Home page for accesible maths 2 Random Variables and Summary Measures 2.1 Random variables 2.3 Discrete Random Variables

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2.2 Cumulative distribution function

For all random variables, $X$ , we may consider the probability of events of the form

\displaystyle\{X(\omega)\leq x\}

for fixed $x$ , and examine how this varies as $x$ changes. The probability that the random variable $X$ is less than or equal to some value $x$ ,

\displaystyle F_{X}(x)=\operatorname{\mathsf{P}}\left({X\leq x}\right)=\sum_{% \omega\in\Omega\colon X(\omega)\leq x}\operatorname{\mathsf{P}}\left({\omega}% \right),

is the cumulative distribution function (CDF) of the random variable $X$ , evaluated at $x$ .

Recall: upper case for the random variable, lower case for a value.

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

1.

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

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

The Survivor function: The survivor function of a random variable $X$ is defined as $S_{X}(x)=\operatorname{\mathsf{P}}\left({X>x}\right)$ . Using the law of complementary events

\displaystyle S_{X}(x)=\operatorname{\mathsf{P}}\left({X>x}\right)=1-% \operatorname{\mathsf{P}}\left({X\leq x}\right)=1-F_{X}(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 $\operatorname{\mathsf{P}}\left({X\leq b}\right)=\operatorname{\mathsf{P}}\left% ({X\leq a}\right)+\operatorname{\mathsf{P}}\left({a<X\leq b}\right)$ so the probability of the interval event is

\displaystyle\operatorname{\mathsf{P}}\left({a<X\leq b}\right)=\operatorname{% \mathsf{P}}\left({X\leq b}\right)-\operatorname{\mathsf{P}}\left({X\leq a}% \right)=F_{X}(b)-F_{X}(a).

Example 2.2.1.

Explain why each of the following functions is not a valid cdf for a random variable $X$ satisfying $0<X<\infty$ ?

(a)

$-\frac{1}{1+x}$
(b)

$\frac{1}{1+x}$
(c)

$\frac{1}{1+(x-5)^{2}}$
(d)

$2-\frac{1}{2+x}$
(e)

$\frac{1}{2}-\frac{1}{2+x}$

Solution. (a) Negative; (b) Decreasing; (c) Decreasing from $x=5$ ; (d) Greater than ${\color[rgb]{0.76,0.01,0}1}$ ; (e) $\lim_{x\rightarrow\infty}F_{X}(x)\neq 1.$

The cumulative distribution function of a random variable is the fundamental quantity from which all other important properties of the random variable can be derived. To prove the following statement, the additivity axiom from Chapter 1 must be extended (to ‘countable additivity’). This will be covered in Year 3, however the result is useful for this year (and for very keen students, a stand-alone proof is given in Appendix B).

Lemma.

For any $x\in\mathbb{R}$ we have

\displaystyle\operatorname{\mathsf{P}}\left({X=x}\right)=F_{X}(x)-\lim_{i% \rightarrow\infty}F(x-1/i).

So, if $F_{X}$ is continuous at $x$ then by definition $\lim_{i\rightarrow\infty}F_{X}(x-1/i)=F_{X}(x)$ so $\operatorname{\mathsf{P}}\left({X=x}\right)=0$ ; however, if $F_{X}(x)$ is discontinuous at $x$ then $\operatorname{\mathsf{P}}\left({X=x}\right)>0$ . This motivates two types of random variables. Continuous random variables have a cdf which is continuous at all $x$ values. Discrete random variables have a cdf which is horizontal (so continuous) except at a number of ‘jump points’. Quiz: Can you think of a third type of cdf, and hence of random variable?

Example 2.2.2.

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

\displaystyle F_{X}(x)=\left\{\begin{array}[]{ll}0&\quad x\leq 0\\ x^{2}/4&\quad 0<x\leq 2\\ 1&\quad x>2\end{array}\right.

Obtain the following probabilities:

(a)

$\operatorname{\mathsf{P}}\left({X\leq 1}\right)$ ,
(b)

$\operatorname{\mathsf{P}}\left({X>1}\right)$ ,
(c)

$\operatorname{\mathsf{P}}\left({X=1}\right)$ ,
(d)

$\operatorname{\mathsf{P}}\left({X<0.5}\right)$ ,
(e)

$\operatorname{\mathsf{P}}\left({0.5<X\leq 1}\right)$ .

Solution.

(a)

$\operatorname{\mathsf{P}}\left({X\leq 1}\right)={\color[rgb]{0.76,0.01,0}F(1)=% 1/4,}$
(b)

$\operatorname{\mathsf{P}}\left({X>1}\right)={\color[rgb]{0.76,0.01,0}1-% \operatorname{\mathsf{P}}\left({X\leq 1}\right)=3/4,}$
(c)

$\operatorname{\mathsf{P}}\left({X=1}\right)={\color[rgb]{0.76,0.01,0}0,}$
(d)

$\operatorname{\mathsf{P}}\left({X<1/2}\right)={\color[rgb]{0.76,0.01,0}F(1/2)=% 1/16,}$
(e)

$\operatorname{\mathsf{P}}\left({1/2<X\leq 1}\right)={\color[rgb]{0.76,0.01,0}% \operatorname{\mathsf{P}}\left({X\leq 1}\right)-\operatorname{\mathsf{P}}\left% ({X\leq 1/2}\right)=1/4-1/16=3/16.}$