Probability Mass Function(PMF) and Cumulative Distribution Function(CDF) Explained

PMF and CDF

1. PMF (Probability Mass Function):

The PMF gives the probability that a discrete random variable takes on a particular value. Think of it as answering: "What is the chance of getting this specific outcome?" For a random variable \( X \), the PMF is defined as:

\( P(X = x) = p(x) \)

In simpler terms, it tells us the likelihood of specific values.

2. CDF (Cumulative Distribution Function):

The CDF shows the probability that a random variable \( X \) will take on a value less than or equal to a given value. Think of it as "What are the chances that our outcome is less than or equal to this value?"

\( F(x) = P(X \leq x) \)

Relationship between PMF and CDF:

The CDF is essentially the accumulation of probabilities given by the PMF. For a discrete random variable, the CDF at any point \( x \) is the sum of all PMF values less than or equal to \( x \).

\( F(x) = \sum_{k \leq x} P(X = k) \)

This means the CDF can be obtained by adding up the probabilities from the PMF.

Example:

Let’s consider a simple example: a 3-sided die with outcomes {1, 2, 3}. The PMF is:

\( P(X = 1) = \frac{1}{3}, \quad P(X = 2) = \frac{1}{3}, \quad P(X = 3) = \frac{1}{3} \)

The corresponding CDF will be:

• \( F(1) = P(X \leq 1) = \frac{1}{3} \)
• \( F(2) = P(X \leq 2) = \frac{2}{3} \)
• \( F(3) = P(X \leq 3) = 1 \)

Summary:

• The PMF gives the probability of a specific outcome.
• The CDF gives the cumulative probability up to a certain value.
• The CDF is obtained by summing up the PMF values.