
Out[28]: