Page Description Equation
127 probability mass function If $X$ can assume at most a countable number of values $x_1,x_2,\ldots$ then it is a discrete random variable, and the probability mass function is defined as given. \[ p\left(x\right)=\Pr{\left\{X=x\right\}} \]
128 consequences for the probability mass function of a discrete random variable $X$ which can assume at most the countable number of values $x_1,x_2,\ldots$ \[ \begin{array}{ll} p\left( x_i \right) \geq 0 & i = 1,2,3,\ldots\\ p\left(x\right)=0 & \mathrm{all\ other\ values\ of\ x}\\ \sum\limits_{ i = 1 }^{ \infty } p \left( x_i \right) = 1 \end{array} \]
129 cumulative distribution function \[ F\left(a\right)=\sum_{x:x\le a} p\left(x\right) \]
129 constant cumulative distribution function. Holds if the CDF is constant, $x_1 \lt x_2 \lt x_2 \lt \cdots,$ and $x_{i-1} \le a \le x_i$ \[ p\left(a\right)=F\left(x_i\right)-F\left(x_{i-1}\right) \]
130 \[ \mu=E\left[X\right]=\sum_{x:p\left(x\right)>0} x p\left(x\right) \]
137 expectation of a discrete random variable also called the mean of $X$ and first moment of $X,$ and center of mass. This number is a constant.
134 expectation of a function of a discrete random variable \[ E \left[ g \left( X \right) \right] = \sum_{ x : p \left( x \right) \gt 0} g \left( x \right) p \left( x \right) \]
137 sum rule for expectation, $a$ and $b$ constants. \[ E \left[ aX + b \right] = aE \left[ X \right] + b \]
137 $n$th moment of $X$, power rule for expectation \[ E\left[X^n\right]=\sum_{x:p\left(x\right)>0}{x^np\left(x\right)} \]
139 variance for discrete random variable It is the second moment of $X-\mu$ or the moment of inertia \[ \begin{array}{l} \Var\left(X\right) &= E\left[\left( X - \mu^2 \right)\right]\\ &= E\left[X^2\right] - \mu^2 \end{array} \]
139 standard deviation \[ SD\left(X\right)=\sqrt{\Var\left(X\right)} \]
139 Bernoulli random variable $X$ can only assume two values, $0$ or $1.$ The parameter $p\in\left(0,1\right).$ The value $0$ is used for failure, $1$ for success. \[ p \left( i \right) = \left\{ \begin{array}{cl} 1-p & i=0\\ p & i=1 \end{array} \right. \]
140 binomial random variable, probability mass function \[ p\left(i\right)=\sum_{i=0}^{n}{\binom{n}{i}p^i\left(1-p\right)^{n-i}} \]
144
145		 binomial random variable, expectation \[ E \left[ X \right] = np \]
145 binomial random variable, variance \[ \begin{array}{l} \Var\left(X\right) &= np \left(1 - p \right)\\ &= npq \end{array} \]
147 Stirling's approximation approximation of $n!$ for larger $n.$ \[ \begin{array}{l} n! &\approx n^{n+1/2} e^{-n} \sqrt{2 \pi}\\ &= n^n e^{-n} \sqrt{2 \pi n} \end{array} \]
149 Poisson random variable, probbility mass function If $X$ takes values $0, 1, 2, \ldots$ such that $\Pr{\left\{X=k\right\}}=\frac{e^{-\lambda}\lambda^k}{k!}, \lambda>0,$ then $X$ is a Poisson random variable with parameter $\lambda.$ \[ p\left(k\right)=\frac{e^{-\lambda}\lambda^k}{k!},\ \lambda>0 \]
Poisson random variable, expectation \[ E\left[X\right]=\lambda \]
Poisson random variable, variance \[ \Var \left( X \right) = \lambda \]
4.7 Poisson approximation of a binomial random variable If $np=\lambda$ remains constant as $n\rightarrow\infty.$ \[ p\left(k\right)\approx\frac{e^{-\lambda}\lambda^k}{k!},\ \lambda \gt 0 \]
geometric random variable, probability mass function \[ p\left(k\right)=\left(1-p\right)^{k-1}p,\quad k=1,2,3,\ldots \]
geometric random variable, expectation \[ E\left[X\right]=\frac{1}{p} \]
negative binomial random variable, probability mass function \[ p\left(k\right)=\binom{k-1}{r-1}p^r\left(1-p\right)^{k-r},\quad k\geq r \]
negative binomial random variable, expectation \[ E\left[X\right]=\frac{r}{p} \]
negative binomial random variable, variance \[ \Var \left( X \right) = \frac{ r \left(1-p \right)} { p^2 } \]
hypergeometric random variable
The Zeta (or Zipf) distribution
4.9 properties of the cumulative distribution function $F$ is a Distribution Function of the random variable $X$ if the conditions holds.
  1. $F \left( x \right) = \Pr{\left( X \le x \right)}$
  2. $F$ is non-decreasing
  3. $0 \lt F \lt 1$
  4. \( \begin{array}{ll} \lim\limits_{ x \rightarrow - \infty} F \left( x \right) = 0\\ \lim\limits_{ x \rightarrow \infty} F \left( x \right) = 1\\ \lim\limits_{ x \rightarrow a+ } F \left( x \right) = F \left(a \right)\\ \lim\limits_{ x \rightarrow a- } F \left(x \right) = F \left(a \right) \end{array} \)