Formulário
Teoria da probabilidade
\[ \Pr[A \cup B] = \Pr[A] + \Pr[B] - \Pr[A \cap B] \qquad \Pr[A \mid B] = \dfrac{\Pr[A \cap B]}{\Pr[B]} \] \[ \begin{aligned} \Pr[A \cup B] = \Pr[A] + \Pr[B], & \quad \text{para $A$ e $B$ eventos disjuntos.} \\ \Pr[A \cap B] = \Pr[A] \Pr[B], & \quad \text{para $A$ e $B$ eventos independentes.} \end{aligned} \] \[ \Pr[B] = \sum_{i} \Pr[B \mid A_i] \Pr[A_i], \quad \text{onde } \{ A_i \} \text{ são eventos que particionam o espaço amostral.} \]Variáveis aleatórias
\[ \begin{alignedat}{2} \text{PMF} & \qquad & p_X(x) & \qquad \text{Função massa de probabilidade} \\ \text{PDF} & \qquad & f_X(x) & \qquad \text{Função densidade de probabilidade} \\ \text{CDF} & \qquad & F_X(x) & \qquad \text{Função distribuição cumulativa} \end{alignedat} \] \[ p_X(x) = \Pr[X = x] \qquad \Pr[a \leq X \leq b] = \int_{a^-}^{b^+} f_X(x) \dif x \qquad F_X(x) = \Pr[X \leq x] = \int_{-\infty}^{x^+} f_X(u) \dif u \] \[ f_{X}(x) = \int_{-\infty}^\infty f_{X,Y}(x,y) \dif y \qquad \qquad f_{X}(x \mid Y=y) = \frac{f_{X,Y}(x,y)}{f_Y(y)} \] \[ f_X(x) = \sum_{i} f_{X}(x \mid A_i) \Pr[A_i], \quad \text{onde $\{ A_i \}$ são eventos que particionam o espaço amostral.} \]Distribuição normal
\[ X \sim \mathrm{N}(\mu, \sigma^2) \quad \iff \quad f_X(x) = \dfrac{1}{\sqrt{2 \pi \sigma^2}} \ee^{-\frac{(x - \mu)^2}{2\sigma^2}} \] \[ \Phi(x) = \dfrac{1}{\sqrt{2 \pi}} \int_{-\infty}^x \ee^{-\frac{u^2}{2}} \dif u \qquad \mathrm{Q}(x) = 1 - \Phi(x) \] \[ X \sim \mathrm{N}(\mu, \sigma^2) \quad \implies \quad \Pr[a \leq X \leq b] = \Phi \left( \frac{b - \mu}{\sigma} \right) - \Phi \left( \frac{a - \mu}{\sigma} \right) \]Valor esperado
\[ \EV[g(X)] = \sum_{x \in S_X} g(x) p_\mathrm{X}(x) \qquad \EV[g(X)] = \int_{-\infty}^{\infty} g(x) f_\mathrm{X}(x) \dif x \] \[ \EV[g(X, Y)] = \sum_{x \in S_X} \sum_{y \in S_Y} g(x, y) p_\mathrm{X, Y}(x, y) \qquad \EV[g(X, Y)] = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(x, y) f_\mathrm{X, Y}(x, y) \dif x \dif y \] \[ \mu_X = \EV[X] \qquad \sigma_X^2 = \var[X] = \EV[(X - \mu_X)^2] = \EV[X^2] - \EV[X]^2 \] \[ \cov[X, Y] = \EV[(X - \mu_X)(Y - \mu_Y)] = \EV[XY] - \EV[X] \EV[Y] \qquad \rho_{X,Y} = \frac{\cov[X, Y]}{\sqrt{\var[X] \var[Y]}} \] \[ \var[X + Y] = \var[X] + \var[Y] + 2\cov[X, Y] \\ \] \[ \EV[X] = \sum_{i} \EV[X \mid A_i] \Pr[A_i], \quad \text{onde $\{ A_i \}$ são eventos que particionam o espaço amostral.} \]Vetores aleatórios
\[ \vec{\mu}_{\vec{X}} = \EV[ \vec{X} ] = \mat{\EV[X_1] \\ \EV[X_2] \\ \vdots \\ \EV[X_n]} \qquad C_{\vec{X}} = \EV[ (\vec{X} - \vec{\mu}_{\vec{Y}}) (\vec{X} - \vec{\mu}_{\vec{Y}})^\tr] = \mat{ \var[X_1] & \cov[X_1, X_2] & \cdots & \cov[X_1, X_n] \\ \cov[X_2, X_1] & \var[X_2] & \cdots & \cov[X_2, X_n] \\ \vdots & \vdots & \ddots & \vdots \\ \cov[X_n, X_1] & \cov[X_n, X_2] & \cdots & \var[X_n] } \] \[ \vec{Y} = A \vec{X} + \vec{b} \quad \implies \quad \vec{\mu}_{\vec{Y}} = A \vec{\mu}_{\vec{X}} + \vec{b}, \quad C_{\vec{Y}} = A C_{\vec{X}} A^\tr \] \[ \vec{X} \sim \mathrm{N}(\vec{\mu}, C) \quad \iff \quad f_{\vec{X}}(\vec{x}) = \frac{1}{(2 \pi)^{n/2} \sqrt{\det C}} \exp \left( -\frac{1}{2} (\vec{x} - \vec{\mu})^\tr C^{-1} (\vec{x} - \vec{\mu}) \right) \]Processos estocásticos
\[ \mu_X(t) = \EV[X(t)] \qquad C_X(t_1, t_2) = \cov[X(t_1), X(t_2)] = \EV[ X(t_1) X(t_2) ] - \EV[X(t_1)] \EV[X(t_2)] \]Processos estocásticos estacionários no sentido amplo
\[ S_X(f) = \Fourier \{ R_X(\tau) \} \qquad \mu_Y = \hat{h}(0) \mu_X \qquad S_Y(f) = |\hat{h}(f)|^2 S_X(f) \] \[ P_X = \EV[X^2(t)] = R_X(0) = \int_{-\infty}^{\infty} S_X(f) \dif f \]Séries
\[ \sum_{k=0}^{\infty} r^k = \frac{1}{1 - r}, \qquad \qquad \sum_{k=0}^{\infty} kr^k = \frac{r}{(1 - r)^2}, \qquad \qquad \sum_{k=0}^{\infty} k^2r^k = \frac{r(1+r)}{(1 - r)^3}. \qquad \qquad (|r| < 1) \]Identidades trigonométricas
\[ \cos(a \pm b) = \cos a \cos b \mp \sin a \sin b \qquad \sin(a \pm b) = \sin a \cos b \mp \sin b \cos a \] \[ \cos a \cos b = \frac{1}{2} [ \cos (a-b) + \cos (a+b) ] \qquad \sin a \sin b = \frac{1}{2} [ \cos (a-b) - \cos (a+b) ] \] \[ \cos^2 a = \frac{1}{2} (1 + \cos 2a) \qquad \sin^2 a = \frac{1}{2} (1 - \cos 2a) \]Fórmula de Euler
\[ \ee^{\iu\theta} = \cos \theta + \iu \sin \theta \qquad \cos \theta = \frac{1}{2} (\mathrm{e}^{\iu\theta} + \mathrm{e}^{-\iu\theta}) \qquad \sin \theta = \frac{1}{2\iu} (\mathrm{e}^{\iu\theta} - \mathrm{e}^{-\iu\theta}) \]Sinais básicos
\[ \rect(x) = \begin{cases} 1, & \text{se } |x| < 1/2, \\ 0, & \text{caso contrário.} \end{cases} \qquad \tri(x) = \begin{cases} 1 - |x|, & \text{se } |x| < 1, \\ 0, & \text{caso contrário.} \end{cases} \qquad \sinc(x) = \frac{\sin(\pi x)}{\pi x} \] \[ \sign(x) = \begin{cases} 1, & x > 0, \\ 0, & x = 0, \\ -1, & x < 0. \end{cases} \qquad \step(x) = \begin{cases} 1, & x > 0, \\ 0, & x < 0. \end{cases} \]Definição e propriedades da transformada de Fourier em tempo contínuo
\[ \vphantom{\int_{-\frac{1}{2}}^{\frac{1}{2}}} \vphantom{\sum_{n=-\infty}^{\infty}} \hat{x}(f) = \Fourier \{ x(t) \} = \int_{-\infty}^{\infty} x(t) \ee^{-\iu 2 \pi f t} \dif t \qquad x(t) = \Fourier^{-1} \{ \hat{x}(f) \} = \int_{-\infty}^{\infty} \hat{x}(f) \ee^{\iu 2 \pi t f} \dif f \] \[ \Fourier \{ a x_1(t) + b x_2(t) \} = a \Fourier \{ x_1(t) \} + b \Fourier \{ x_2(t) \} \] \[ \Fourier \{ x(t - t_0) \} = \Fourier \{ x(t) \} \ee^{-\iu 2 \pi t_0 f} \qquad \Fourier \{ x(t) \ee^{\iu 2 \pi f_0 t} \} = \hat{x}(f - f_0) \qquad \Fourier \{ x(-t) \} = \hat{x}^*(f) \] \[ \Fourier \{ x_1(t) x_2(t) \} = \Fourier \{ x_1(t) \} \star \Fourier \{ x_2(t) \} \qquad \Fourier \{ x_1(t) \star x_2(t) \} = \Fourier \{ x_1(t) \} \Fourier \{ x_2(t) \} \] \[ \Fourier \{ \hat{x}(t) \} = x(-f) \] \[ \Fourier \{ x(at) \} = \dfrac{1}{|a|} \hat{x} \left( \dfrac{f}{a} \right) \] \[ \Fourier \left \{ \dfrac{\dif^n}{\dif t^n} x(t) \right \} = (\iu 2 \pi f)^n \hat{x}(f) \qquad \Fourier \{ t^n x(t) \} = \left( \dfrac{\iu}{2\pi} \right)^n \dfrac{\dif^n}{\dif f^n} \hat{x}(f) \]
Pares transformados de Fourier em tempo contínuo
$\text{Sejam $t_0 \in \mathbb{R}$, $f_0 \in \mathbb{R}$ e $T \in \mathbb{R}$.}$ \[ \def\arraystretch{2} \begin{alignedat}{2} x(t) & \qquad \hat{x}(f) & \qquad \\ \hline 1 & \qquad \delta(f) & \qquad (1) \\ \delta(t) & \qquad 1 & \qquad (2) \\ \ee^{\iu 2\pi f_0 t} & \qquad \delta(f - f_0) & \qquad (3) \\ \delta(t - t_0) & \qquad \ee^{-\iu 2\pi t_0 f} & \qquad (4) \\ \cos(2 \pi f_0 t) & \qquad \dfrac{1}{2} \delta(f - f_0) + \dfrac{1}{2} \delta(f + f_0) & \qquad (5) \\ \sin(2 \pi f_0 t) & \qquad \dfrac{1}{2\iu} \delta(f - f_0) - \dfrac{1}{2\iu} \delta(f + f_0) & \qquad (6) \\ \rect(t/t_0) & \qquad t_0 \sinc(t_0 f) & \qquad (7) \\ \sinc(t/t_0) & \qquad t_0 \rect(t_0 f) & \qquad (8) \\ \tri(t/t_0) & \qquad t_0 \sinc^2(t_0 f) & \qquad (9) \\ \sinc^2 (t/t_0) & \qquad t_0 \tri(t_0 f) & \qquad (10) \\ \sign(t) & \qquad \dfrac{1}{\iu \pi f} & \qquad (11) \\ \dfrac{1}{t} & \qquad -\iu \pi \sign(f) & \qquad (12) \\ \step(t) & \qquad \dfrac{1}{\iu 2 \pi f} + \dfrac{1}{2} \delta(f) & \qquad (13) \\ \mathrm{J}_0(t/t_0) & \qquad \dfrac{2 t_0 \rect(\pi t_0 f)}{\sqrt{1 - (2 \pi t_0 f)^2}} & \qquad (14) \\ \displaystyle{\sum_{n=-\infty}^{\infty} \delta(t - n T)} & \qquad \dfrac{1}{T} \displaystyle{\sum_{n=-\infty}^{\infty} \delta \left (f - \frac{n}{T} \right)} & \qquad (15) \\ \ee^{-t/t_0} \step(t) & \qquad \dfrac{t_0}{1 + \iu 2 \pi t_0 f} & \qquad (16) \\ \ee^{-|t/t_0|} & \qquad \dfrac{2 t_0}{1 + (2 \pi t_0 f)^2} & \qquad (17) \\ \ee^{-\frac{1}{2}(t/t_0)^2} & \qquad t_0 \sqrt{2 \pi} \ee^{-\frac{1}{2} (2 \pi t_0 f)^2} & \qquad (18) \end{alignedat} \]