Common probability distributions [Jaeyoung Wiki]

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====== Common probability distributions ======

===== Gaussian/Normal =====

  * 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: $\sigma^2$


==== Calculating CDF ====

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

<code matlab>
p = normcdf(x, mu, sigma)
</code>

====Calculating inverse CDF or quantile====

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

<code matlab>
q = norminv(1 - alpha, mu, sigma)
</code>


===== Binomial distribution =====

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

===== Bernoulli =====

  * 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: $p(1-p)$


===== Poisson =====

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

===== Exponential =====

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