Random Variables and Probability Distributions

Introduction to Random Variables

In many experiments, we are interested in a numerical characteristic associated with outcomes of a random experiment.

A random variable (RV) is a function that assigns a numerical value to each outcome of a random experiment.

Example: Consider tossing a fair die twice and defining \( X \) as the number of times an odd number appears.

Sample Space and Classification of \( X \)

The sample space consists of \( 6 \times 6 = 36 \) outcomes. The odd numbers are \( \{1, 3, 5\} \) and even numbers are \( \{2, 4, 6\} \).

Classification of \( X \):

• \( X = 0 \) (Both numbers are even) → 9 outcomes
• \( X = 1 \) (One odd, one even) → 18 outcomes
• \( X = 2 \) (Both numbers are odd) → 9 outcomes

Probability Mass Function (PMF)

The PMF gives \( P(X = x) \) for each possible value of \( X \):

• \( P(X=0) = \frac{9}{36} = \frac{1}{4} \)
• \( P(X=1) = \frac{18}{36} = \frac{1}{2} \)
• \( P(X=2) = \frac{9}{36} = \frac{1}{4} \)

Cumulative Distribution Function (CDF)

The CDF is defined as:

\[
F(x) = P(X \leq x)
\]

• \( F(0) = P(X \leq 0) = P(0) = \frac{1}{4} \)
• \( F(1) = P(X \leq 1) = P(0) + P(1) = \frac{3}{4} \)
• \( F(2) = P(X \leq 2) = P(0) + P(1) + P(2) = 1 \)

Summary

• A random variable represents numerical values associated with a random experiment.
• The PMF describes the probability distribution of \( X \).
• The CDF provides cumulative probabilities.

PDF

Random Variables and Probability Distributions

Video

 

Follow PostNetwork Academy

• Website: www.postnetwork.co
• YouTube: YouTube Channel
• Facebook: Facebook Page
• LinkedIn: LinkedIn Page

Thank You!