Logarithm : The logarithm $\log_{b} x$ of a number $x$ with respect to the basis $b$ is the number such that $b^{\log_{b} x}=x.$

Product : A product is the result of a multiplication.

Sum : A sum is the result of an addition.

Equation : An equation is a mathematical statement in which two expressions are written with an equal sign in between. A solution of an equation is a set of variables that makes the statement a true statement.

Half : A Half is the number equal to $\frac{1}{2}=0.5.$

Integral : The integral of a function $f$ between $a$ and $b$ is defined as $\int\limits_{a}^{b}f(x)dx$ $=\lim \limits_{n\to\infty} \frac{(b-a)}{n}\sum\limits_{k=1}^{n}f(a+\frac{k(b-a)}{n}).$ For example $\int\limits_{a}^{b}xdx=\frac{b^2}{2}-\frac{a^2}{2}.$ The fundamental theorem of calculus provides a simple way to calculate integrals using antiderivatives.

Interval : An interval is a part of the number line between two numbers $a$ and $b.$ $a$ and $b$ can but do not have to be part of the interval. If $a$ and $b$ are part of the interval the interval is a closed interval. If $a$ and $b$ are not part fo the interval the interval is an open interval. If either $a$ or $b$ but not both are part of the interval the interval is called half-open or half-closed. $a$ can be equal to $-\infty$ and $b$ can be equal to $\infty$.

Random variable : A random variable $X$ describes all the possible outcomes of a random experiment. The probability distribution of $X$ describes how likely all these outcomes are.

Squared : $x$ squared refers to the number $x^2=x\cdot x.$ For example 3 squared equals 9.

Third : A third either refers to the third object in an ordering or to the number $\frac{1}{3}=0.\overline{3}.$

Uniformly distributed : A random variable $X$ is uniformly distributed on the interval $[a,b]$ if it only takes values in $[a,b]$ and any value within the interval is equally likely in the sense of $P[X\lt x]=\frac{x-a}{b-a}.$ for $a\leq x\leq b.$ A uniformly distributed random variable has a mean $E[X]=\frac{a+b}{2}$ and a variance of $var[X]=\frac{(b-a)^2}{12}.$

Variance : The variance $var(X)$ of a random variable $X$ is defined by $var(X)=E[(X-E[X])^2]=E[X^2]-E^2[X].$ The variance is the square of the standard deviation of a random variable and is a measure for how far outcomes of $X$ will typically deviate from the mean $E[X].$

Count : To count means to find out the number of objects by adding one for every object.