Identically distributed : Random variables $X_1,X_2,...$ are identically distributed random variables if they all have the same distribution.

Independence : Random variables $X,$ $Y$ are called independent if $P[X\in A, Y\in B]=P[X \in A]P[Y \in B].$ Independent identically distributed random variables feature prominently in the law of large numbers and the central limit theorem. Vectors $x_1,x_2\ldots, x_n$ are called linearly independent if $\lambda_1 x_1+\lambda_2 x_2+\ldots+\lambda_n x_n=0$ implies $\lambda_1=\lambda_2=\ldots=\lambda_n=0.$

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 $\infty$.

Law of large numbers : According to the law of large numbers $\lim \limits_{n \to \infty} \frac{1}{n}\sum_{i=1}^{n}X_i=E[X]$ for independent identical distributed random variables $X_1, X_2,...$ This means that the average outcome of an independently repeated random experiment converges to the expected value. For example the law of large numbers implies that the average of a large number of dice throws converges to 3.5.

Limit : The limit of a function $f$ for $x$ converging to $x_0$ or $\lim\limits_{x\to 0}f(x)$ is a number $y$ such that for every $\epsilon\gt 0$ there is a $\delta\gt 0$ with $|f(x)-y|\lt\epsilon$ for all $|x-x_0|\lt\delta.$ This means that if $x$ only gets close enough to $x_0$ it will get and stay arbitrarily close to $y.$

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.

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 $var[X]=\frac{(b-a)^2}{12}.$

Fraction : A fraction is a number that can be written as $\frac{a}{b}$ with an integer $a$ and a natural numbers $b.$ A fraction $\frac{a}{b}$ is the result of the division $a\div b.$ Examples of fractions are $\frac{1}{2}, -\frac{2}{3}$ and $\frac{3}{2}.$

Reduced fraction : A reduced fraction (or simplified fraction or fraction in simplified form or fraction in simplest form) is a fraction $\frac{a}{b}$ such that $a$ and $b$ do not have any common factors bigger than 1. For example the reduced fraction of the fraction $\frac{4}{6}$ is $\frac{2}{3}.$

Cos : The cosine of an angle $0\leq\alpha\leq 180^{\circ}$ or $\cos \alpha$ is defined by finding a right triangle with angle $\alpha$ and dividing the length of the leg adjacent to $\alpha$ by the length of the hypotenuse. For arbitrary angles the cosine function can be extended in a periodic way by inscribing the right triangle in a circle. The cosine can be used to calculate unknown side lengths in a triangle via the cosine law and in the case of right triangles via its definition. The Pythagorean theorem implies that $\cos^2 \alpha+\sin^2 \alpha=1.$

Derivative : A derivative of $f$ also written as $f^{\prime}$ or $\frac{d}{dx}f$ or $\frac{df}{dx}$ is defined by $f^{\prime}(x)=\lim \limits_{h\to 0}\frac{f(x+h)-f(x)}{h}.$ The derivative $f^{\prime}(x)$ can be interpreted as the speed at which the function value is changing or as the slope of the tangent in the graph of $f$ at $x$. For example the derivative of a constant function is 0, the derivative of $f(x)=x$ is 1 and the derivative of $x^2$ is $2x.$ All local extrema of differentiable functions $f$ are roots of the derivative of $f.$

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.

Integration : Calculating an integral is called integration.

Power : A power is a number of the form $a^b.$ $b$ is called the exponent of the power and $a^b$ is called a power of $a$. For natural numbers $b$ the number $a^b$ is an abbreviation for successively multiplying $a$ by itself $b$ times. For example $2^3=2\cdot 2\cdot 2=8.$ For fractional exponents $b=\frac{p}{q}$ the number $a^{\frac{p}{q}}$ is defined as $\sqrt[q]{a^p}.$ For arbitrary real exponents $b$ the power $a^b$ is defined as the limit of $a^{b_n}$ with rational $b_n$ that converge towards $b.$

Variable substitution : In integration variable substitution refers to the following rule that can sometimes be used to make calculation of an integral easier: $\int \limits_{a}^{b}f(g(x))g^{\prime}(x)dx$ $=\int \limits_{g(a)}^{g(b)}f(x)dx.$ The rule is a consequence of the fundamental theorem of calculus in conjunction with the chain rule of differentiation.