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

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

$\mathbf{F}\mathbf{\left(}{\mathit{x}}_{\mathbf{0}}\mathbf{\right)}\mathbf{=}\mathit{P}\mathbf{(}\mathit{X}\mathbf{\le }{\mathit{x}}_{\mathbf{0}}\mathbf{)}\mathbf{=}{\mathbf{\int }}_{\mathbf{-}\mathbf{\infty }}^{{\mathbf{x}}_{\mathbf{0}}}\mathbf{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{d}\mathit{t}$

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

$\mathrm{F}\left(x_0\right)=P(X\le x_0)=$$\sum _{x\le x_0}\mathrm{P}(X=x)$

• F(x₀) is the probability that X is a value less than or equal to x₀

• Notice the use of uppercase $\text{F}$ for the c.d.f. but lowercase $\text{f}$  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 $x\in \mathrm{ℝ}$

• 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.

$\mathrm{F}\left(x\right)=\{\begin{array}{ll}0& x<0\\ 0.5x^2& 0\le x\le 1\\ 0.5& 1\le x\le 1.5\\ x-1& 1.5\le x\le 2\\ 1& x>2\end{array}$

• 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?

• $\mathrm{P}(a\le X\le b)=\mathrm{F}\left(b\right)-\mathrm{F}\left(a\right)$

• Although $\mathrm{P}(X=k)$, 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

$\mathrm{F}\left(x\right)={\int }_{-\infty }^x\mathrm{f}\left(t\right)dt$

• Ensure you define F(x) fully for$x\in \mathrm{ℝ}$  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. $x\le a$ and $x>a$

  • For $x>a$:

$\mathrm{F}\left(x\right)={\int }_{-\infty }^x\mathrm{f}\left(t\right)\mathrm{d}t={\int }_{-\infty }^a\mathrm{f}\left(t\right)\mathrm{d}t+{\int }_a^x\mathrm{f}\left(t\right)\mathrm{d}t=\mathrm{F}\left(a\right)+{\int }_a^x\mathrm{f}\left(t\right)\mathrm{d}t$

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

$\mathrm{F}\left(x\right)=\mathrm{F}\left(a\right)+{\int }_a^x\mathrm{f}\left(t\right)\mathrm{d}t$

• See part (b) in the Worked Example below

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

$\mathrm{f}\left(x\right)=\frac{d}{dx}\mathrm{F}\left(x\right)$

• 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  $\mathrm{F}\left(p\right)=\frac{n}{100}$