cdf.Rmd

# Cumulative Distribution Functions {#cdf}

## Theory {-}

The p.m.f. \@ref(def:pmf-function) is one way to describe a random variable, but 
it is not the only way. The **cumulative distribution function** is a different 
representation that contains the same information as the p.m.f.

```{definition cdf-function, name="Cumulative Distribution Function"}
The **cumulative distribution function** (c.d.f.) is a function that returns the probability 
that a random variable is _less than or equal_ to a particular value:
\begin{equation} 
F(x) \overset{\text{def}}{=} P(X \leq x). 
(\#eq:cdf)
\end{equation}
It is called "cumulative" because it includes all the probability up to (and including) $x$.
```

```{example, name="Calculating the C.D.F."}
Let's calculate the c.d.f. of $X$, the number of diamonds among the community cards, using the 
p.m.f. that we calculated in Lesson \@ref(rv). Note that 
$F(x)$ is the sum of all the probabilities up to $x$. So, for example, 
\begin{align*}
F(2.8) = P(X \leq 2.8) &= f(0) + f(1) + f(2) \\
&= .2546 + .4243 + .2496 \\
&= .9284.
\end{align*}
There is no simple formula for $F(x)$. However, we can describe it piecewise.
\[ F(x) = \begin{cases} 0 & x \leq 0 \\
.2546 & 0 \leq x < 1 \\
.6788 & 1 \leq x < 2 \\
.9284 & 2 \leq x < 3 \\
.9926 & 3 \leq x < 4 \\
.9997 & 4 \leq x < 5 \\
1.0 & x \geq 5
\end{cases}. \]
Note that the c.d.f. of $X$ has the following properties:

- It is constant between integers. Because it is impossible for the random variable $X$ 
to assume non-integer values, $F(1.2) = P(X \leq 1.2)$ and $F(1) = P(X \leq 1)$ must be the same.
- The value of $F(x)$ increases from 0 to 1 as $x$ increases. This makes sense because as $x$ increases, 
we accumulate more and more probability.
```

```{example, name="Graphing the C.D.F."}
The properties of the c.d.f. become clearer on a graph, like the one below. 
Because the random variable $X$ cannot take on decimal values, the c.d.f. of $X$ does not change 
between integers, giving it its step-function appearance.
```

```{r cdf-graph, echo=FALSE, fig.show = "hold", fig.align = "default", fig.cap="Graph of a cumulative distribution function"}
plot(0:5, phyper(0:5, 11, 37, 5), pch=19, xlab="x", ylab="F(x)", xlim=c(-0.5, 5.5), ylim=c(-0.05, 1.05))
points(0:5, phyper(-1:4, 11, 37, 5), xlab="x", ylab="f(x)")
lines(c(-1, 0), c(0, 0), lwd=2)
for(x in 0:5) {
  p <- phyper(x, 11, 37, 5)
  lines(c(x, x + 1), c(p, p), lwd=2)
}
```

Note that the c.d.f. $F(x)$ can be evaluated at all values $x$, not just at integer values. 

```{example, name="Using the C.D.F."}
Some probabilities are easier to calculate using the c.d.f. than using the p.m.f. 
For example, the probability that Alice gets a flush, $P(X \geq 3)$, can be calculated 
by using the complement rule (\@ref(thm:complement)) and looking up the appropriate 
probability directly from the c.d.f. $F(x)$.

\begin{align*} 
P(X \geq 3) &= 1 - P(X < 3) \\
&= 1 - P(X \leq 2) \\
&= 1 - F(2) \\
&= 1 - .9284 \\
&= .0716
\end{align*}

Remember that the c.d.f. $F(x)$ always includes the probability of $x$. Since 
$P(X < 3)$ should not include the probability of $3$, we use $F(2)$ instead of 
$F(3)$.

At the beginning of this lesson, we mentioned that the c.d.f. contains the exact same 
information as the p.m.f., no more and no less. Therefore, it should be possible 
to recover the p.m.f. from the c.d.f. 

For example, how would we calculate $f(3) = P(X = 3)$, if we only knew the c.d.f. $F(x)$? We could
subtract $P(X \leq 2)$ from $P(X \leq 3)$ to get just the probability that it is equal to 3.
\begin{align*}
f(3) = P(X = 3) &= P(X \leq 3) - P(X \leq 2) \\
&= F(3) - F(2) \\
&= .9926 - .9284 \\
&= .0642,
\end{align*}
which agrees with the p.m.f. from Lesson \@ref(rv).
```

## Examples {-}

1. Continuing with the example from the lesson, let $Y$ be the number of Jacks among the community cards. 
Calculate and graph the c.d.f. of $Y$.

2. Consider a random variable $Z$ with c.d.f. given by the formula
\begin{align*} 
F(x) &= \begin{cases} 1 - 3^{-\lfloor x \rfloor} & x \geq 0 \\ 0 & \text{otherwise} \end{cases}.
\end{align*}
(Note that $\lfloor x \rfloor$ denotes the _floor_ operator, which rounds $x$ down to the nearest integer. 
So $\lfloor 3.9 \rfloor = 3$ and $\lfloor 7.1 \rfloor = 7$.) 

    Graph the c.d.f. $F(x)$. Then, use it to calculate:
    a. $P(Z > 3)$
    b. $P(Z = 2)$