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Continuous Distributions



Normal Distribution
Uniform Continuous Distribution
Exponential Distribution






Normal Distribution

Normal (Gaussian) Distribution Symmetric bell curve around mean μ X can take any real value from -∞ to +∞ BELL-SHAPED PROBABILITY DENSITY Example: Human heights with μ=170 cm, σ=10 cm PROBABILITY DENSITY FUNCTION: f(x) μ μ-σ μ+σ μ-2σ μ+2σ Peak at μ 68% Value (x) Density Key Properties: • Continuous distribution • Symmetric bell curve • Mean = Median = Mode • 68% within μ±σ • 95% within μ±2σ • E(X) = μ • Var(X) = σ² X ~ N(μ, σ²) Example shown: bell curve centered at mean μ with standard deviation σ PROBABILITY DENSITY FUNCTION: f(x) = (1/(σ√(2π))) · exp(-(x-μ)²/(2σ²)) for -∞ < x < +∞ where μ is the mean and σ² is the variance

Uniform Continuous Distribution

Continuous Uniform Distribution All values in [a,b] equally likely X can take any real value in the interval [a, b] CONSTANT PROBABILITY DENSITY Example: Random time between 0 and 10 minutes (a=0, b=10) PROBABILITY DENSITY FUNCTION: f(x) a b 1/(b-a) f(x) = 1/(b-a) = 1/10 = 0.1 Value (x) Density Key Properties: • Continuous distribution • Defined on [a, b] • Constant density • f(x) = 1/(b-a) • E(X) = (a+b)/2 • Var(X) = (b-a)²/12 X ~ Uniform(a, b) Example shown: uniform on [0,10] with constant density 0.1 PROBABILITY DENSITY FUNCTION: f(x) = 1/(b-a) for a ≤ x ≤ b, and f(x) = 0 otherwise Area under the curve equals 1: (b-a) × 1/(b-a) = 1

Exponential Distribution

Exponential Distribution Waiting time between events at rate λ X can take any non-negative real value (0 to +∞) EXPONENTIAL DECAY FROM PEAK AT ZERO Example: Time until next customer arrival with λ=0.5 per minute PROBABILITY DENSITY FUNCTION: f(x) 0 1/λ Peak = λ Rapid decay: most events happen soon Memoryless: P(X > s+t | X > s) = P(X > t) Time (x) Density Key Properties: • Continuous distribution • Defined on [0, ∞) • Memoryless property • Peak at x = 0 • Rate parameter λ • E(X) = 1/λ • Var(X) = 1/λ² X ~ Exp(λ) Example shown: exponential decay with rate λ, mean waiting time 1/λ PROBABILITY DENSITY FUNCTION: f(x) = λ · exp(-λx) for x ≥ 0, and f(x) = 0 for x < 0 where λ > 0 is the rate parameter (events per unit time)