🚀 Data Science Roadmap 2026

📘 Phase 2: Mathematics for Data Science

📖 Topic 6: Probability Distributions — Discrete, Continuous, PMF, PDF & CDF

Welcome back! 👋

In the previous lesson, you learned Bayes' Theorem, which helps us update probabilities when new evidence becomes available.

Now we'll learn Probability Distributions.

Probability distributions are extremely important in Data Science because they help us understand how values are distributed and how likely different outcomes are.

They are used in:

✅ Statistical analysis

✅ Machine Learning

✅ Hypothesis testing

✅ A/B testing

✅ Forecasting

✅ Risk analysis

✅ Data simulation

🔹 1. What is a Probability Distribution?

A probability distribution describes how the probabilities of different possible outcomes are distributed.

For example, when rolling a fair die:

1 → 1/6

2 → 1/6

3 → 1/6

4 → 1/6

5 → 1/6

6 → 1/6

Every possible outcome has an associated probability.

The sum of all probabilities must equal: 1 = 100%

🔹 2. Two Main Types of Probability Distributions

Probability distributions can broadly be divided into:

1️⃣ Discrete Distribution

Used when outcomes are countable.

Examples: Number of customers, Number of defective products, Number of emails, Number of heads in coin tosses

2️⃣ Continuous Distribution

Used when values can take any value within a range.

Examples: Height, Weight, Temperature, Time, Salary

🔹 3. Discrete Random Variable

A discrete random variable takes countable values.

Example: Number of customers arriving at a store: 0, 1, 2, 3, 4, 5, ...

Another example: Number of defective products in a batch.

🔹 4. Continuous Random Variable

A continuous random variable can take infinitely many possible values within a range.

For example: someone's height could be: 170 cm, 170.1 cm, 170.15 cm, 170.157 cm...

There are infinitely many possible values.

🔹 5. PMF — Probability Mass Function ⭐

PMF stands for: Probability Mass Function

It is used for discrete random variables.

PMF tells us the probability of a specific outcome.

For example, when rolling a fair die:

P(X=3) = 1/6

Important Rule:

The probabilities of all possible outcomes must add up to 1:

∑P(X=x) = 1

🔹 6. PDF — Probability Density Function ⭐

PDF stands for: Probability Density Function

It is used for continuous random variables.

Unlike PMF, the PDF does not directly give the probability of a single exact value.

Instead, the area under the PDF curve over an interval represents probability.

For example: P(170 < Height < 180) is represented by the area under the PDF between 170 and 180.

Important Point:

For a continuous variable:

P(X=x) = 0

for any exact single value under the usual continuous probability model.

This doesn't mean the value is impossible. It means probability is assigned to intervals, not individual points.

🔹 7. CDF — Cumulative Distribution Function ⭐

CDF stands for: Cumulative Distribution Function

It tells us the probability that a random variable is less than or equal to a particular value.

Formula:

F(x) = P(X ≤ x)

Example: Suppose X = Test Score

Then: F(80) = P(X ≤ 80)