Moments

$E(aX+b)=aE(X)+b$

$Var(X)=E(X^2)-\{E(X)\}^2$

$Var(aX+b)=a^2Var(X)$

$Cov(X,Y)=E(XY)-E(X)E(Y)$

$Cov(X,X)=Var(X)$

$Cov(aX+b,cY+d)=acCov(X,Y)$

$Cov(X+Y,Z)=Cov(X,Z)+Cov(Y,Z)$

Definition of distributions

• $\chi^2(v)=Gamma(v/2, 2)$
• $Z\sim(0,1), W\sim \chi^2(\nu) \rightarrow T=\frac{Z}{\sqrt{W/\nu}}\sim t(\nu)$
• $W_1 \sim \chi^2(\nu_1), W_2 \sim \chi^2(\nu_2) \rightarrow F=\frac{W_1/\nu_1}{W_2/\nu_2}\sim F (\nu_1, \nu_2)$

Properties of some distributions

• $X \sim Possion(\lambda)$

  $E(X)=Var(X)=\lambda$

  $E[X(X-1)(X-2)...(X-k)]=\lambda^{k+1}$

  $X_i \sim Poisson(\lambda_i)$

  $\sum^n_iX_i \sim Poisson(\sum^n_i\lambda_i)$

• $Gamma(\alpha_1, \beta) + Gamma(\alpha_2, \beta)=Gamma(\alpha_1+\alpha_2, \beta)$

• $Y \sim Exponential(\theta)=Gamma(1, \theta)$

  $Y_N =:Y_1 +...+Y_n=Gamma(n,\theta)$

  $cY_N \sim Gamma(n,c\theta)$

• $Z \sim N(0,1)$

  $Z^2 \sim \chi^2(1)=Gamma(1/2,2)$

  $Z_1 \sim N(\mu_1, \sigma^2), Z_2 \sim N(\mu_2, \sigma^2)$

  $Z_1 \pm Z_2 \sim N(\mu_1 \pm \mu_2, \sigma^2)$

• $X \sim Geo(p)$

  $\sum^r_iX_i \sim NegBin(r,p)$

CDF Method

• $Z \sim N(0,1), Z^2\sim Gamma(1/2, 2)=\chi^2(1)$