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Ann Example

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Ann Example

Series Results

$\begin{eqnarray*} \sum _{x=0}^{\infty } q^x & =& \frac{1}{1-q}, \ |q|<1. \\ \frac{d}{dx}\left(\sum _{x=0}^{\infty } q^x \right) & =& \sum _{x=0}^{\infty } xq^{x-1}=\sum _{x=1}^{\infty } xq^{x-1} = \frac{1}{(1-q)^2}, \ |q|<1. \\ \sum _{x=0}^{N} q^x & =& \frac{1-q^{N+1}}{1-q}. \\ \sum _{n=0}^{\infty } \frac{x^n}{n!} & =& 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots = e^x \end{eqnarray*}$

Expectation

Discrete Random Variables

$\begin{eqnarray*} E[g(X)] & =& \sum _{x \in S} g(x) Pr(X=x) \\ Var(X) & =& E[X^2] - E[X]^2. \end{eqnarray*}$

Continuous Random Variables

$\begin{eqnarray*} E[g(X)] & =& \int _{-\infty }^{\infty } g(x) f_X(x) dx \\ Var(X) & =& E[X^2]-E[X]^2. \end{eqnarray*}$

Families of Discrete Random Variables I

Bernoulli random variables, $X \sim Bern(p)$

$\begin{eqnarray*} Pr(X=x) & =& \left\{ \begin{array}{cc} p, & x=1, \\ 1-p, & x=0. \end{array} \right. \end{eqnarray*}$ \[ E[X] = p \qquad Var(X) = p(1-p). \]

Binomial random variables, $X \sim Bin(n,p)$

$\begin{eqnarray*} Pr(X=x) & =& \left( \begin{array}{c} n \\ x \end{array} \right) p^x (1-p)^{n-x},\ x=0,1,\ldots ,n, \end{eqnarray*}$ \[ E[X] = np \qquad Var(X) = np(1-p). \]

Poisson random variables, $X \sim Po(\lambda ), \ \lambda {\gt}0$

$\begin{eqnarray*} Pr(X=x) & =& \frac{\lambda ^x e^{-\lambda }}{x!}, \ x=0,1,2,\ldots . \end{eqnarray*}$ \[ E[X] = \lambda \qquad Var(X) = \lambda . \]

Families of Discrete Random Variables II

Geometric random variables, $X \sim Geom(p)$

$\begin{eqnarray*} Pr(X=x) & =& p(1-p)^{x-1}, \ x=1,2,\ldots . \end{eqnarray*}$ \[ E[X] = \frac{1}{p} \qquad Var(X) = \frac{1-p}{p^2}. \]

Negative Binomial random variables, $X \sim NBin(r,p)$

$\begin{eqnarray*} Pr(X=x) & =& \left( \begin{array}{cc} x-1 \\ r-1 \end{array} \right) p^r (1-p)^{n-r}, \ x=r,r+1,\ldots . \end{eqnarray*}$ \[ E[X] = \frac{r}{p} \qquad Var(X) = \frac{r(1-p)}{p^2}. \]

Discrete Uniform random variables, $X \sim U(1,2,\ldots ,n)$

$\begin{eqnarray*} Pr(X=x) & =& \frac{1}{n}, \ x=1,2,\ldots ,n. \end{eqnarray*}$ \[ E[X] = (n+1)/2 \qquad Var(X) = (n^2-1)/12. \]

Families of Continuous Random Variables I

Uniform random variables, $X \sim U(a,b)$,

$\begin{eqnarray*} f_X(x) & =& \frac{1}{b-a}, \ a < x < b. \end{eqnarray*}$ \[ E[X]=\frac{a+b}{2} \qquad Var(X) = \frac{(b-a)^2}{12}. \]

Exponential random variables, $X \sim Exp(\lambda )$,

$\begin{eqnarray*} f_X(x)& =& \lambda e^{-\lambda x}, \ x>0, \ \lambda >0. \end{eqnarray*}$ \[ E[X]= \frac{1}{\lambda } \qquad Var(X) = \frac{1}{\lambda ^2}. \]

Normal random variables, $X \sim N(\mu ,\sigma ^2)$,

\[ f_X(x) = \frac{1}{\sqrt{2\pi \sigma ^2}} e^{-\frac{(x-\mu )^2}{2\sigma ^2}}, \ -\infty {\lt} x {\lt} \infty , \ -\infty {\lt} \mu {\lt} \infty , \ \sigma {\gt}0. \] \[ E[X]= \mu \qquad Var(X) = \sigma ^2. \]

Families of Continuous Random Variables II

Gamma random variables, $X \sim Ga(n,\lambda )$,

\[ f_X(x) = \frac{\lambda ^{n}}{\Gamma (n)} x^{n-1} e^{-\lambda x}, \qquad x{\gt}0, \ n{\gt}0,\ \lambda {\gt}0, \ \Gamma (n)=(n-1)! \] \[ E[X] = \frac{n}{\lambda } \qquad Var(X) = \frac{n}{\lambda ^2}. \]

Beta random variables, $X \sim Beta(a,b)$,

$\begin{eqnarray*} f_X(x) & =& \frac{1}{B(a,b)} x^{a-1}(1-x)^{b-1} \\ & =& \frac{\Gamma (a+b)}{\Gamma (a)\Gamma (b)} x^{a-1}(1-x)^{b-1},\qquad 0<x<1, \ a>0, \ b >0. \end{eqnarray*}$ \[ E[X]=\frac{a}{a+b} \qquad Var(X) = \frac{ab}{(a+b)^2(a+b+1)}. \]

Column example

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