MA2102

Probability and Statistics

Lecture-11

Types of Random Variables

Let $X$ be a random variable defined on $(\Omega,\mathscr{F},P)$ , and $(\mathbb{R},\mathscr{B},P^X)$ be induced probability space by $X$ , then $CDF$ $F_X$ will be either continuous everywhere or it will have countably many (finite/countably infinite) (jump)discontinuities where the sum of jump lengths at the point of discontinuities always $\le$ 1. We will use these properties to classify random variables.

import matplotlib.pyplot as plt
import numpy as np
x=np.linspace(-10,10,100)
fig, ax = plt.subplots(3,1,figsize=(15,10))
# Continuous type example CDF Plot
ax[0].plot(x,1/2+(1/np.pi)*np.arctan(x),color='blue')
ax[0].set_title("(i) Continuous type",fontsize=10,color='blue')

# Discrete type exmaple CDF Plot
x1=[0,1,2,3]
y1=[0,1/8,1/2,7/8]
x2=[0,1,2,3]
y2=[1/8,1/2,7/8,1]
ax[1].scatter(x1,y1,facecolors='none', edgecolors='black')
ax[1].scatter(x2,y2,facecolors='black', edgecolors='black')
ax[1].set_xlim(-2,5)
ax[1].plot([-2,0],[0,0],color='purple')
ax[1].plot([0,1],[1/8,1/8],color='purple')
ax[1].plot([1,2],[1/2,1/2],color='purple')
ax[1].plot([2,3],[7/8,7/8],color='purple')
ax[1].plot([3,5],[1,1],color='purple')
ax[1].set_title("(ii) Descrete type",fontsize=10,color='purple')

# Mixture Type example CDF Plot
ax[2].plot([-3,1],[0,0],color='navy')
xx=np.linspace(1,2,50)
ax[2].plot(xx,(xx-1)*(xx-1)/2,color='navy')
ax[2].plot([2,3],[1/2,1/2],color='navy')
ax[2].plot([3,5],[1,1],color='navy')
ax[2].scatter([3],[1/2],facecolors='none', edgecolors='black')
ax[2].scatter([3],[1],facecolors='black', edgecolors='black')
ax[2].plot([3,5],[1,1],color='navy')
ax[2].set_title("(iii) Mixture type",fontsize=10,color='navy')
# ax[0].set_xticks([])

plt.show()

Definition: A random variable $X$ is said to be of discrete type if it's $CDF$ has countably many discontinuities with sum of jump at the point of discontinuities exactly equals to $1$ .