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?"

$F(x)=P(X\le x)$

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 $x$.

$F(x)=\sum _{k\le x}P(X=k)$

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:

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

The corresponding CDF will be:

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

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.