Updated 3 years ago
MA2102
Probability and Statistics
Lecture-11
Types of Random Variables
Let be a random variable defined on , and be induced probability space by , then 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 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 is said to be of discrete type if it's has countably many discontinuities with sum of jump at the point of discontinuities exactly equals to .