**moments math**collection.

## Random Variables

For a space of elementary events, say \Omega=\{\omega\}, a random variable X is a real number function X=X(\omega) deﬁned on the set \Omega.

Essentially, X may be considered to be a quantity which takes its values (say x_i) from a subset R of real numbers.

We note that *iff* X is a random variable, a function g(X) is also random.

Random variables are further quantiﬁed and classiﬁed on the basis of their *distribution functions*.

- Distribution Law
A rule (tabular, functional, graphical, etc) which permits one to ﬁnd the probabilities of an event (a.k.a the random variable) is the distribution law for the random variable.

## Distribution Functions

Every random variable is deﬁned in terms of it’s probabilities, i.e they are characterized by the likelihood of having a particular value.

Mathematically, the *cumulative distribution function* of a random variable X is the function F(x) whose value at every point x is equal to the probability of the event {X <x}:

F(x)=P(X<x)

### Properties

- 0 \leq F(x) \leq 1
- \lim_{x\to -\infty}F(x)=F(-\infty)=0 and \lim_{x\to\infty}F(x)=F(\infty)=1
- \forall x_i, x_2>x_1 \implies F(x_2)\geq F(x1)
- P(x_1 \leq X < x_2)=F(x_2)-F(x_1)
- F(x) is left continuous. (i.e., \lim_{x\to(x_0-0)}F(x)=F(x_0))

## Types of Random Variables

On the basis of the above concepts, we now quantify random variables as: X \to (1)

## Expectation

The expectation (expected value) E(X) of a discrete or continuous random variable X is mathematically deﬁned by: E\{X\}= (2)

For the continuous case, it is necessary that the integral or it’s corresponding series converges absolutely.

In generic terms, the expectation is the main characteristic deﬁning the “position” of a random variable, i.e., the number near which its possible values are concentrated.

Similarly, due to the similarity of functions describing random variables and random variables, given a random variable Y related to a random variable X by a functional dependence Y=\phi(X) then we have:

E\{Y\}=E\{\phi(X)\}= (3)

## Variance

The variance, Var{X} is the measure of deviation of a random variable X from the expectation E\{X\} as determined by:

\text{Var}\{X\}=E\{(X-E\{X\})^2\} (4)

The variance characterizes the spread in values of the random variable X about its expectation.

## Graphical Preliminaries

Having introduced the density function and the distribution function, it is trivial to interpret the following curves in the ﬁgure below and note, that the probability P(X\leq x)=F(x) may be represented as an area between the density function f(t) and the x-axis on the interval -\infty<t\leq x

Often there is given (frequently in %) a probability value \alpha.

If P(X > x) = \alpha holds, the corresponding value of the abscissa x = x_\alpha is called the quantile or the fractile of order \alpha

This means the area under the density function f(t) to the right of x = x_\alpha is equal to \alpha.

**Remark:** In the literature, the area to the left of x = x_\alpha is also used for the definition of quantile.

In mathematical statistics, for small values of \alpha, e.g., \alpha= 5\% or \alpha= 1\%, is also used the notion significance level of ﬁrst type or type 1 error rate.

## References

Bronshtein, I.N., K.A. Semendyayev, G. Musiol, and H. Mühlig. 2015. *Handbook of Mathematics*. Springer Berlin Heidelberg. https://books.google.co.in/books?id=5L6BBwAAQBAJ.

Bronshtein et al. (2015)↩