Distribuição Uniforme
\[\begin{gathered} f\left(x \mid \theta_1, \theta_2\right)=\frac{1}{\theta_2-\theta_1}, \theta_2>\theta_1 \\ E[x]=\frac{a+b}{2} \\ \text{Var}[x]=\frac{(b-a)^2}{12} \\ f_n\left(x \mid \theta_1, \theta_2\right)=\frac{1}{\left(\theta_2-\theta_1\right)^n} \end{gathered}\]Distribuição Binomial
k = número de sucessos
\[\begin{gathered} f(x \mid \theta, k)=\left(\begin{array}{c} n \\ k \end{array}\right) \theta^k(1-\theta)^{n-k} \\ E[x]=p \leftrightarrow \theta \\ E\left[x^2\right]=n^2 p^2+n p(1-p) \leftrightarrow n^2 \theta^2+n \theta(1-\theta) \\ \text{Var}[x]=n p(1-p) \leftrightarrow n \theta(1-\theta) \\ f_n(x \mid \theta)=\prod_{i=1}^n\left(\begin{array}{c} n \\ x_i \end{array}\right) \theta^y(1-\theta)^{n-y}, y=\sum_1^n x_i \end{gathered}\]Distribuição Geométrica
$k=$ fracasso
\[\begin{gathered} f(x \mid \theta)=(1-\theta)^3 \theta \\ E[x]=\frac{1-p}{p} \leftrightarrow \frac{1-\theta}{\theta} \\ \text{Var}[x]=\frac{1-p}{p^2} \leftrightarrow \frac{1-\theta}{\theta^2} \end{gathered}\]Distribuição de Bernoulli
\[\begin{gathered} f(x \theta)-\theta^x(1-\theta)^{1-x}, x \in\{0,1\} \\ E[x]-p \leftrightarrow \theta \\ \text{Var}[x]=p(1-p) \leftrightarrow \theta(1-\theta) \\ f_n(x \mid \theta)=\theta^y(1-\theta)^{n-y}, y=\sum_1^n x_i \end{gathered}\]Distribuição Exponencial
\[\begin{gathered} f(x \mid \theta)=\theta e^{-\theta z}, \theta>0, x>0 \\ E[x]=\frac{1}{\lambda} \leftrightarrow \frac{1}{\theta} \\ V a r|x|=\frac{1}{p^2} \leftrightarrow \frac{1}{\theta^2} \\ f_n(x \mid \theta)=\theta^n e^{-\theta y}, y=\sum_1^n x_i \end{gathered}\]Distribuição de Poisson
\[\begin{gathered} f(x \theta)=\frac{e^{-\theta} \theta^x}{x !}, \theta>0 \\ E[x]-\lambda \leftrightarrow \theta \\ \text{Var}[x-\lambda \leftrightarrow \theta \\ f_n(x \mid \theta)=\frac{e^{-r \theta} \theta^y}{\prod x_{i} !}, y=\sum_1^n x_i \end{gathered}\]Distribuiçăo $\text{Normal}\left(\mu, \sigma^2\right)$
\[\begin{gathered} \int\left(x \mid \mu, \sigma^2\right)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{i s-\left.\mu\right|^2}{s r^2}} \\ E[x]=\mu \\ \text{Var}(x)=\sigma^2 \end{gathered}\] \[Z=\frac{X \quad \mu}{\sigma}\]$Z \sim(0,1)$
\[\text{Pr}(Z \leq c)=\Phi(c)\]Propriedade
Seja $X_1, X_2, \ldots X_i$ i.i.d onde $X_i \sim\left(\mu, \sigma^2\right)$, entāo $X_n=\frac{1}{n} \sum_{i=1}^n X_i$ tem distribuiçäo:
\[\begin{gathered} E\left(\frac{1}{n} \sum_{i=1}^n X_i\right)-\sum_{i=1}^n \frac{1}{n} E\left(X_i\right)=n \frac{1}{n} \mu-\mu \\ \text{Var}\left(\sum_{i=1}^n \frac{1}{n} X_i\right)=\sum_{i=1}^n\left(\frac{1}{n}\right)^2 \text{Var}\left(X_i\right)=\sum_{i=1}^n\left(\frac{1}{n}\right)^2 \sigma^2=n\left(\frac{1}{n}\right)^2 \sigma^2=\frac{\sigma^2}{n} \\ \bar{X}_n \sim\left(\mu, \frac{\sigma^2}{n}\right) \end{gathered}\]