Home page for accesible maths 6 Continuous random variables 6.1 Introduction to Continuous Variables 6.3 Probability Density Function

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6.2 Cumulative Distribution Function

The (cumulative) distribution function, cdf, of a (continuous or discrete) random variable, $X$ , is defined, for all real values of $x$ , by

F_{X}(x)={\rm P}(X\leq x),

the probability that the random variable $X$ takes a value less than or equal to $x$ . When $F_{X}$ is continuous, we have a continuous random variable. An example of $F_{X}$ for a continuous random variable is displayed in Figure 6.2.

Properties of $F_{X}(x)$ :

1.

$0\leq F_{X}(x)\leq 1$ , with $\lim_{x\rightarrow-\infty}F_{X}(x)={\color[rgb]{1,1,1}0}$ and $\lim_{x\rightarrow\infty}F_{X}(x)={\color[rgb]{1,1,1}1}$ ,
2.

$F_{X}(x)$ is non-decreasing function of $x$ . Why?

The distribution function is particularly useful for continuous random variables as we often want to know the probability of events that can be related by the laws of probability into probability statements about the event $\{X\leq x\}$ for some $x$ .

Probabilities of Intervals:

Often the probability of the random variable $X$ falling in the interval $(a,b]$ is of interest for some real numbers $a, b$ with $a<b$ . This corresponds to the event $\{a<X\leq b\}$ . By using the law of total probability ${\rm P}(X\leq b)={\rm P}(X\leq a)+{\rm P}(a<X\leq b)$ so the probability of the interval event is

{\rm P}(a<X\leq b)={\rm P}(X\leq b)-{\rm P}(X\leq a)=F_{X}(b)-F_{X}(a).

As ${\rm P}(X=x)=0$ for all $x$ , for any continuous random variable $X$

{\rm P}(X\leq x)={\rm P}(X<x).

Exercise 6.1.

Let $X$ be a random variable with cumulative distribution function

F_{X}(x)=\left\{\begin{array}[]{rcl}0&\mbox{if}&x\leq 0,\\ x&\mbox{if}&0<x\leq 1\\ 1&\mbox{if}&x>1.\end{array}\right.

Obtain the following probabilities:

a.

${{\rm P}(X\leq 0.5)}=$ $\color[rgb]{1,1,1}{F_{X}(0.5)=0.5}$ ,
b.

${{\rm P}(X>0.5)}=$ $\color[rgb]{1,1,1}{1-{\rm P}(X\leq 0.5)=1-F_{X}(0.5)=0.5}$ ,
c.

${{\rm P}(X=0.5)=}\color[rgb]{1,1,1}{0}$ ,
d.

${{\rm P}(X<.9)=}\color[rgb]{1,1,1}{0.9}$ ,
e.

${{\rm P}(0.5<X\leq 0.9)}=\color[rgb]{1,1,1}{{\rm P}(X\leq.9)-{\rm P}(X\leq 0.5% )=0.9-0.5=0.4}$ .

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6.2 Cumulative Distribution Function

Properties of FX⁢(x):

Probabilities of Intervals:

Exercise 6.1.

Properties of $F_{X}(x)$ :