| Dist. | PMF/PDF | CDF | Mean | Variance | Inv. CDF |
|---|---|---|---|---|---|
| Bernoulli\((p)\) | \(p(x)=p^x(1-p)^{1-x},\ x=0,1\) | \(F(x)=\begin{cases}0,&x<0\\1-p,&0\le x<1\\1,&x\ge1\end{cases}\) | \(p\) | \(p(1-p)\) | |
| Binomial\((n,p)\) | \(p(x)=\binom{n}{x}p^x(1-p)^{n-x},\ x=0,\ldots,n\) | \(F(x)=\sum_{i=0}^{\lfloor x\rfloor}\binom{n}{i}p^i(1-p)^{n-i}\) | \(np\) | \(np(1-p)\) | |
| Poisson\((\lambda)\) | \(p(x)=\dfrac{\lambda^x e^{-\lambda}}{x!},\ x=0,1,2,\ldots\) | \(F(x)=e^{-\lambda}\sum_{i=0}^{\lfloor x\rfloor}\dfrac{\lambda^i}{i!}\) | \(\lambda\) | \(\lambda\) | |
| Geometric\((p)\) | \(p(x)=p(1-p)^{x-1},\ x=1,2,\ldots\) | \(F(x)=1-(1-p)^{\lfloor x\rfloor}\) | \(\dfrac{1}{p}\) | \(\dfrac{1-p}{p^2}\) | \(x=\left\lceil\dfrac{\log(1-u)}{\log(1-p)}\right\rceil\) |
| Negative Binomial\((r,p)\) | \(p(x)=\binom{x-1}{r-1}p^r(1-p)^{x-r},\ x=r,r+1,\ldots\) | \(F(x)=\sum_{i=r}^{\lfloor x\rfloor}\binom{i-1}{r-1}p^r(1-p)^{i-r}\) | \(\dfrac{r}{p}\) | \(\dfrac{r(1-p)}{p^2}\) | |
| Simple PMF | \(P(X=x_i)=p_i,\ i=1,\ldots,k\) | \(F(x)=\sum_{x_i\le x}p_i\) | \(\sum_i x_ip_i\) | \(\sum_i x_i^2p_i-\left(\sum_i x_ip_i\right)^2\) | Choose \(x_i\) if \(F(x_{i-1})<u\le F(x_i)\) |
| Logarithmic\((\theta)\) | \(p(x)=\dfrac{-1}{\log(1-\theta)}\dfrac{\theta^x}{x},\ x=1,2,\ldots\) | \(F(x)=\dfrac{-1}{\log(1-\theta)}\sum_{i=1}^{\lfloor x\rfloor}\dfrac{\theta^i}{i}\) | \(\dfrac{-\theta}{(1-\theta)\log(1-\theta)}\) | \(\dfrac{-\theta(\theta+\log(1-\theta))}{(1-\theta)^2[\log(1-\theta)]^2}\) | Choose \(x\) if \(F(x-1)<u\le F(x)\) |
Bernoulli:
\(X\) is the outcome of a single
success/failure trial (e.g., one coin toss).
Binomial:
\(X\) is the number of successes in
\(n\) independent Bernoulli trials
(e.g., number of heads in \(n\) coin
tosses).
Poisson:
\(X\) is the number of events occurring
in a fixed interval (e.g., number of phone calls per hour).
Geometric:
\(X\) is the trial on which the first
success occurs (e.g., number of coin tosses until the first head).
Negative Binomial:
\(X\) is the trial on which the \(r^{\text{th}}\) success occurs (e.g.,
number of coin tosses until the fifth head).
Simple PMF:
\(X\) is a discrete random variable
with arbitrary finite probabilities (e.g., outcomes of a loaded
die).
Logarithmic:
\(X\) is a count random variable with a
logarithmic distribution (e.g., species abundance or zero-truncated
count data).
| Dist. | PMF/PDF | CDF | Mean | Variance | Inv. CDF |
|---|---|---|---|---|---|
| Standard Uniform \(U(0,1)\) | \(f(x)=1,\ 0<x<1\) | \(F(x)=x,\ 0<x<1\) | \(\dfrac{1}{2}\) | \(\dfrac{1}{12}\) | \(x=u\) |
| Uniform\((a,b)\) | \(f(x)=\dfrac{1}{b-a},\ a<x<b\) | \(F(x)=\dfrac{x-a}{b-a},\ a<x<b\) | \(\dfrac{a+b}{2}\) | \(\dfrac{(b-a)^2}{12}\) | \(x=a+(b-a)u\) |
| Polynomial | \(f(x)=kx^{k-1},\ 0<x<1,\ k>0\) | \(F(x)=x^k\) | \(\dfrac{k}{k+1}\) | \(\dfrac{k}{(k+2)(k+1)^2}\) | \(x=u^{1/k}\) |
| Exponential\((\lambda)\) | \(f(x)=\lambda e^{-\lambda x},\ x\ge0\) | \(F(x)=1-e^{-\lambda x}\) | \(\dfrac{1}{\lambda}\) | \(\dfrac{1}{\lambda^2}\) | \(x=-\dfrac{1}{\lambda}\log(1-u)\) |
| Rayleigh \(\left(\sigma=\dfrac{1}{\sqrt2}\right)\) | \(f(x)=2xe^{-x^2},\ x\ge0\) | \(F(x)=1-e^{-x^2}\) | \(\dfrac{\sqrt{\pi}}{2}\) | \(1-\dfrac{\pi}{4}\) | \(x=\sqrt{-\log(1-u)}\) |
| Beta\((\alpha,\beta)\) | \(f(x)=\dfrac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1},\ 0<x<1\) | \(F(x)=I_x(\alpha,\beta)\) | \(\dfrac{\alpha}{\alpha+\beta}\) | \(\dfrac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\) | |
| \(\sin(x)\) on \(\left[0,\dfrac{\pi}{2}\right]\) | \(f(x)=\sin(x),\ 0\le x\le\dfrac{\pi}{2}\) | \(F(x)=1-\cos(x)\) | \(1\) | \(\pi-3\) | \(x=\cos^{-1}(1-u)\) |
| \(\cos(x)\) on \(\left[0,\dfrac{\pi}{2}\right]\) | \(f(x)=\cos(x),\ 0\le x\le\dfrac{\pi}{2}\) | \(F(x)=\sin(x)\) | \(\dfrac{\pi}{2}-1\) | \(2-\dfrac{\pi^2}{4}\) | \(x=\sin^{-1}(u)\) |
Standard Uniform \(U(0,1)\):
\(X\) is equally likely to fall
anywhere on \((0,1)\) (e.g., output of
a random number generator).
Uniform\((a,b)\):
\(X\) is equally likely to fall
anywhere on \((a,b)\) (e.g., random
arrival time within an interval).
Polynomial:
\(X\) has density proportional to \(x^{k-1}\) on \((0,1)\) (e.g., inverse CDF sampling
example).
Exponential\((\lambda)\):
\(X\) is the waiting time until the
first event in a Poisson process (e.g., time until the next customer
arrives).
Beta\((\alpha,\beta)\):
\(X\) is a random proportion or
probability on \((0,1)\) (e.g.,
Bayesian prior for a coin’s bias).
CDF \[ F_X(x)=P(X\le x) \]
MSE \[ \operatorname{MSE}(\hat{\theta}) = E[(\hat{\theta}-\theta)^2] = \operatorname{Var}(\hat{\theta}) +\operatorname{Bias}(\hat{\theta})^2 \]
Linear Transformation
If
\[ X\sim U(a,b), \]
then
\[ Y=\alpha X+\beta \sim U(\alpha a+\beta,\ \alpha b+\beta), \qquad (\alpha>0) \]
ECDF \[ \hat F_n(x_0) = \frac1n\sum_{i=1}^nI(X_i\le x_0) \]
\[ E[\hat F_n(x_0)] = F(x_0)=p \]
\[ \operatorname{Var}(\hat F_n(x_0)) = \frac1nF(x_0)\bigl(1-F(x_0)\bigr) \]
\[ I(X_i\le x_0)\sim\mathrm{Bernoulli}(p), \qquad \sum_{i=1}^nI(X_i\le x_0)\sim\mathrm{Binomial}(n,p) \]
Consistent Estimator \[ \hat\theta_n\xrightarrow{P}\theta \qquad\Longleftrightarrow\qquad P(|\hat\theta_n-\theta|>\varepsilon)\rightarrow0 \]
Inverse CDF \[ U\sim U(0,1) \quad\Longrightarrow\quad X=F^{-1}(U) \]
Rejection Sampling \[ f(x)\le Mg(x) \]
\[ 0\le\frac{f(x)}{Mg(x)}\le1 \]
\[ U<\frac{f(X)}{Mg(X)} \Longrightarrow \text{Accept} \]
Recursive Geometric PMF \[ p(1)=p, \qquad p(x)=(1-p)p(x-1),\ x\ge2 \]
Recursive Logarithmic PMF \[ p(1)=\frac{-\theta}{\log(1-\theta)}, \qquad p(x)=\frac{x-1}{x}\theta\,p(x-1),\ x\ge2 \]