uniform.Rmd

# Uniform Distribution {#uniform}

## Motivation {-}

How do we model a random variable that is equally likely to take on any real 
number between $a$ and $b$? 


## Theory {-}

```{definition uniform, name="Uniform Distribution"}
A random variable $X$ is said to follow a $\text{Uniform}(a, b)$ distribution if 
its p.d.f. is 
\[ f(x) = \begin{cases} \frac{1}{b-a} & a \leq x \leq b \\ 0 & \text{otherwise} \end{cases}, \]
or equivalently, if its c.d.f. is 
\[ F(x) = \begin{cases} 0 & x < a \\ \frac{x-a}{b-a} & a \leq x \leq b \\ 1 & x > b \end{cases}. \]
```

The p.d.f. and c.d.f. are graphed below.

```{r uniform-graph, echo=FALSE, fig.show = "hold", fig.align = "center",fig.cap="PDF and CDF of the Uniform Distribution", fig.asp=1.2}
par(mfrow=c(2, 1), mgp=c(1, 1, 0))

t <- seq(0, 2, by=0.1)
plot(t, rep(1, length(t)), type="l", lwd=3, col="blue", 
     xaxt="n", yaxt="n", bty="n",
     xlab="x", ylab="Probability Density Function f(x)", 
     xlim=c(-0.5, 2.5), ylim=c(-0.05, 1.1))
axis(1, pos=0, at=c(0, 2), labels=c("a", "b"))
axis(2, pos=0, at=c(0, 1), labels=c("", expression(frac(1, b - a))))
lines(c(-0.5, 2.5), c(0, 0))
lines(c(0, 0), c(-0.05, 1.1))
lines(c(-0.5, 0), c(0, 0), lwd=3, col="blue")
lines(c(2, 2.5), c(0, 0), lwd=3, col="blue")

plot(t, t / 2, type="l", lwd=3, col="red",
     xaxt="n", yaxt="n", bty="n",
     xlab="x", ylab="Cumulative Distribution Function F(x)", 
     xlim=c(-0.5, 2.5), ylim=c(-0.05, 1.1))
axis(1, pos=0, at=c(0, 2), labels=c("a", "b"))
axis(2, pos=0, at=c(0, 1), labels=c("", 1))
lines(c(-0.5, 2.5), c(0, 0))
lines(c(0, 0), c(-0.05, 1.1))
lines(c(-0.5, 0), c(0, 0), lwd=3, col="red")
lines(c(2, 2.5), c(1, 1), lwd=3, col="red")
```

Why is the p.d.f. of a $\text{Uniform}(a, b)$ random variable what it is?

- Since we want all values between $a$ and $b$ to be equally likely, the p.d.f. must be constant 
between $a$ and $b$.
- This constant is chosen so that the total area under the p.d.f. (i.e., the total probability) 
is 1. Since the p.d.f. is a rectangle of width $b-a$, the height must be $\frac{1}{b-a}$ to make 
the total area 1.

```{example fencing}
You buy fencing for a square enclosure. The length of fencing that you buy is uniformly 
distributed between 0 and 4 meters. What is the probability that the enclosed area will be 
larger than $0.5$ square meters?
```

```{solution}
Let $L$ be a $\text{Uniform}(a=0, b=4)$ random variable. Note that $L$ represents the perimeter of 
the square enclosure, so $L/4$ is the length of a side and the area is 
\[ A = \left( \frac{L}{4} \right)^2 = \frac{L^2}{16}. \]
We want to calculate $P(\frac{L^2}{16} > 0.5)$. To do this, we rearrange the expression inside the 
probability to isolate $L$, at which point we can calculate the probability by integrating the p.d.f. 
of $L$ over the appropriate range.
\begin{align*}
P(\frac{L^2}{16} > 0.5) &= P(L > \sqrt{8}) \\
&= \int_{\sqrt{8}}^{4} \frac{1}{4 - 0}\,dx \\
&= \frac{4 - \sqrt{8}}{4} \\
&\approx .293.
\end{align*}
```


## Essential Practice {-}

1. A point is chosen uniformly along the length of a stick, and the stick is broken at that point. 
What is the probability the left segment is more than twice as long as the right segment?

    (Hint: Assume the length of the stick is 1. Let $X$ be the point at which the stick is 
    broken, and observe that $X$ is the length of the left segment.)

2. You inflate a spherical balloon in a single breath. If the volume of air you exhale in a 
single breath (in cubic inches) is $\text{Uniform}(a=36\pi, b=288\pi)$ random variable, what 
is the probability that the radius of the balloon is less than 5 inches?