1 Uniform Distribution
The continuous uniform distribution has constant probability density over an interval.
Definition
\(X \sim \text{Uniform}(a, b)\) with PDF: $$ f_X(x) = \begin{cases} \frac{1}{b-a} & a < x < b \\ 0 & \text{otherwise} \end{cases} $$ Mean: \(\frac{a+b}{2}\), Variance: \(\frac{(b-a)^2}{12}\)
Interactive: Uniform Explorer
2 Exponential Distribution
Models waiting times and has the memoryless property.
Definition
\(X \sim \text{Exponential}(\lambda)\) with PDF: $$ f_X(x) = \lambda e^{-\lambda x}, \quad x > 0 $$ Mean: \(\frac{1}{\lambda}\), Variance: \(\frac{1}{\lambda^2}\)
Interactive: Exponential Explorer
Memoryless Property: P(X > a+b | X > a) = P(X > b)
3 Normal (Gaussian) Distribution
The most important continuous distribution, central to the Central Limit Theorem.
Definition
\(X \sim N(\mu, \sigma^2)\) with PDF: $$ f_X(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} $$
Interactive: Normal Explorer
68-95-99.7 Rule:
- • 68% within μ ± σ
- • 95% within μ ± 2σ
- • 99.7% within μ ± 3σ