Random Variables and Distributions

Random Variables and PMF vs PDF

A random variable turns random outcomes into numbers, letting us do math with chance. It maps qualitative results to quantitative values.

For continuous variables, $P(a \le X \le b) = \int_a^b f(x)\, dx$.

Common pitfall: For a continuous variable, the probability of any single exact value is zero ($P(X = c) = 0$). Probability lives in intervals, not points; a point has zero width and zero area.

Expectation and Variance

Two core numbers summarise a random variable:

• Expected value ($E[X]$): The long-run average, or centre of mass. Discrete: $\sum x\, P(X = x)$. Continuous: $\int x f(x)\, dx$.
• Variance ($\operatorname{Var}(X)$): Expected squared deviation $E[(X - \mu)^2]$, measuring spread. Standard deviation is its square root.

Expectation is linear: $E[aX + b] = a\,E[X] + b$, and $E[X + Y] = E[X] + E[Y]$ for any variables, whether independent or not.

Variance is not linear: $\operatorname{Var}(aX) = a^2\operatorname{Var}(X)$. Crucially, variances add ($\operatorname{Var}(X + Y) = \operatorname{Var}(X) + \operatorname{Var}(Y)$) only for independent variables.