IMSE 317 Discrete CDF

class: middle, center

.large[Cumulative distribution functions]

<br>

![:scale 60%](../images/cumulation-cover.png)

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# [Cumulative Distribution Function (CDF)](https://en.wikipedia.org/wiki/Cumulative_distribution_function)

The .red[cumulative distribution function] of a RV $X$ is defined by

$$F_X(x)=\text{P}(X \leq x), \;\;\;\;\;\text{for all $x$.}$$

It is the probability that the value of $X$ is .red[at most] $x$.

Loosely speaking, the CDF .red[accumulates] the probability .red[up to] the value $x$.

If RV $X$ is discrete,

$$F_X(x)=\text{P}(X \leq x)=\sum_{k \leq x} p_X(k)$$

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# Finding CDF

$X$: Roll a die 🎲 and record the number

What is the CDF of $X$?

$$\text{CDF def:} \;\;\;\; F_X(x)=\text{P}(X \leq x), \;\;\;\;\;\text{for all $x$.}$$

.left[![:scale 45%](../images/pmf-roll1die.png)]

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.center[![:scale 90%](../images/cdf-roll1die.png)]

$$
\small{
F_X(x) = 
\begin{cases}
    0, & \text{if $ x < 1$,} \\\
    1/6, & \text{if } 1 \leq x < 2, \\\
    2/6, & \text{if } 2 \leq x < 3, \\\
    3/6, & \text{if } 3 \leq x < 4, \\\
    4/6, & \text{if } 4 \leq x < 5, \\\
    5/6, & \text{if } 5 \leq x < 6, \\\
    1,   & \text{if } x \geq 6. \\\
\end{cases}
}
$$

For discrete RVs the CDF $F_X(x)$ is a .green[step function] of $x$.

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$X$: Toss a coin twice and record the number of heads.

What is the CDF of $X$?

$$
\small{
p_X(x) = 
\begin{cases}
    0.25, & \text{if $x = 0$ or 2}, \\\
    0.5, & \text{if $x = 1$,} \\\
    0,              & \text{otherwise.}
\end{cases}
}
$$

.center[![:scale 70%](../images/cdf-binomial-n2.png)]

$$
\small{
F_X(x) = 
\begin{cases}
    0, & \text{if $x < 0$,} \\\
    0.25, & \text{if } 0 \leq x < 1, \\\
    0.75, & \text{if } 1 \leq x < 2, \\\
    1,   & \text{if } x \geq 2. \\\
\end{cases}
}
$$

---

If $X$ only takes integer values, 
its PMF can be obtained from CDF by .red[differencing].

$$
\small{
\begin{aligned}
p_X(k)&=\text{P}(X \leq k) - \text{P}(X \leq k-1) \\\
&= F_X(k) - F_X(k-1) \\\
\end{aligned}
}
$$

.center[![:scale 70%](../images/cdf-binomial-n2.png)]

$$
\small{
\begin{aligned}
p_X(0)&=F_X(0)-F_X(-1)=0.25-0=0.25 \\\
\\\
p_X(1)&=F_X(1)-F_X(0)=0.75-0.25=0.5 \\\
\\\
p_X(2)&=F_X(2)-F_X(1)=1-0.75=0.25 \\\
\end{aligned}
}
$$

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# Properties of CDF $F_X(x)$

$F_X(x)$ is .red[nondecreasing].

$$
\begin{aligned}
\text{If } a &\leq b, \\\
\\\
\text{then } F_X(a) &\leq F_X(b).
\end{aligned}
$$

$F_X(x)$ tends to $0$ as $x\rightarrow -\infty$, and to $1$ as $x \rightarrow +\infty$.

.center[![:scale 45%](../images/cdf-example.png)]