3.2.2 The Cumulative Distribution Function

• Fixed Value $x$
  • Objective: Compute the probability that the observed value of $X$ will be at most $x$
• Example Scenario
  • Let $X$ = number of beds occupied in a hospital’s emergency room at a certain time of day
  • Probability mass function (pmf) of $X$ is given by

x	0	1	2	3	4
p(x)	.20	.25	.30	.15	.10

Then the probability that at most two beds are occupied is $P\left(X\le 2\right)=p\left(0\right)+p\left(1\right)+p\left(2\right)=.75$

• Probability Conditions
  • $X\le 2.7$ if and only if $X\le 2$
  • $P(X\le 2.7)=0.75$
  • $P(X\le 2.999)=0.75$
• Smallest Possible Value of $X$
  • Since 0 is the smallest possible value:
    • $P(X\le -1.5)=0$
    • $P(X\le -10)=0$
  • For any negative number $x$:
    • $P(X\le x)=0$
• Largest Possible Value of $X$
  • Since 4 is the largest possible value:
    • $P(X\le 4)=1$
    • $P(X\le 9.8)=1$
    • And so on
  • Very importantly,

P(X < 2) &= p(0) + p(1) = 0.45 < 0.75 \\ &= P(X \leq 2). \end{align}

• Probability Comparison
  • $P(X<x)<P(X\le x)$
    • This holds whenever $x$ is a possible value of $X$
  • Reason: $P(X\le x)$ includes the probability mass at $x=2$, while $P(X<x)$ does not
• Well-Defined Probability
  • $P(X\le x)$ is a well-defined and computable probability for any number $x$

cumulative distribution function (cdf)

• Cumulative Distribution Function (CDF)
  • Definition of $F(x)$ for a discrete random variable $X$ with probability mass function (pmf) $p(x)$:

F(x) = P(X \leq x) = \sum_{y : y \leq x} p(y) \tag{3.3}

• Interpretation
  • For any number $x$:
    • $F(x)$ is the probability that the observed value of $X$ will be at most $x$

EX 3.13 flash drives

For $X$ a discrete rv, the then the probability of having to examine at graph of $F\left(x\right)$ will have a jump at every possible value of $X$ and will be flat between possible values. Such a graph is called a step function.

EX 3.14 (EX 3.12 continued)

In examples thus far, the cdf has been derived from the pmf. This process can be reversed to obtain the pmf from the cdf whenever the latter function is available. For example, consider again the rv of EX 3.7 number of computers in use (the number of computers being used in a lab); possible $X$ values are $0,1,\dots ,6$ . Then

$$\begin{array}{rl}p(3)& =P(X=3)\\ & =\left[p(0)+p(1)+p(2)+p(3)\right]-\left[p(0)+p(1)+p(2)\right]\\ & =P(X\le 3)-P(X\le 2)\\ & =F(3)-F(2)\end{array}$$

More generally, the probability that $X$ falls in a specified interval is easily obtained from the cdf. For example,

$$\begin{array}{rl}P(2\le X\le 4)& =p(2)+p(3)+p(4)\\ & =\left[p(0)+\cdots +p(4)\right]-\left[p(0)+p(1)\right]\\ & =P(X\le 4)-P(X\le 1)\\ & =F(4)-F(1)\end{array}$$

Notice that $P\left(2\le X\le 4\right)\ne F\left(4\right)-F\left(2\right)$ This is because the $X$ value 2 is included in $2\le X\le 4$ , so we do not want to subtract out its probability. However, $P\left(2<X\le 4\right)=F\left(4\right)-F\left(2\right)$ because $X=2$ is not in the interval $2<X\le 4$ .

Proposition

For any two numbers $a$ and $b$ with $a\le b$ , $P\left(a\le X\le b\right)=F\left(b\right)-F\left(a-\right)$ where "$a-$" represents the largest possible $X$ value that is strictly less than $a$ . In particular, if the only possible values are integers and if $a$ and $b$ are integers, then

$$\begin{array}{rl}P\left(a\le X\le b\right)& =P\left(X=a\text{ or }a+1\text{ or ... or }b\right)\\ & =F\left(b\right)-F\left(a-1\right)\end{array}$$

Taking $a=b$ yields $P\left(X=a\right)=F\left(a\right)-F\left(a-1\right)$ in this case.

The reason for subtracting $F\left(a-\right)$ rather than $F\left(a\right)$ is that we want to include $P\left(X=a\right);F\left(b\right)-F\left(a\right)$ gives $P\left(a<X\le b\right)$ . This proposition will be used extensively when computing binomial and Poisson probabilities in Sections 3.4 and 3.6.

• EX 3.15 days of sick leave