Probability mass function

The probability mass function (PMF) is a function that is used to find the probabilities of specific events, where the outcomes can be counted or listed, like rolling dice/flipping coins. The PMF is used explicitly for discrete random variables, where the possible values are distinct and separate. In contrast, the probability density function (PDF) is used for continuous random variables.

Formal definition

Let X be a discrete random variable with possible outcomes as $X= \{$$x_{1}$,$x_{2}$,$x_{3}$, ..., $\}$ then the probabilities of each outcome are calculated using the probability mass function where the probability of an outcome $x_{i}$ is calculated as

where $f(x)$is a PMF that outputs the probability of a discrete value of X.

Properties of a PMF

1. The probability of a discrete random event is always between 0 and 1 inclusive.$0 \le P_{x}(x) \le 1$

2. Let n be the total outcomes of the discrete random variable X. Then$\sum_{i=1}^ n P(X=x_{i}) = P(X=x_{1})+P(X=x_{2}) + ... +P(X=x_{n})= 1$The total probabilities of all possible discrete outcomes add up to 1.

Example 1

Take the example of rolling two six-sided dice. Let X be the sum of two dice, then X could take on any value. ${\{2,3,4,5,6,7,8,9,10,11,12}\}$

The outcomes can be shown in the table:

To simply find the PMF for this set of experiments we need to calculate the probability for all possible values, i.e., 2,3,4..,12. The possible values that PMF can take for different values of X can be tabulated as shown below:

In this table, each column represents a possible sum of two dice rolls, ranging from 2 to 12. The heading P(X = x) denotes the likelihood of obtaining that particular sum. For example, the sum of 2 can only be obtained by rolling a 1 on each dice, which has a probability of 1/36. The sum of 7 has the highest probability of 6/36 since there are six combinations (e.g., 1+6, 2+5, 3+4, etc.) that result in a sum of 7.

Important observations

1. The probability of each outcome lies between 0 and 1 inclusive.

2. The sum of all possible outcomes equals 1 i.e $P(X=2) + P(X=3) + P(X=4) + P(X=5) + ... P(X=12) = 1$

Example 2

Find the missing probabilities given in the PMF table.

Recall the property of PMF that the sum of probability equals 1. $\sum_{i=1}^ n P(X=x_{i}) = 1$

$\frac{1}{2} + x + 2x = 1 , x =1/6$.

$P(X=3) = x = 1/6$ and $P(X=4) = 2x = 2/6$

Summary

In conclusion, the Probability Mass Function is a fundamental concept crucial in probability theory. It is used to analyze and quantify the probabilities associated with discrete random variables. For continuous random variables, we use the probability density function.