# Mathematical Preliminaries

### Common Statistical Deﬁnitions

Published:
This is part of the mo­ments math col­lec­tion.

## Random Variables

For a space of el­e­men­tary events, say , a ran­dom vari­able is a real num­ber func­tion de­ﬁned on the set .

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

We note that iff is a ran­dom vari­able, a func­tion 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 is the func­tion whose value at every point is equal to the prob­a­bil­ity of the event :

### Properties

• and
• ,
• is left con­tin­u­ous. (i.e., )

## Types of RandomVariables

On the ba­sis of the above con­cepts, we now quan­tify ran­dom vari­ables as:

## Expectation

The ex­pec­ta­tion (expected value) of a dis­crete or con­tin­u­ous ran­dom vari­able is math­e­mat­i­cally de­ﬁned by:

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 re­lated to a ran­dom vari­able by a func­tional de­pen­dence then we have:

## Variance

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

(1)

The vari­ance char­ac­ter­izes the spread in val­ues of the ran­dom vari­able 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 may be rep­re­sented as an area be­tween the den­sity func­tion and the -axis on the in­ter­val

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

If holds, the cor­re­spond­ing value of the ab­scissa is called the quan­tile or the frac­tile of or­der

This means the area un­der the den­sity func­tion to the right of is equal to .

Remark: In the lit­er­a­ture, the area to the left of is also used for the de­f­i­n­i­tion of quan­tile.

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

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.

1. Bronshtein et al. (2015)