Distribution of continuous random variable

The number of values of continuous random variables is indefinite, even when they vary in a small interval (from 0 to 1, for example). So probability distributions of continuous random variables differ a lot from that of discrete ones. In this page, we investigate some of their characteristics.

Cumulative distribution function

Cumulative distribution function `F(X)` (or cdf for brief) for continuous random variables is defined similarly as the case of discrete one:

`F(x)=P(X<=x)`(12)

cdf for continuous random variables has nearly the same basic characteristics with the discrete one :

`0<=F(x)<=1`
`F(x)` is a non-decreasing function
`F(-oo)=0`
`F(oo)=1`
`P(a<=X<=b)=F(b)-F(a)`

Probability density function

Probability density function (or briefly pdf) `f(x)` of a continuous random variable can be defined as :

`f(x)=(dF(x))/dx`

(13)

From the definition (13) and characteristics of cdf, we can derive some characteristics of pdf :

`f(x)>=0` (because `F(x)` is non-decreasing function)
`P(X=x)=0`

Characteristics of distribution of continuous random variable

Because `f(x)` is the derivative of `F(x)`, so `F(x)` is the integral of `f(x)` :

`F(x)=P(X>=x)=int_(-oo)^x f(t)dt`

(14)

Combine (14) and characteristics of cdf :

`F(oo)=P(X>=x)=int_(-oo)^oo f(x)dx`

(15)

and

`P(a<=X<=b)=int_(a)^b f(x)dx=F(b)-F(a)`

(16)

Upon geometric properties of integral, we can represent values in formula (14) and (16) by areas as shown on Fig. 1a and Fig. 1b.

Fig. 1 Geometrical representation of formula (14) (1a) and formula (16) (1b)

Expected value and variance of continuous random variable

Expected value of a continuous random variable `X`, symbolized as `mu` or `E(X)`, is calculated by:

`mu=E(x)=int_(-oo)^oo xf(x)dx`

(17)

Variance of a continuous random variable, denoted as `sigma^2` or `V(X)`, is determined by :

`sigma^2=int_(-oo)^oo (x-mu)^2f(x)dx=int_(-oo)^oo x^2f(x)dx\ -\ mu^2`

(18)

This web page was last updated on 02 December 2018.