Understanding the Joint Probability Mass Function (PMF) for Independent Bernoulli Random Variables

In the field of probability and statistics, it is crucial to understand the relationship between different random variables, especially when dealing with discrete variables. One important concept is the Joint Probability Mass Function (PMF), which describes the joint probabilities of two or more random variables. In this article, we will explore the values of the joint PMF ( f_{XY}(x, y) ) for two independent Bernoulli random variables ( X ) and ( Y ).

Introduction to Bernoulli Random Variables

Bernoulli random variables are a fundamental concept in probability theory. A Bernoulli random variable ( X ) takes on the value 1 with a probability ( p ) and the value 0 with a probability ( 1 - p ). The probability mass function (PMF) for a Bernoulli random variable is given by:

(P(X 1) p)

(P(X 0) 1 - p)

Similarly, for another Bernoulli random variable ( Y ), we have:

(P(Y 1) q)

(P(Y 0) 1 - q)

Joint PMF for Independent Bernoulli Random Variables

When ( X ) and ( Y ) are independent, the joint PMF ( f_{XY}(x, y) ) can be expressed as the product of the individual PMFs of ( X ) and ( Y ).

Let's denote the probabilities as follows:

(p P(X 1) 1/3)

(q P(Y 1) 1/2)

The joint PMF table for ( X ) and ( Y ) can be represented in a contingency table, where ( X ) takes values 1 and 0, and ( Y ) takes values 1 and 0.

Y 1 Y 0 Total X 1 p(1 - q) pq p X 0 (1 - p)q (1 - p)(1 - q) 1 - p Total q 1 - q 1

The Importance of the Contingency Table

The contingency table provides a clear and concise way to summarize the joint distribution of the random variables ( X ) and ( Y ). It is particularly useful in situations where these variables are independent, as it allows us to visualize the probabilities easily.

In the given contingency table, the probabilities are:

For ( X 1 ) and ( Y 1 ):

(P(X 1, Y 1) pq (1/3)(1/2) 1/6)

For ( X 1 ) and ( Y 0 ):

(P(X 1, Y 0) p(1 - q) (1/3)(1 - 1/2) 1/6)

For ( X 0 ) and ( Y 1 ):

(P(X 0, Y 1) (1 - p)q (1 - 1/3)(1/2) 1/3)

For ( X 0 ) and ( Y 0 ):

(P(X 0, Y 0) (1 - p)(1 - q) (1 - 1/3)(1 - 1/2) 1/6)

Analysis of Dependent Variables

For a more in-depth understanding, it is important to contrast the case of independent variables with the case where ( X ) and ( Y ) are dependent. In the article linked below, we analyze the contingency table for two dependent Bernoulli random variables. Analysis of Another Contingency Table Where the Variables Are Dependent

Conclusion

The concept of the joint PMF for independent Bernoulli random variables is fundamental to understanding the behavior and interactions of discrete random variables. By constructing and analyzing the contingency table, we obtain a clear picture of the joint distribution and the independence properties of the variables.