Probability Distributions

A probability distribution describes how the values of a random variable are distributed. It tells you which outcomes are likely, which are unlikely, and how probabilities are spread across all possible values.

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Random Variables

A random variable assigns a numerical value to each outcome of a random experiment.

Type	Description	Examples
Discrete	Takes a countable number of values	Number of heads in 10 coin flips, dice rolls
Continuous	Takes any value within an interval	Height, weight, temperature

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Discrete Probability Distributions

Probability Mass Function (PMF)

For a discrete random variable X, the PMF gives the probability of each specific value:

P(X = x) = f(x)

Requirements:

• 0 ≤ f(x) ≤ 1 for all x
• Σ f(x) = 1 (probabilities sum to 1)

Expected Value (Mean)

E(X) = μ = Σ xᵢ × P(X = xᵢ)

Variance

Var(X) = σ² = Σ (xᵢ − μ)² × P(X = xᵢ)

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Common Discrete Distributions

Bernoulli Distribution

A single trial with two outcomes (success/failure):