Probability Distributions

Discrete Distributions

Binomial Distribution

Models the number of successes in n independent trials, each with probability p of success.

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

Example: A quality inspector tests 10 items, each with a 5% defect rate. What is the probability of finding exactly 2 defectives?

P(X = 2) = C(10,2)(0.05)²(0.95)⁸ = 45 × 0.0025 × 0.6634 = 0.0746 ≈ 7.5%

Poisson Distribution

Models the number of events in a fixed interval when events occur at constant average rate λ.

$P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$

Example: A call center receives an average of 3 calls per minute. What is the probability of exactly 5 calls in a given minute?

P(X = 5) = e⁻³ · 3⁵ / 5! = 0.0498 · 243 / 120 = 0.1008 ≈ 10.1%

Continuous Distributions

Normal (Gaussian) Distribution

$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

Key Properties:

• Bell-shaped and symmetric about the mean μ
• Mean = median = mode
• Total area under the curve = 1
• Characterized entirely by μ (location) and σ (spread)

Standard Normal Distribution (z-distribution): $z = \frac{x - \mu}{\sigma}$

The z-score tells you how many standard deviations x is from the mean.

Empirical Rule (68-95-99.7)

Example: Bolt diameters are normally distributed with μ = 10.0 mm and σ = 0.1 mm. What percentage falls between 9.8 mm and 10.2 mm?

9.8 mm = μ - 2σ, 10.2 mm = μ + 2σ → 95.45% fall in this range.

Uniform Distribution

All values in [a, b] are equally likely.

Exponential Distribution

Models the time between events in a Poisson process with rate λ.

$f(x) = \lambda e^{-\lambda x} \quad \text{for } x \geq 0$

Memoryless property: P(X > s + t | X > s) = P(X > t). The probability of an event occurring in the next t units does not depend on how long you have already waited.