Common probability distributions

• Continuous
• Parameters
  • $\mu \in \mathbb{R}$ (mean)
  • $\sigma^2 > 0$ (variance)
• Support: $x \in \mathbb{R}$
• PDF: $\frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$
• Mean/expectation: $\mu$
• Variance: $\sigma^2$

$P(X < x)$ for $X \sim \mathcal{N}(mu, sigma)$

p = normcdf(x, mu, sigma)

$\mathbb{P}(X \leq q) = 1 - alpha$ for $X \sim \mathcal{N}(mu, sigma)$

q = norminv(1 - alpha, mu, sigma)

• Discrete
• Parameters
  • $n \in \mathbb{N}_0$ (number of trials)
  • $p \in [0, 1]$ (probability of success of single trial)
• Support: $\{0, 1, \ldots, n\}$
• PMF: ${n \choose k} p^k (1-p)^{n-k}$
• Mean: $np$
• Variance: $np(p-1)$

• Discrete
• Special case of binomial for $n=1$
• Parameter: $p \in [0, 1]$ (probability of success)
• Support: $\{0, 1\}$ (either 0 or 1)
• PMF: $p^k(1-p)^{1-k}$
• Mean/expectation: $p$
• Variance: $p(1-p)$

• Discrete
• Parameter: $\lambda > 0$
• Support: $\mathbb{N}_0$ (0, 1, …)
• PMF: $\frac{\lambda^ke^{-\lambda}}{k!}$
• Mean/expectation: $\lambda$
• Variance: $\lambda$

• Continuous
• Parameter: $\lambda > 0$ (rate)
• Support: $[0, \infty]$
• Mean: $\frac{1}{\lambda}$
• Variance: $\frac{1}{\lambda^2}$