Cumulative Distribution Function

What is the cumulative distribution function (c.d.f.)?

For a continuous random variable,X , with probability density function f(x) the cumulative distribution function (c.d.f.) is defined as

$F (x_{0}) = P (X \leq x_{0}) = \int_{- \infty}^{x_{0}} f (t) d t$

Compare this to the cumulative distribution function for a discrete random variable

$F (x_{0}) = P (X \leq x_{0}) =$ $\sum_{x \leq x_{0}} P (X = x)$

F(x₀) is the probability that X is a value less than or equal to x₀
Notice the use of uppercase $F$ for the c.d.f. but lowercase $f$ for the p.d.f.
On the graph of the p.d.f. y= f(x) this would be the area under the graph up to the (vertical) line x=x₀
F(x) should be defined for all values of $x \in ℝ$
The graph of the c.d.f. y = F(x) will
- start on the x-axis (i.e. start at a probability of 0)
- end at x = 1 (i.e. finish at a probability of 1)
- will be continuous function, even when defined piecewise

e.g.

$F (x) = \{\begin{cases} 0 & x < 0 \\ 0.5 x^{2} & 0 \leq x \leq 1 \\ 0.5 & 1 \leq x \leq 1.5 \\ x - 1 & 1.5 \leq x \leq 2 \\ 1 & x > 2 \end{cases}$

BSM6-gbH_1-4-1-ial-fig1-cdf-graph

The horizontal lines at F(x) = 0 and F(x) = 1 may not always be shown

How do I find probabilities using the cumulative frequency distribution?

$P (a \leq X \leq b) = F (b) - F (a)$
Although $P (X = k)$ , for all values of k , F(k) is not necessarily zero

How do I find the cumulative frequency distribution (c.d.f.) from the probability density function (p.d.f.) and vice versa?

To find the c.d.f.,F(x) , from the p.d.f.,f(x), integrate

$F (x) = \int_{- \infty}^{x} f (t) d t$

- Ensure you define F(x) fully for $x \in ℝ$ so include values of x for which F(x) = 0 and values of x for which F(x) = 1
- For piecewise functions as well as integrating you will need to add on the value of the c.d.f. at the end of the previous part
  - Suppose there are two sections to a p.d.f. $x \leq a$ and $x > a$
  - For $x > a$ :

$F (x) = \int_{- \infty}^{x} f (t) d t = \int_{- \infty}^{a} f (t) d t + \int_{a}^{x} f (t) d t = F (a) + \int_{a}^{x} f (t) d t$

- Therefore the c.d.f can be calculated for the interval a < x < b by using

$F (x) = F (a) + \int_{a}^{x} f (t) d t$

- - See part (b) in the Worked Example below

To find the p.d.f from the c.d.f., differentiate

$f (x) = \frac{d}{d x} F (x)$

Any part of a c.d.f that is constant corresponds to the p.d.f. for that part being zero (the derivative of a constant is zero)

How do I find the median, quartiles and percentiles using the cumulative frequency distribution (c.d.f.)?

For piecewise functions, first identify the section the required value lies in
- To do this find the upper limit of each section of the c.d.f.
To find the median, $m$ , solve the equation F(m) = 0.5
- The median is sometimes referred to as the second quartile, Q₂
To find the lower quartile, Q₁, solve the equation F(Q₁) = 0.25
To find the upper quartile,Q₃ , solve the equation F(Q₃ ) = 0.75
To find the n^th percentile, solve the equation $F (p) = \frac{n}{100}$

Worked example

The continuous random variable,

X

, has cumulative distribution function

$F (x) = \{\begin{cases} 0 & x < 0 \\ \frac{1}{4} x (4 - x) & 0 \leq x \leq 2 \\ 1 & x > 2 \end{cases}$

Find

(i)

P (X > 1.5)

(ii)

P (0.5 \leq X \leq 1)

(iii)

The lower quartile of $X$ .

(b) The continuous random variable, $X$ , has probability density function

$f (x) = \{\begin{cases} 0.5 x & 0 \leq x \leq 1 \\ 0.5 & 1 \leq x \leq 2.5 \\ 0 & otherwise \end{cases}$

Find the cumulative frequency distribution, $F (x)$ .

The continuous random variable,

X

, has cumulative distribution function

$F (x) = \{\begin{cases} 0 & x < 0 \\ \frac{1}{4} x (4 - x) & 0 \leq x \leq 2 \\ 1 & x > 2 \end{cases}$

Find

(i)

P (X > 1.5)

(ii)

P (0.5 \leq X \leq 1)

(iii)

The lower quartile of $X$ .

1-4-1-ial-fig2-we-solution-part-1

(b) The continuous random variable, $X$ , has probability density function

$f (x) = \{\begin{cases} 0.5 x & 0 \leq x \leq 1 \\ 0.5 & 1 \leq x \leq 2.5 \\ 0 & otherwise \end{cases}$

Find the cumulative frequency distribution, $F (x)$ .

1-4-1-ial-fig2-we-solution-part-2

Examiner Tip

Remember that P(X=k) = 0 , for any value of k, is zero
- This can be easily missed when working with c.d.f. rather than a p.d.f.
A quick check you can do is verify that your c.d.f. is continuous
- The value of the c.d.f. at the upper limit of one section should equal the value of the c.d.f at the lower limit of the next section

Cumulative Distribution Function (Edexcel International A Level Maths: Statistics 2)

Revision Note