Discrete Random Variables

From Events to Numbers

A random variable $X$ is a function that assigns a number to each outcome in $\Omega$. A discrete random variable takes countably many values.

Probability Mass Function (PMF)

$p(x) = P(X = x)$ — the probability of each value. Must satisfy: $\sum_x p(x) = 1, \quad p(x) \geq 0$

Expectation

$E[X] = \sum_x x \cdot P(X = x)$

The long-run average value of $X$ over many repetitions.

Variance

$\text{Var}(X) = E[X^2] - (E[X])^2 = \sum_x x^2 \cdot P(X=x) - \left(\sum_x x \cdot P(X=x)\right)^2$

Variance measures spread around the mean.

# Fair die: values 1-6, each with prob 1/6
values = [1, 2, 3, 4, 5, 6]
probs  = [1/6] * 6

mean = sum(x * p for x, p in zip(values, probs))
var  = sum(x**2 * p for x, p in zip(values, probs)) - mean**2

print(round(mean, 4))  # 3.5
print(round(var, 4))   # 2.9167

Your Task

Implement discrete_stats(values, probs) that prints $E[X]$ and $\text{Var}(X)$, each rounded to 4 decimal places.