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):

P(X = 1) = p       (success)
P(X = 0) = 1 − p   (failure)

Mean = p, Variance = p(1 − p)

Binomial Distribution

The number of successes in n independent Bernoulli trials:

P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ

• Mean = np
• Variance = np(1 − p)

Example: Probability of getting exactly 3 heads in 5 coin flips (p = 0.5):

P(X = 3) = C(5,3) × 0.5³ × 0.5² = 10 × 0.125 × 0.25 = 0.3125

Poisson Distribution

Models the number of events occurring in a fixed interval of time or space, given a known average rate λ:

P(X = k) = (e⁻λ × λᵏ) / k!

• Mean = λ
• Variance = λ

Example: A call centre receives an average of 4 calls per minute. What is the probability of receiving exactly 6 calls in a minute?

P(X = 6) = (e⁻⁴ × 4⁶) / 6! ≈ 0.1042

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Continuous Probability Distributions

Probability Density Function (PDF)

For a continuous random variable, the PDF f(x) describes the relative likelihood of values. Probabilities are found by integrating: