Continuous Random Variables

Density Instead of Mass

A continuous random variable is described by a probability density function (PDF) $f(x)$, where:

$P(a \leq X \leq b) = \int_a^b f(x)\, dx$

Note that $P(X = x) = 0$ for any single point — only intervals have positive probability.

Cumulative Distribution Function

$F(x) = P(X \leq x) = \int_{-\infty}^x f(t)\, dt$

The Uniform Distribution

$X \sim \text{Uniform}(a, b)$ assigns equal density to every point in $[a, b]$:

$f(x) = \frac{1}{b-a}, \quad a \leq x \leq b$

$E[X] = \frac{a+b}{2}, \qquad \text{Var}(X) = \frac{(b-a)^2}{12}$

a, b = 0, 1
mean = (a + b) / 2
var  = (b - a)**2 / 12

print(round(mean, 4))  # 0.5
print(round(var, 4))   # 0.0833

Your Task

Implement uniform_stats(a, b) that prints $E[X]$ and $\text{Var}(X)$ of $\text{Uniform}(a, b)$, each rounded to 4 decimal places.