Random Variables and Distributions

Numbers Governed by Chance

A random variable assigns a number to each outcome of a random experiment — turning "heads or tails" into "number of heads." This lets us do arithmetic and calculus with chance. Random variables come in two kinds:

• Discrete — takes countable, separate values (a die roll, a count). Its probability mass function gives $P(X = x)$ for each value, and the masses sum to 1.
• Continuous — takes any value in a range (a height, a time). Its probability density function $f(x)$ gives probabilities via areas: $P(a \le X \le b) = \int_a^b f(x)\, dx$, and the total area is 1.

A key subtlety for continuous variables: the probability of any single exact value is zero — probability lives in intervals (areas under the density), not at points. This is the integral calculus of Unit 6 applied to chance.

Two numbers summarise a random variable:

• Expected value (mean) $E[X]$ — the long-run average, a probability-weighted sum $\sum x\, P(X = x)$ for discrete (or $\int x f(x)\, dx$ for continuous). It is the "centre of mass" of the distribution.
• Variance $\operatorname{Var}(X) = E[(X - \mu)^2]$ — the expected squared deviation, measuring spread, with standard deviation its square root.

Expectation is linear: $E[aX + b] = a\,E[X] + b$, and $E[X + Y] = E[X] + E[Y]$ for any $X, Y$ (independent or not). Variance is not linear — $\operatorname{Var}(aX) = a^2\operatorname{Var}(X)$, and variances add only for independent variables. Random variables are the bridge from raw probability to statistics and the modelling of real quantities.

Common pitfall: thinking a continuous random variable has a positive probability at a single exact value. For a continuous distribution, $P(X = c) = 0$ for any specific $c$ — probability is the area under the density over an interval, and a single point has zero width, hence zero area. Only for discrete variables does an individual value carry positive probability.