1.4.1 Cumulative Distribution Function

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

Exam 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

Edexcel International A Level Maths: Statistics 2

Revision Notes

Cumulative Distribution Function

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

How do I find probabilities using the cumulative frequency distribution?

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

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

Worked example

Exam Tip

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Author: Paul