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click Inference Time Series Numerical Methods


 * $$ P_r^n = \frac{n!}{(n-r)!} $$
 * $$ C_n^k = {n \choose k} = \frac{n!}{k!(n-k)!}.$$
 * $$\int f'(x)g(x)\,dx = f(x)g(x) - \int f(x)g'(x)\,dx$$



\int_{a}^{b} f(\phi(t)) \phi'(t)\,dt = \int_{\phi(a)}^{\phi(b)} f(x)\,dx $$ x = φ(t) yields dx / dt = φ'(t) dx = φ'(t) dt
 * $$\int {f'(x)\over f(x)}\,dx= \ln{\left|f(x)\right|} + C $$
 * $$\int {f'(x) f(x)}\,dx= {1 \over 2} [ f(x) ]^2 + C $$
 * $$\int [f(x)]^n f'(x)\,dx = {[f(x)]^{n+1} \over n+1} + C \qquad\mbox{(for } n\neq -1\mbox{)}\,\! $$
 * $$\int (ax + b)^n dx$$ || $$ = \frac{(ax + b)^{n+1}}{a(n + 1)} \qquad\mbox{(for } n\neq -1\mbox{)}\,\!$$
 * $$\int\frac{1}{ax + b} dx$$ || $$ = \frac{1}{a}\ln\left|ax + b\right|$$
 * $$\int\ln cx\;dx = x\ln cx - x$$
 * $$\int x^n\,{\rm d}x = \frac{x^{n+1}}{n+1} + C\qquad\mbox{ if }n \ne -1$$
 * $$\int \ln {x}\,dx = x \ln {x} - x + C$$
 * $$\int \log_b {x}\,dx = x\log_b {x} - x\log_b {e} + C$$
 * $$\int e^x\,dx = e^x + C$$
 * $$\int a^x\,dx = \frac{a^x}{\ln{a}} + C$$
 * $$\int \sin{x}\, dx = -\cos{x} + C$$
 * $$\int \cos{x}\, dx = \sin{x} + C$$
 * Reimann $$ \lim_{n->\infty} { \sum_{k=1}^{n}{f(k/n)} \over n } = \int_{0}^{1} f(x)dx$$

Derivs

 * $$ \frac{d}{dx}f(g(x)) = f'(g(x)) g'(x)$$


 * $$\left({fg}\right)' = f'g + fg'$$ $$\left({f \over g}\right)' = {f'g - fg' \over g^2}, \qquad g \ne 0$$


 * $${d \over dx} |x| = {|x| \over x} = \sgn x $$


 * $${d \over dx} c^x = {c^x \ln c }, c > 0$$
 * $${d \over dx} e^x = e^x$$
 * $${d \over dx} \log_c x = {1 \over x \ln c},\qquad c > 0, c \ne 1$$
 * $${d \over dx} \ln x = {1 \over x},\qquad x > 0$$
 * $${d \over dx} \ln |x| = {1 \over x}$$
 * $${d \over dx} x^x = x^x(1+\ln x)$$


 * $${d \over dx} \sin x = \cos x$$
 * $${d \over dx} \cos x = -\sin x$$

Series
Taylor: $$ f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+\cdots = \sum_{n=0}^{\infin} \frac{f^{(n)}(a)}{n!} (x-a)^{n} $$


 * 1) $$\sum_{j=1}^m=1+2+3+\cdots+m=\frac{m(m+1)}{2} $$
 * 2) $$\sum_{j=1}^mj^2=1^2 + 2^2 + 3^2 + \cdots + m^2 = \frac{m(m+1)(2m+1)}{6} $$


 * 3) $$ e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots,$$


 * 4)$$ \sin(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}+\cdots,$$
 * 5)$$ \cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots = \sum_{j=0}^\infty \frac{(-1)^j x^{2j}}{(2j)!}$$
 * 6)$$ \frac{1}{1-x} = \sum_{n=0}^\infty x^n = 1 + x + x^2 + x^3 + \cdots,$$ $$|x|<1$$
 * 7)$$ ln(1-x) = -x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} - \cdots = - \sum_{j=1}^\infty \frac{x^j}{j} $$
 * 8)$$ (a+b)^n=\sum_{k=0}^n {n \choose k} a^kb^{n-k} $$
 * $$ \sum_{n=1}^{\infty} np^{n-1} = \frac {1} { (1-p)^2 }

$$

Probability

 * $$ E[X]=\int\limits_{-\infty}^{\infty} x(fx)dx $$
 * $$ E(g(x))=\sum{g(x)f(x)}$$
 * $$E[E(X|Y)] = \sum_{y} E(X|Y=y)*P(Y=y)$$
 * $$V[X]=E[V[X|Y]] + V[E[X|Y]] $$
 * $$T=X+Y, f_T(t)=\sum_{x}{f_x(x)f_y(t-x)}$$
 * $$ \operatorname{var}(X) = E(X^2) - [E(X)]^2 $$.
 * $$f(k;\lambda)=\frac{e^{-\lambda} \lambda^k}{k!},\,\!$$ e = 2.71828, k number of occurrences, λ>0 = expected number of occurrences that occur during the given interval
 * $$P(A|B)   =       \frac{P(B | A)\, P(A)}{P(B)} $$
 * $$E[X+Y]=E[X] + E[Y]$$
 * $$Cov(X,X) = V[X]$$
 * $$f_{X,Y}(x,y) = f_{X} (X) f_{Y} (Y) $$ X independent of Y, cov(X,Y) = 0 does not mean independent
 * $$ E[(( e^{cy} )^2)]=E[e^{2cy}]=M_y (t=2c) $$
 * $$ \Gamma( \frac{1}{2} ) = \sqrt{\pi} $$
 * $$ median P(X<m) = \frac{1}{2}$$

LIM

 * $$\lim_{n\to\infty}\left(1-{\lambda \over n}\right)^n=e^{-\lambda}.$$
 * Euler's $$re^{ix} = r\cos x + ri \sin x \,\!$$

Normal


\varphi_{\mu,\sigma}(x) = \frac{1}{\sigma\sqrt{2\pi}} \,\exp\biggl( -\frac{(x- \mu)^2}{2\sigma^2}\biggr) = \frac{1}{\sigma} \varphi\left(\frac{x - \mu}{\sigma}\right),\quad x\in\mathbb{R}, $$
 * Moment Gen$$M_X(t)= \exp\left(\mu\,t+\frac{\sigma^2 t^2}{2}\right)$$
 * Characteristic $$\chi_X(t)=\exp\left(\mu\,i\,t-\frac{\sigma^2 t^2}{2}\right)$$

Characteristic

 * $$\varphi_{X+Y}(t)=E\left(e^{it(X+Y)}\right)=E\left(e^{itX}e^{itY}\right)=E\left(e^{itX}\right)E\left(e^{itY}\right)=\varphi_X(t) \varphi_Y(t)$$

Log-Normal

 * $$f(x) = \frac{1}{x\sigma\sqrt{2\pi}}\exp\left(-\frac{\left[\ln(x)-\mu\right]^2}{2\sigma^2}\right)$$
 * $$F(x) = \frac{1}{2}+\frac{1}{2} \mathrm{erf}\left[\frac{\ln(x)-\mu}{\sigma\sqrt{2}}\right]$$
 * $$Mean=e^{\mu+\sigma^2/2}$$
 * $$Var=(e^{\sigma^2}\!\!-1) e^{2\mu+\sigma^2}$$
 * $$moment \mu_k=e^{k\mu+k^2\sigma^2/2}$$

Jacobian
$$(U,V)=g(X,Y)=(X+Y, X-Y); g^{-1}(U,V)=(\frac{1}{2}(U+V), \frac{1}{2}(U-V))$$ $$f_{U,V}(U,V)=f_{X,Y}(g^{-1}(U,V))*|J_{g^{-1}(U,V)}| $$