17  Cumulative Distribution Functions and Quantile Functions

17.1 Cumulative distribution functions

Example 17.1 Maggie and Seamus are babies who have just turned one. At their one-year visits to their pediatrician:

• Maggie is 76cm tall and in the 75th percentile of height for girls.
• Seamus is 72cm tall and in the 10th percentile of height for boys.

Explain what these percentiles mean.

• Roughly, the value $x$ is the $p$th percentile of a distribution of a random variable $X$ if $p$ percent of values of the variable are less than or equal to $x$: $\text{P}(X\le x)=p$.
• The cumulative distribution function (cdf) of a random variable fills in the blank for any given $x$: $x$ is the (blank) percentile. That is, for an input $x$, the cdf outputs $\text{P}(X\le x)$.
• The cumulative distribution function (cdf) (of a random variable $X$ defined on a probability space with probability measure $\text{P}$) is the function, $F_X:\mathbb{R}\mapsto [0,1]$, defined by $F_X(x)=\text{P}(X\le x)$. A cdf is defined for all real numbers $x$ regardless of whether $x$ is a possible value of $X$.

Example 17.2

According to data on students who took the SAT in 2018-2019, 1400 was the 94th percentile of SAT scores, while 1000 was the 40th percentile. Let $X$ be the SAT score of a randomly selected student (from this cohort), and let $F_X$ be the cdf of $X$. Evaluate the cdf for each of the following. For the purposes of this exercise, interpret these quantities in terms of actual SAT scores, which take values in 400, 410, 420, $\dots$, 1590, 1600.

1. $F_X(1400)$
   
   
2. $F_X(1405)$
   
   
3. $F_X(1000)$
   
   
4. $F_X(1003.7)$
   
   
5. $F_X(-3.1)$
   
   
6. $F_X(390)$
   
   
7. $F_X(399.5)$
   
   
8. $F_X(1600)$
   
   
9. $F_X(1610)$
   
   
10. $F_X(2307.4)$
    
    
11. $F_X(1400)-F_X(1000)$
    
    

Example 17.3 Let $X$ be a random variable with the Exponential(1) distribution. Then the pdf of $f_X$ is

$$f_X(x)=\{\begin{array}{ll}{e}^{-x},& x>0,\\ 0,& \text{otherwise.}\end{array}$$

1. Find the cdf of $X$, and sketch a plot of it.
   
   
   
   
   
2. Evaluate and interpret $F_X(1)$, and draw a picture depicting it.
   
   
   
   
   
3. Evaluate and interpret $F_X(2)-F_X(1)$, and draw a picture depicting it.
   
   
   
   
   
4. Evaluate and interpret $F_X(2)$, and draw a picture depicting it.
   
   
   
   
   
5. Find $\text{P}(1<X<2.5)$ without integrating again.
   
   
   
   
   
6. Suppose we had been given the cdf instead of the pdf. How could we find the pdf?
   
   
   
   
   

Example 17.4

Let $X$ be the number of heads in 3 flips of a fair coin.

1. Find the cdf of $X$ and sketch a plot of it.
   
   
   
   
   
2. Let $Y$ be the number of tails in 3 flips of a fair coin. Find the cdf of $Y$.
   
   
   
   
   

• A cdf is defined for all values of $x$, regardless if $x$ is a possible value of the RV.
• A cdf is a non-decreasing function: if $x_1\le x_2$ then $F_X(x_1)\le F_X(x_2)$.
• A cdf approaches 0 as the input approaches $-\mathrm{\infty }$: $\underset{x\to -\mathrm{\infty }}{lim}{F}_X(x)=0$
• A cdf approaches 1 as the input approaches $\mathrm{\infty }$: $\underset{x\to \mathrm{\infty }}{lim}{F}_X(x)=1$
• The cdf of a discrete random variable is a step function.
  • The steps occur at the possible values of the random variable.
  • The height of a particular step corresponds to the probability of that value, given by the pmf.
• The cdf of a continuous random variable is a continuous function.
  • The cdf of a continuous random variable is obtained by integrating the pdf, so
  • The pdf of a continuous random variable is obtained by differentiating the cdf $$F_X^{\prime }=f_X\text{if }X\text{ is continuous}$$
• For any random variable $X$ with cdf $F_X$

$$F_X(b)-F_X(a)=\text{P}(a<X\le b)$$

Whether the inequalities in the above event are strict ($<$) or not ($\le$) matters for discrete random variables, but not for continuous. - Random variables $X$ and $Y$ have the same distribution if their cdfs are the same, that is, if $F_X(u)=F_Y(u)$ for all $u$. - That is, two random variables have the same distribution if all the percentiles are the same.

17.2 Quantile functions

• The quantile function (essentially the inverse cdf) fills in the following blank for a given $p\in [0,1]$: the $100p$th percentile is (blank).
• For example, evaluating the quantile function at $p=0.25$ outputs the 25th percentile.

Example 17.5 In the meeting problem, suppose arrival times (minutes) follow a Normal(30, 10) distribution. Let $Q$ be the quantile function.

In addition to finding the values below, identify how they are represented in the standard Normal spinner.

1. Find $Q(0.16)$.
   
   
2. Find $Q(0.25)$.
   
   
3. Find $Q(0.5)$.
   
   
4. Find $Q(0.25)$.
   
   
5. Find $Q(0.975)$.
   
   

• For a continuous random variable with cdf $F$, the quantile function $Q:[0,1]\mapsto \mathbb{R}$ is the inverse of the cdf, $Q(p)=F^{-1}(p)$.

Example 17.6

Let $X$ have an Exponential(1) distribution. Recall that the cdf is $F_X(x)=1-e^{-x},x>0$.

1. Find the 25th percentile.
   
   
   
   
   
2. Find the quantile function $Q_X$.
   
   
   
   
   
3. Remember that if $U$ has a Uniform(0, 1) distribution then $X=-\log (1-U)$ has an Exponential(1) distribution. How does the transformation of $U$ relate to the quantile function? What insight does this give you into constructing spinners?
   
   
   
   
   

• The quantile function can be used to create a spinner for a distribution. Basically, the values on the outside boundary of the spinner are scaled based on the quantile function (which is determined by the cdf). Intervals corresponding to regions of higher density (“more likely”) values are stretched out on the spinner boundary; intervals corresponding regions of lower density (“less likely” values) are shrunk.
• The foundation of all spinners is the Uniform(0, 1) spinner.

Example 17.7 Recall Example 16.3 where Regina’s arrival time $X$ (in hours) had pdf

$$f_X(x)=2x,0<x<1.$$

1. Find the cdf of $X$.
   
   
   
   
   
2. Find the quantile function of $X$.
   
   
   
   
   
3. Construct a spinner for simulating values of $X$ (hours) according to its distribution.
   
   
   
   
   
4. If we measure $X$ in minutes, how would the spinner change?
   
   
   
   
   

• Universality of the Uniform (or “one spinner to rule them all”). Let $F$ be a cdf and $Q$ its corresponding quantile function. Let $U$ have a Uniform(0, 1) distribution and define the random variable $X=Q(U)$. Then the cdf of $X$ is $F$.
• Universality of the uniform might look complicated but all it basically says is that you can construct a spinner by putting the 25th percentile 25% of the way around, the 75th percentile 75% of the way around, etc.
• Actually, universality of the uniform says we don’t have to create a new spinner. We can just spin the Uniform(0, 1) spinner and transform each resulting value by plugging it into the quantile function.