Understanding and Calculating Marginal Probability Mass Functions and Expected Values

In probability theory, understanding the concepts of marginal probability mass functions (PMFs) and expected values is crucial for analyzing complex distributions. In this article, we will explore the steps to identify and calculate these functions, along with a detailed example to help clarify the process.

Identifying and Calculating the Marginal PMF

To begin, let's consider a joint probability mass function (PMF) for two discrete random variables, X and Y. The joint PMF, denoted by (P_{X,Y}(x, y)), represents the probability that X takes value x and Y takes value y. The marginal PMF of X, denoted as (P_X(x)), is found by summing the joint PMF over all possible values of Y. Similarly, the marginal PMF of Y, denoted as (P_Y(y)), is found by summing the joint PMF over all possible values of X.

Steps to Find the Marginal PMFs

Identify the Joint PMF: Start with the joint PMF for the variables X and Y. For example, consider a joint PMF given in the table: 01 2 10.20.00.3 20.10.20.2 Here, the probabilities are given for each combination of X and Y. Calculate the Marginal PMF of X: The marginal PMF of X is calculated by summing the joint PMF over all possible values of Y:

[P_X(0) P_{X,Y}(0,0) P_{X,Y}(0,1) P_{X,Y}(0,2) 0.2 0 0 0.2]

[P_X(1) P_{X,Y}(1,0) P_{X,Y}(1,1) P_{X,Y}(1,2) 0 0 0.3 0.3]

[P_X(2) P_{X,Y}(2,0) P_{X,Y}(2,1) P_{X,Y}(2,2) 0.1 0.2 0.2 0.5]

Calculate the Marginal PMF of Y: The marginal PMF of Y is calculated by summing the joint PMF over all possible values of X:

[P_Y(1) P_{X,Y}(0,1) P_{X,Y}(1,1) P_{X,Y}(2,1) 0 0 0.2 0.2]

[P_Y(2) P_{X,Y}(0,2) P_{X,Y}(1,2) P_{X,Y}(2,2) 0.3 0.2 0.2 0.7]

Note that the sum of probabilities in each marginal PMF should equal 1.

Calculating Expected Values

Once the marginal PMFs are determined, the expected values of X and Y can be calculated using the probabilities in the marginal distributions.

Steps to Find the Expected Values

Find the expected value of X: The expected value (E[X]) is found by summing the product of each possible value of X and its corresponding probability:

[E[X] 0 cdot P_X(0) 1 cdot P_X(1) 2 cdot P_X(2) 0 cdot 0.2 1 cdot 0.3 2 cdot 0.5 0 0.3 1.0 1.3]

Find the expected value of Y: Similarly, the expected value (E[Y]) is found by summing the product of each possible value of Y and its corresponding probability:

[E[Y] 1 cdot P_Y(1) 2 cdot P_Y(2) 1 cdot 0.2 2 cdot 0.7 0.2 1.4 1.6]

Conclusion

In summary, to find the marginal probability mass functions and expected values, follow the steps of identifying the joint PMF, calculating the marginal PMFs by summing over the other variable, and then calculating the expected values using the marginal PMFs.

For example, given the joint PMF table provided, the marginal PMFs and expected values would be:

Marginal PMFs

01 2 P_X(x)0.20.30.5 P_Y(y)0.20.20.7

Expected Values

[E[X] 1.3], [E[Y] 1.6]

Checking for Independence

To check if X and Y are independent, compare the joint PMF with the product of the marginal PMFs:

0 1 2 10.2 0.0 0.3 20.1 0.2 0.2 Product of Marginals0.04 0.06 0.15

Since (P_{X,Y}(0,1) 0.2 eq 0.15 P_X(0) cdot P_Y(1)), X and Y are not independent.