Appendix 01: Common Distributions#
Distribution |
Common Context |
PMF / PDF |
Support of X |
Parameter Range |
E[X] |
Var[X] |
|---|---|---|---|---|---|---|
Bernoulli(p) |
Single binary trial (e.g., coin toss) |
\(P(X = x) = p^x(1 - p)^{1 - x}\) |
\(x \in \{0, 1\}\) |
\(0 \le p \le 1\) |
\(p\) |
\(p(1 - p)\) |
Binomial(n, p) |
Number of successes in n trials |
\(P(X = k) = \binom{n}{k}p^k(1 - p)^{n - k}\) |
\(k \in \{0, 1, \dots, n\}\) |
\(n \in \mathbb{N},\ 0 \le p \le 1\) |
\(np\) |
\(np(1 - p)\) |
Geometric(p) |
Trials until first success |
\(P(X = k) = (1 - p)^{k - 1}p\) |
\(k \in \{1, 2, \dots\}\) |
\(0 < p \le 1\) |
\(\frac{1}{p}\) |
\(\frac{1 - p}{p^2}\) |
Exponential(λ) |
Time until an event |
\(f(x) = \lambda e^{-\lambda x}\) |
\(x \in [0, \infty)\) |
\(\lambda > 0\) |
\(\frac{1}{\lambda}\) |
\(\frac{1}{\lambda^2}\) |
Normal(μ, σ²) |
Measurement errors, natural phenomena |
\(f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}\) |
\(x \in \mathbb{R}\) |
\(\mu \in \mathbb{R},\ \sigma^2 > 0\) |
\(\mu\) |
\(\sigma^2\) |
Uniform(a, b) |
Equal probability over interval |
\(f(x) = \frac{1}{b - a}\) |
\(x \in [a, b]\) |
\(a < b,\ a, b \in \mathbb{R}\) |
\(\frac{a + b}{2}\) |
\(\frac{(b - a)^2}{12}\) |