Publish Date: June 10, 2024

Conditional Probability Mass Function (PMF) of More than Two Discrete Random Variables

Introduction

In probability theory, the Conditional Probability Mass Function (PMF) provides the probability distribution of a discrete random variable given that certain conditions are satisfied by other random variables. When dealing with more than two discrete random variables, the conditional PMF describes the probability of one variable given specific values of the other variables.

Definition

Let be discrete random variables. The conditional PMF of given is defined as:

This can be computed using the joint PMF and the marginal PMF :

where is not zero.

Calculation

To calculate the conditional PMF:

Determine the Joint PMF: The joint PMF of is:
Determine the Marginal PMF: The marginal PMF of is:
Apply the Formula: Substitute the joint and marginal PMFs into the conditional PMF formula:

Meaning and Importance

Understanding conditional PMFs is crucial for:

Modeling Dependencies: It helps in understanding the dependency of one variable on the others in a probabilistic sense.
Bayesian Inference: Conditional probabilities are foundational in Bayesian statistics, allowing updates of beliefs based on new evidence.
Data Analysis: In multivariate data analysis, conditional PMFs help in understanding relationships between different variables.

Example: Coin Toss

Consider a scenario with three coins. Let , , and be the outcomes of the three coins, respectively. Define for heads and for tails (similarly for and ). Assume the results of the coin tosses are independent.

The joint PMF can be defined as follows:

for all , because each coin has two possible outcomes, and all combinations are equally likely.

To find the conditional PMF of given and :

Joint PMF:
Marginal PMF:
Conditional PMF:

So, given and , the probability that is 0.5, and the probability that is also 0.5.

This result is expected because the outcomes are independent. Hence, knowing the outcomes of and does not affect the probability distribution of .

So, given that , the probability that is 0.5, and the probability that is also 0.5.