The Mathematical Descriptions

1. Probability Mass Function (PMF)

The PMF tells you the exact probability that a discrete random variable is exactly equal to some value. Because the variable is discrete, you can think of the PMF as assigning a "mass" of probability to specific, isolated points.

Notation:

Rules:

1. Every individual probability must be between 0 and 1: $0 \le p(x) \le 1$
2. The sum of all probabilities across all possible outcomes must equal exactly 1: $\sum p(x) = 1$

Example: The PMF of rolling a fair 6-sided die is $p(x) = 1/6$ for $x \in \{1, 2, 3, 4, 5, 6\}$.

2. Probability Density Function (PDF)

The PDF is the continuous equivalent of the PMF, but it requires a paradigm shift. For a continuous variable (like exact human height), the probability of someone being exactly 180.00000000... cm tall is technically zero!

Instead of looking at exact points, the PDF describes the probability density. To find an actual probability, you must calculate the area under the curve over a specific range using calculus (integration).

Notation:

To find the probability of falling between $a$ and $b$:

Rules:

1. Density can never be negative: $f(x) \ge 0$
2. The total area under the entire curve must equal exactly 1: $\int_{-\infty}^{\infty} f(x) dx = 1$

3. Cumulative Distribution Function (CDF)

The CDF is arguably the most practical function. It answers the question: "What is the probability that my variable will be less than or equal to a specific value $x$?"

It is the running total (cumulative sum or integral) of the probabilities up to that point.

Notation:

Rules:

1. It always starts at 0: $\lim_{x \to -\infty} F(x) = 0$
2. It always ends at 1: $\lim_{x \to \infty} F(x) = 1$
3. It is monotonically non-decreasing (it never goes down, it only goes up or stays flat).

Interactive Visualization: PDF vs CDF

Switch between the PDF and CDF views below. Notice how the Area under the curve in the PDF view perfectly matches the exact Y-axis height in the CDF view!

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