2 Glossary

$𝖯\left(\right)$ probability.

$\mathrm{\Omega }$ sample space.

$A,B\subseteq \mathrm{\Omega }$ subsets.

$A\cup B$ union.

$A\cap B$ intersection.

$A^C$ complement.

$A\setminus B=A\cap B^C$.

$𝖯\left(A\right)$ probability of event $A$.

$𝖯(A\mid B)$ probability of event $A$ given event $B$.

A random variable is an indicator of the outcome of a probability experiment that always takes numerical values. $F_R(r)$ is the cumulative distribution function (cdf) of a discrete random variable $R$ evaluated at $r$.

$F_X(x)$ is the cumulative distribution function of a continuous random variable $X$ evaluated at $x$, i.e. $𝖯\left(X\le x\right)$.

$𝖤\left[X\right]$ is the expectation of random variable $X$.

$𝖤\left[g(X)\right]$ is the expectation of the function $g(X)$ of the random variable $X$.

$\mathrm{𝖵𝖺𝗋}\left[X\right]$ is the variance of the random variable $X$.

$\mathrm{𝖲𝗍𝖽𝖣𝖾𝗏}\left[X\right]$ is the standard deviation of the random variable $X$.

A continuous random variable is a variable whose set of possible values is uncountable.

${\overline{F}}_X(x)$ is the survivor function of the random variable $X$ evaluated at $x$, i.e. $𝖯\left(X>x\right)$.

$f_X(x)$ is the probability density function (pdf) of random variable $X$ evaluated at $x$.

${\mu }_X$ is the expectation of variable $X$.

${\sigma }_X$ is the standard deviation of variable $X$.

${\mu }_r=𝖤\left[{\left(\frac{X-{\mu }_X}{{\sigma }_X}\right)}^r\right]$ is the $r$-th standardized central moment.

${\mu }_3$ is the coefficient of skewness.

${\mu }_4-3$ is the kurtosis.

$x_p$ is the $100p\%$ quantile of random variable, i.e. $F_X(x_p)=p$.

$x_{0.5}$ is the median.

$x_{0.75}-x_{0.25}$ is the inter quartile range.

$X\sim \mathrm{𝖴𝗇𝗂𝖿}(a,b)$, shows the random variable $X$ follows the Uniform distribution on the interval $[a,b]$.

$X\sim \mathrm{𝖤𝗑𝗉}(\beta )$, shows the random variable $X$ follows the Exponential distribution with rate $\beta$ and mean ${\beta }^{-1}$.

$X\sim \mathrm{Gamma}(\alpha ,\beta )$ shows the random variable $X$ follows the Gamma distribution with rate $\beta$ and shape $\alpha$ and mean $\alpha /\beta$.

$X\sim \mathrm{Normal}(\mu ,{\sigma }^2)$, usually written as $X\sim 𝖭(\mu ,{\sigma }^2)$, shows the random variable $X$ follows a Normal distribution with mean $\mu$ and standard deviation $\sigma$.

$\mathrm{\Phi }$ is the cumulative distribution function for the standard Normal distribution $N(0,1)$.

The distribution of a random variable is either the name e.g. $\mathrm{Exponential}(\beta )$, the probability density function $f$ or the cumulative distribution function $F$.

The probability integral transform is a result that allows one to transform from a Uniform random variable to a random variable with any specified cumulative distribution function.

$F_{XY}(x,y)$ is the joint cumulative distribution function of two random variables $X$ and $Y$ evaluated at $(x,y)$, i.e. $𝖯(X\le x,Y\le y)$.

$p_{XY}(x,y)$ is the joint probability mass function (pmf) of two discrete random variables $X$ and $Y$, i.e. $𝖯(X=x,Y=y)$.

$f_{XY}(x,y)$ is the joint probability density function of two continuous random variables $X$ and $Y$ evaluated at $(x,y)$.

$X\mid Y=y$ is the conditional distribution of $X$ given $Y=y$.

$p_{X\mid Y}(x\mid y)$ is the conditional probability mass function of $X$ given $Y=y$.

$f_{X\mid Y}(x\mid y)$ is the conditional probability density function of $X$ given $Y=y$.

$𝖤\left[g(X,Y)\right]$ is the expectation of the function $g(X,Y)$ of the random variables $X$ and $Y$.

$𝖤[X\mid Y=y]$ is the conditional expectation of $X$ given $Y=y$.

$\mathrm{𝖵𝖺𝗋}[X\mid Y=y]$ is the conditional variance of $X$ given $Y=y$.

$\mathrm{𝖢𝗈𝗏}[X,Y]$ is the covariance between $X$ and $Y$.

$\rho =\mathrm{𝖢𝗈𝗋𝗋}[X,Y]$ is the correlation between $X$ and $Y$.

$𝑿={(X_1,\mathrm{\dots },X_n)}^{\prime }$ is a random vector. It is a column vector.

$𝖤\left[𝑿\right]={(𝖤\left[X_1\right],\mathrm{\dots },𝖤\left[X_n\right])}^{\prime }$ is the mean vector $𝑿$.

$\mathrm{𝖵𝖺𝗋}\left[𝑿\right]$ is the variance matrix of $𝑿$, also called the variance-covariance matrix.

$𝑿\sim {\mathrm{MVN}}_d(𝝁,\mathrm{\Sigma })$, shows the random vector $𝑿$ follows the multivariate Normal distribution of $d$ dimensions with mean vector $𝝁$ and variance matrix $\mathrm{\Sigma }$.

${\overline{X}}_n$ is the average of $X_1,\mathrm{\dots },X_n$, i.e. ${\overline{X}}_n=\frac{1}{n}{\sum }_{i=1}^n{X}_i$.

R is a statistical software package and programming language.