The Uniform Distribution

A uniform distribution is simply one where every value in a range is equally likely, like your wait for a bus that comes every 20 minutes when you show up at a random time. "Uniform" means the same everywhere: the density curve is perfectly flat. So the chance of landing in a piece of the range is just how big that piece is. The formulas look like three different things, but each is simple. $f(x) = \frac{1}{b - a}$ (the height, chosen so the rectangle's area comes out to 1), $\mu = \frac{a + b}{2}$ (the middle of the interval, where it balances) and $\sigma = \frac{b - a}{\sqrt{12}}$ (the spread, which grows with the width). The $\sqrt{12}$ is the only odd-looking part, and you'll see where it comes from in the practice problems.

Probabilities are proportions of the interval. This is the simplest density there is, so it's a good place to see the "probability is area" idea from the last lesson in action. A rectangle's area is width times height, so $P(c \le X \le d) = (d - c)\cdot\frac{1}{b - a} = \frac{d - c}{b - a}$ for $a \le c \le d \le b$. Notice that only the width of the window matters, not where it sits.

Percentiles are just as easy: the $k$th percentile sits a fraction $\frac{k}{100}$ of the way from $a$ to $b$.

Random number generators produce uniform values, which simulations then transform into every other distribution; rounding errors are often modeled as uniform.