Distribuciones de probabilidad
Normal
\[ X \sim \text{Normal}(\mu, \sigma) \]
\[ p(x \mid \mu, \sigma) = \frac{1}{\sqrt{2\pi} \sigma}e^{-\frac{(x - \mu) ^ 2}{2\sigma^2}} \]
\[ P(X \le x) = \int_{-\infty}^{x}{p(x | \mu, \sigma) dx} \]
o también
\[ P(X \le x) = \frac{1}{2} \left[1 + \text{erf}\left(\frac{x - \mu}{\sigma\sqrt{2}} \right) \right] \]
con
\[ \text{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^{x} {e^{-t^2}dt} \]
- \(X \in \mathbb{R}\)
- \(\mu \in \mathbb{R}\)
- \(\sigma > 0\)
- \(\mathbb{E}(X) = \mu\)
- \(\mathbb{V}(X) = \sigma^2\)
T-Student
\[ X \sim \text{StudentT}(\nu) \]
\[ p(x \mid \nu) = \frac{\Gamma(\frac{\nu + 1}{2})}{\Gamma(\frac{\nu}{2})} \left(\frac{1}{\pi\nu}\right) ^ {\frac{1}{2}} \left[1 + \frac{x^2}{\nu}\right]^{-\frac{\nu + 1}{2}} \]
\[ P(X \le x) = \frac{1}{2} + x \Gamma\left(\frac{\nu + 1}{2}\right) \frac{{}_2F_1\left(\frac{1}{2}, \frac{v + 1}{2}, \frac{3}{2}, \frac{-x^2}{\nu} \right)} {\sqrt{\pi \nu} \Gamma(\frac{\nu}{2})} \]
donde \({}_2F_1\) es la función hipergeométrica.
- \(X \in \mathbb{R}\)
- \(\nu > 0\)
- \(\mathbb{E}(X) = 0\) si \(\nu > 1\)
- \(\mathbb{V}(X) = \nu / (\nu - 2)\) si \(\nu > 2\)
Gamma
Parametrización 1
\[ X \sim \text{Gamma}(k, \theta) \]
\[ p(x \mid k, \theta) = \frac{1}{\Gamma(k)\theta^k}x^{k-1}e^{-\frac{x}{\theta}} \]
\[ P(X \le x) = \frac{1}{\Gamma(k)} \gamma \left(k, \frac{x}{\theta}\right) \]
- \(X > 0\)
- \(k > 0\)
- \(\theta > 0\)
- \(\mathbb{E}(X) = k\theta\)
- \(\mathbb{V}(X) = k\theta^2\)
Parametrización 2
\[ X \sim \text{Gamma}(\alpha, \beta) \]
\[ p(x | \alpha, \beta) = \frac{1}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}\beta^\alpha \]
\[ P(X \le x) = \frac{1}{\Gamma(\alpha)} \gamma(\alpha, \beta x) \]
- \(X > 0\)
- \(\alpha > 0\)
- \(\beta > 0\)
- \(\mathbb{E}(X) = \alpha/\beta\)
- \(\mathbb{V}(X) = \alpha/\beta^2\)
Exponencial
\[ X \sim \text{Exponencial}(\lambda) \]
\[ p(x \mid \lambda) = \lambda e^{-\lambda x} \]
\[ P(X \le x) = 1 - e^{-\lambda x} \]
- \(X > 0\)
- \(\lambda > 0\)
- \(\mathbb{E}(X) = 1 / \lambda\)
- \(\mathbb{V}(X) = 1 / \lambda ^ 2\)
Es un caso particular de \(\text{Gamma}(\alpha, \beta)\) con \(\alpha = 1\) y \(\beta = \lambda\)
Beta
\[ X \sim \text{Beta}(a, b) \]
\[ p(x \mid a, b) = \frac{x^{a-1} (1-x)^{b-1}}{B(a, b)} \]
\[ B(a, b) = \int_0^1 x^{a-1} (1-x)^{b-1} dx = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} \]
\[ \Gamma(x) = \int_0^\infty u^{x-1} e^{-u} du \]
- \(X \in (0, 1)\)
- \(\displaystyle \mathbb{E}(X) = \frac{a}{a + b}\)
- \(\displaystyle \mathbb{V}(X) = \frac{ab}{(a + b) ^ 2 (a + b + 1)}\)
Binomial
\[ X \sim \text{Binomial}(\theta, n) \]
\[ p(x \mid \theta, n) = {n \choose x} \theta^x (1 - \theta)^{(n - x)} \]
\[ P(X \le x) = \sum_{i = 0} ^ {x} {n \choose i} \theta^i (1 - \theta)^{(n - i)} \]
- \(X \in \{0, 1, 2, \cdots, n\}\)
- \(\theta \in [0, 1]\)
- \(n \in \{0, 1, 2, \cdots \}\)
- \(\mathbb{E}(X) = n \theta\)
- \(\mathbb{V}(X) = n \theta (1 - \theta)\)
Poisson
\[ X \sim \text{Poisson}(\lambda) \]
\[ p(x \mid \lambda) = \frac{\lambda^x e^{-\lambda}}{x!} \]
\[ P(X \le x) = e^{-\lambda} \sum_{i = 0} ^ {x} \frac{\lambda^i}{i!} \]
- \(X \in \{0, 1, 2, \cdots\}\)
- \(\lambda > 0\)
- \(\mathbb{E}(X) = \lambda\)
- \(\mathbb{V}(X) = \lambda\)
Binomial Negativa
\[ X \sim \text{BinomialNegativa}(r, p) \]
\[ p(x \mid k, p)= \binom{x + r - 1}{x}(1 - p)^x p^r \]
- \(X \in \{0, 1, 2, \cdots \}\)
- \(r \in \{1, 2, 3, \cdots \}\)
- \(p \in [0, 1]\)
- \(\mathbb{E}(X) = r(1 - p) / p\)
- \(\mathbb{V}(X) = r(1 - p) / p^2\)