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

In probability theory and statistics, the conditional probability mass function (PMF) of two discrete random variables and describes the probability distribution of one random variable, given that the other random variable takes on a specific value.

Definition

The conditional PMF of given is defined as:

for . Here:

• is the joint PMF of and .
• is the marginal PMF of .

Explanation

1. Joint PMF: This gives the probability that and simultaneously take on specific values.
2. Marginal PMF: This gives the probability that takes on a specific value, summing over all possible values of .

Example: Conditional PMF of Coin Tosses

Consider a scenario where we toss a coin twice. Define two discrete random variables:

• : The number of heads in the first toss.
• : The total number of heads in both tosses.

Joint PMF

The sample space for two coin tosses is {HH, HT, TH, TT}, where ‘H’ stands for heads and ‘T’ stands for tails.

The joint PMF is:

Marginal PMF

The marginal PMF is found by summing over all possible values of :

So,

Conditional PMF

We now compute the conditional PMF of given :

1. For :

2. For :

3. For :

Summary

Here’s the conditional PMF summarized:

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