# 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

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

• F(x0) is the probability that X is a value less than or equal to x0
• Notice the use of uppercase  for the c.d.f. but lowercase  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
• 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.

• 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?

• Although , 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

• Ensure you define F(x) fully for  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. and
• For :

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

• See part (b) in the Worked Example below
• To find the p.d.f from the c.d.f., differentiate

• 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, 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 nth percentile, solve the equation

#### Worked example

a)
The continuous random variable,  , has cumulative distribution function

Find

(i)
(ii)
(iii)

The lower quartile of .

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

Find the cumulative frequency distribution,  .

a)
The continuous random variable,  , has cumulative distribution function

Find

(i)
(ii)
(iii)

The lower quartile of .

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

Find the cumulative frequency distribution,  .

#### 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

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