UA MATH564 概率分布總結

Part zero Continuous Uniform Distribution
Tip 1: $X～U(a,b)$, $F(x)=\frac{x?a}{b?a},f(x)=\frac{1}{b?a}$
Tip 2: First four moments

$EX$$Var(X)$$E{X}^2$$E{X}^3$$E{X}^4$

$\frac{b?a}{2}$	$\frac{(b?a{)}^2}{12}$	$\frac{b^3?a^3}{3(b?a)}$	$\frac{b^4?a^4}{4(b?a)}$	$\frac{b^5?a^5}{5(b?a)}$

Tip 3: $X_{(i)}～Beta(i,n?i+1)$, $(X_{(i)},X_{(j)})～Dirichlet(i,j?i,n?j+1),fori<j$
Tip 4: $X_{(1)},X_{(n)}$ are complete minimal sufficient statistics.
Tip 5: $X_{(1)},X_{(n)}$ are MLE of $a,b$.

Tip 6: A special case $Y～U(0,\theta )$,

$U(0,\theta )$ is Complete. $X_{(n)}$ is complete minimal sufficient statistics.

MLE is $X_{(n)}～\theta ?Beta(n,1)$

MME is $2\bar{X}$ (approximately $N(\theta ,\frac{{\theta }^2}{3n})$)

UMVUE is $\frac{n+1}{n}{X}_{(n)}$

Part a Poisson Distribution
Tip 1: $X～Pois(\lambda ),f(x)=\frac{{\lambda }^x{e}^{?\lambda }}{x!},x=0,1,?$
Tip 2: $M_X(t)=\exp (\lambda (e^t?1))$ and first four moments

$EX$$Var(X)$skewnesskurtosis

$\lambda$	$\lambda$	${\lambda }^{?1/2}$	${\lambda }^{?1}$

Tip 3: Additivity, $X_i～Pois({\lambda }_i),i=1,2,?,n$ and all $X_i$s are independent, and then ${\sum }_{i=1}^n{X}_i～Pois({\sum }_{i=1}^n{\lambda }_i)$
Tip 4: $X_1～Pois({\lambda }_1)$ and $X_2～Pois({\lambda }_2)$ are independent, and then $X_1|X_1+X_2=x～Binom(x,\frac{{\lambda }_1}{{\lambda }_1+{\lambda }_2})$
Tip 5: MLE and MME are both $\bar{X}$, $n\bar{X}～Pois(n\lambda )$
Tip 6: $\bar{X}$ is UMVUE, and complete minimal sufficient statistics.

Part b Bernoulli Distribution
Tip 1: $X～Ber(p),f(x)=p^x(1?p{)}^{1?x},x=0,1$

Tip 2: $M_X(t)=\exp (\lambda (e^t?1))$ and first four moments

$EX$$Var(X)$skewnesskurtosis

$p$	$p(1?p)$	$\frac{1?2p}{\sqrt{p(1?p)}}$	$\frac{6p^2?6p+1}{p(1?p)}$

Tip 3: Additivity, $X_i{～}_{iid}Ber(p),i=1,?,n$, ${\sum }_{i=1}^n{X}_i～Binom(n,p)$
Tip 4: MME and MLE are both $\bar{X}$, $n\bar{X}～Binom(n,p)$
Tip 5: $\bar{X}$ is UMVUE

Part c Negative Binomial Distribution
Tip 1: Number of failed trials to get $r$ successfal trials. $X～NB(r,p),f(x)=C_{r+x?1}^x{p}^r(1?p{)}^x,x=0,1,?$
Tip 2: $M_X(t)={\left(\frac{p{e}^t}{1?(1?p)e^t}\right)}^r$ and first four moments

$EX$$Var(X)$skewnesskurtosis

$\frac{r(1?p)}{p}$	$\frac{r(1?p)}{p^2}$	$\frac{2?p}{\sqrt{r(1?p)}}$	$\frac{6}{r}+\frac{p^2}{r(1?p)}$

Tip 3: Additivity, $X_i{～}_{iid}Geo(p),i=1,?,n$, ${\sum }_{i=1}^n{X}_i～NB(n,p)$
Tip 4: $NB(1,p)$ is $Geo(p)$ (Number of failed trials before the first success)
Tip 5: A special case $Y～NB(r,p)$ where $r$ is known constant. MLE and MME are both $r/(r+\bar{X})$, $n\bar{X}～NB(nr,p)$.

Part d Exponential Distribution
Tip 1: $X～EXP(\lambda ),f(x)=\lambda e^{?\lambda x},x>0,F(x)=e^{?\lambda x},x>0$
Tip 2: $M_X(t)={\left(1?\frac{t}{\lambda }\right)}^{?1}$ and first four moments

$EX$$Var(X)$skewnesskurtosis

$\frac{1}{\lambda }$	$\frac{1}{{\lambda }^2}$	$2$	$6$

Tip 3: $EXP(\lambda ){=}_d\Gamma (1,\lambda )$
Tip 4: Additivity, $X_i{～}_{iid}EXP(\lambda ),i=1,?,n$, ${\sum }_{i=1}^n{X}_i～\Gamma (n,\lambda )$
Tip 5: Scale family, $cX～EXP(\lambda /c)$
Tip 6: $\bar{X}$ is MLE and MME of $1/\lambda$ and also $1/\bar{X}$ is MLE and MME of $\lambda$, $n\bar{X}～\Gamma (n,\lambda )$

Part e Gamma Distribution
Tip 1: $X～\Gamma (\alpha ,\lambda ),f(x)=\frac{{\lambda }^{\alpha }{x}^{\alpha ?1}}{\Gamma (\alpha )}{e}^{?\lambda x},x>0$, $\Gamma (\alpha )={\int }_0^{+\infty }{x}^{\alpha ?1}{e}^{?x}dx$, $\Gamma (n)=(n?1)!,\Gamma (1)=1,\Gamma (1/2)=\sqrt{\pi }$
Tip 2: $M_X(t)={\left(1?\frac{t}{\lambda }\right)}^{?\alpha }$ and first four moments

$EX$$Var(X)$skewnesskurtosis

$\frac{\alpha }{\lambda }$	$\frac{\alpha }{{\lambda }^2}$	$\frac{2}{\sqrt{\alpha }}$	$\frac{6}{\alpha }$

Tip 3: Additivity, $\Gamma ({\alpha }_1,\lambda )+\Gamma ({\alpha }_2,\lambda )=\Gamma ({\alpha }_1+{\alpha }_2,\lambda )$ (Independence)
Tip 4: $\Gamma (1,\lambda )=EXP(\lambda ),\Gamma (\frac{n}{2},\frac{1}{2})={\chi }_n^2$
Tip 5: Scale family, $cX～\Gamma (\alpha ,\lambda /c)$

Part f Beta Distribution
Tip 1: $X～\mathrm{B}(\alpha ,\beta ),f(x)=\frac{\Gamma (\alpha +\beta )}{\Gamma (\alpha )\Gamma (\beta )}{x}^{\alpha ?1}(1?x{)}^{\beta ?1},x\in (0,1)$
Tip 2: $E{X}^k=\frac{(\alpha {)}_k}{(\alpha +\beta {)}_k}$, first two moments

$EX$$Var(X)$

$\frac{\alpha }{\alpha +\beta }$	$\frac{\alpha \beta }{(\alpha +\beta {)}^2(\alpha +\beta +1)}$

Part g Normal Distribution
Tip 1: $X～N(\mu ,{\sigma }^2),f(x)=\frac{1}{\sqrt{2\pi }}{e}^{?\frac{(x?\mu {)}^2}{2{\sigma }^2}}$
Tip 2: $M_X(t)=\exp \left(\mu t+\frac{{\sigma }^2}{2}{t}^2\right)$ and first four moments

$EX$$Var(X)$skewnesskurtosis

$\mu$	${\sigma }^2$	0	0

Tip 3: Additivity\ Linearity
Tip 4: $(\bar{X},S^2)$ are OLS, UMVUE, complete minimal sufficient statistics, $\bar{X}～N(\mu ,{\sigma }^2/n)$, $(n?1)S^2～{\sigma }^2?{\chi }_{n?1}^2$ or $S^2～\Gamma (\frac{n?1}{2},\frac{n?1}{2{\sigma }^2})$
Tip 5: $X_i{～}_{iid}N(0,1),{\sum }_{i=1}^n{X}_i^2～{\chi }_n^2$
Tip 6: $X,Y{～}_{iid}N(0,{\sigma }^2),\sqrt{X^2+Y^2}～Rayleigh({\sigma }^2),X/Y～Cauchy(0,1)$

Part h Log Normal Distribution
Tip 1: $X～LN(\mu ,{\sigma }^2)$, $f(x)=\frac{1}{\sqrt{2\pi }\sigma x}\exp (?\frac{(\ln x?\mu {)}^2}{2{\sigma }^2}),x>0$
Tip 2: $E{X}^k=\exp (\mu k+\frac{{\sigma }^2}{2}k)$
Tip 3: $X_1～LN({\mu }_1,{\sigma }_1^2),X_2～LN({\mu }_2,{\sigma }_2^2)$, $X_1{X}_2～LN({\mu }_1+{\mu }_2,{\sigma }_1^2+{\sigma }_2^2)$ (Independence)
Tip 4: $(\frac{1}{n}{\sum }_{i=1}^n\ln X_i,\frac{1}{n?1}{\sum }_{i=1}^n(\ln X_i?\overline{\ln X}{)}^2)$ are OLS, UMVUE, complete minimal sufficient statistics. The former follows $N(\mu ,{\sigma }^2/n)$ and the latter follows $\Gamma (\frac{n?1}{2},\frac{n?1}{2{\sigma }^2})$

Part i Cauchy Distribution
Tip 1: $X～Cauchy(x_0,\gamma ),f(x)=\frac{1}{\pi }\frac{\gamma }{(x?x_0{)}^2+{\gamma }^2}$, $F(x)=\frac{1}{\pi }\arctan ((x?x_0)/\gamma )+1/2$
Tip 2: No moments
Tip 3: $Cauchy(0,1){=}_{d}t(1)$

Part j Rayleigh Distribution
Tip 1: $X～Ray(\sigma ),f(x)=\frac{x}{{\sigma }^2}{e}^{?\frac{x^2}{2{\sigma }^2}},x\ge 0$
Tip 2: first two moments

$EX$$Var(X)$

$\frac{\pi }{2}\sigma$	$\frac{4?\pi }{2}{\sigma }^2$

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