8 Jan, 2016
Distributions
\[\begin{array}{rcl}
\text{Gamma(n,2)} &=&\chi^2_{2n} \\\\
U \sim \text{Unif(0,1)} &\Rightarrow& -log(U) \sim \text{Exp(1)} \\\\
\frac{\chi^2_p/p}{\chi^2_q/q} &\sim& F_{p,q} \\\\
\frac{\text{N(0,1)}}{\sqrt{\chi^2_p/p}} &=& t_p \\\\
(t_p)^2 &=& F_{1,p} \\\\
\text{N(0,1)}^2 &=& \chi^2_1 \\\\
\sum_{i=1}^n{\text{N(0,1)}^2} &=& \chi^2_n \\\\
\frac{(n-1)S^2}{\sigma^2} &=& \chi^2_{n-1},
\text{ where } S^2 = \frac{\sum_{i=1}^n {x_i-\bar x}}{n-1}\\\\
X \sim \text{Gamma}(\alpha,\text{scale}=\beta) &\Rightarrow& \frac{1}{X} \sim \text{InvGamma}(\alpha,\beta^{-1}) \\\\
\text{F(x)} &\sim& \text{Unif(0,1)} \\\\
\text{1-F(x)} &\sim& \text{Unif(0,1)} \\\\
\text X \sim F_{p,q} &\iff& \frac{(p/q) X}{1+(p/q) X} \sim \text{Beta(p/2, q/2)} \\\\
%\begin{cases}
% X \sim \text{Bin(n,p)} \\\\
% Y\|X \sim \text{Beta(x, n-x+1)}
%\end{cases}
%&\Rightarrow& \text{P(X$\ge$x)} = \text{P(Y$\le\theta$)} \\\\
\sum_{i=1}^n (x_i-\mu)^2 &=& \sum_{i=1}^n (x_i-\bar{x})^2 + n(\bar x-\mu)^2\\\\
\Gamma(1/2) &=& \sqrt{2\pi}\\\\
\sum_{i=1}^k N(\mu_i^2,1) &\sim& {\chi^2}_k(\sum_{i=1}^k \mu_i^2)\\\\
\\\\
\\\\
aX_1 + bX_2 &\sim& N((a+b)\mu,(a^2+b^2)\sigma^2)\\\\
AX + b &\sim& N(\mu+b,~A\Sigma A')\\\\
E[\bf y | \bf x] &=& \mu_y + \Sigma_{yx}\Sigma_{xx}^{-1}(\bf x-\mu_x) \\\\
cov[\bf y | \bf x] &=& \Sigma_{yy} - \Sigma_{yx}\Sigma_{xx}^{-1}\Sigma_{xy} \\\\
\end{array}\]
- If $h(Y) = X$, then $f_Y(y) = f_x[ h(y) ] |h’(y)|$
- If $X = g(U,V)$ and $Y = h(U,V)$, then
\(f_{U,V}(u,v) = f_{X,Y}(g(u,v),h(u,v))\norm{J},\)
where $J$ =
\(\begin{vmatrix}
g_u & g_v \\\\
h_u & h_v \\\\
\end{vmatrix}\)
Order Statistics
- $F_{X_{(j)}}(x) = P(Y\le j) = \displaystyle\sum_{k=j}^n {n\choose k}[F_X(x)]^k[1-F_X(x)]^{n-k}$
- $f_{X_{(j)}}(x) = \displaystyle \frac{n!}{(j-1)!(n-j)!}f_X(x)[F_X(x)]^{j-1}[1-F_X(x)]^{n-j}$
- $f_{X_{(i)},X_{(j)}}(u,v) = \displaystyle\frac{n!}{(i-1)!(j-1-i)!(n-j)!}f_X(u)f_X(v)[F_X(u)]^{i-1}[F_X(v)-F_X(u)]^{j-1-i}[1-F_X(v)]^{n-j}$, for $i \lt j$
- $f_{ X_{(1)} ,…, X_{(n)} } (x_1,…,x_n) = n! f_X(x_1) … f_X(x_n)$
Integrals and finding bounds
Resource for computing integrals to find areas and how to find bounds.