📈 Probability Distribution

Last Updated: Jan 2026

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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

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Probability Distributions Types

Probability distributions are mainly of two types:

1. Discrete Probability Distribution
2. Continuous Probability Distribution

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Discrete Probability Distribution

Used when the random variable takes countable values.

Examples:

• Number of heads
• Number of students
• Dice outcomes

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Discrete Probability Distribution Table

Example: Tossing 2 coins

X (No. of Heads)	Outcomes	P(X)
0	TT	1/4
1	HT, TH	2/4
2	HH	1/4

Check:

1/4 + 2/4 + 1/4 = 1 ✔

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Discrete Distribution Formula

Probability Mass Function (PMF):

P(X = x) = f(x)

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Important Discrete Distributions

• Binomial Distribution
• Bernoulli Distribution
• Poisson Distribution

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Continuous Probability Distribution

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

Examples:

• Height
• Weight
• Time
• Temperature

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Key Difference from Discrete

For continuous variables:

P(X = x) = 0

We always calculate probability over an interval.

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Probability Density Function (PDF)

f(x) \ge 0

∫₋∞^∞ f(x) dx = 1

Probability between a and b:

P(a < X < b) = ∫ₐᵇ f(x) dx

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Important Continuous Distributions

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

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Discrete vs Continuous Distributions

Aspect	Discrete	Continuous
Values	Countable	Infinite in range
Function	PMF	PDF
P(X = x)	Possible	Always 0
Examples	Dice, coins	Height, time

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Cumulative Distribution Function (CDF)

Applicable to both discrete and continuous variables.

Definition:

F(x) = P(X <= x)

Properties:

• 0 ≤ F(x) ≤ 1
• Non-decreasing function
• F(∞) = 1

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