📈 Probability Distribution

Last Updated: Jan 2026

A Probability Distribution describes how probabilities are assigned to different possible values of a random variable.

In simple words:

• It tells which values can occur
• And how likely each value is

This is a core bridge between:

• Probability → Statistics
• Theory → Real-world data
• Math → Programming & ML

🗣 Hinglish Tip: Probability Distribution = value ke saath uska chance attach karna

Probability Distributions Types

Probability distributions are mainly of two types:

1. Discrete Probability Distribution
2. Continuous Probability Distribution

Discrete Probability Distribution

Used when the random variable takes countable values.

Examples:

• Number of heads
• Number of students
• Dice outcomes

Discrete Probability Distribution Table

Example: Tossing 2 coins

Check:

1/4 + 2/4 + 1/4 = 1 $$ \checkmark

Discrete Distribution Formula

Probability Mass Function (PMF):

$P(X = x) = f(x)$

Important Discrete Distributions

• Binomial Distribution
• Bernoulli Distribution
• Poisson Distribution

Continuous Probability Distribution

Used when the random variable takes any value in a range.

Examples:

• Height
• Weight
• Time
• Temperature

Key Difference from Discrete

For continuous variables:

$P(X = x) = 0$

We always calculate probability over an interval.

Probability Density Function (PDF)

$f(x) \ge 0$

$\int_{-\infty}^{\infty} f(x) \, dx = 1$

Probability between a and b:

$P(a < X < b) = \int_{a}^{b} f(x) \, dx$

Important Continuous Distributions

• Uniform Distribution (equal probability)
• Normal (Gaussian) Distribution
• Exponential Distribution

Discrete vs Continuous Distributions

Cumulative Distribution Function (CDF)

Applicable to both discrete and continuous variables.

Definition:

$f(x) = P(X \leq  x)$

Properties:

• $0 \leq F(x) \leq 1$
• Non-decreasing function
• $F(\infty) = 1$

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