Random variables, expectation and variance

So far we talked about probabilities of events. But in data we usually care about NUMBERS: how many items, how much money, how long. A random variable is exactly that — "a number that depends on chance". It has two main descriptors: the expectation (where its center is) and the variance (how widely it wanders around that center). These two concepts are the foundation everything that follows is built on.

E[X] = 3.00 (balance point)Var[X] = 1.20σ = 1.10

Outcome probabilities (drag the weights):

1: 10%

2: 20%

3: 40%

4: 20%

5: 10%

E[X] is the "center of gravity" of the values weighted by probability. Var[X] = E[(X−E[X])²] is the mean squared deviation. For a sum of INDEPENDENT variables both E and Var add up — which yields σ/√n for the mean and the bell of the CLT.

A random variable assigns a number to every random outcome. A die roll → the number shown; a user → how much they spent; a request → how many milliseconds the server took. Before you is a variable with five outcomes; bar height is the probability of each value. Drag the weights — the distribution changes.

you can flip beats with ← → keys

Why an analyst needs this

Expectation and variance are an analyst's working language. "How much will a user bring on average" is E[X]. "How unpredictable is revenue" is σ. Sample-size, confidence-interval and power calculations are the arithmetic of estimator variances.

The variance-addition rule explains why averaging calms noise: add n independent observations, divide by n — the mean's variance comes out n times smaller than a single observation's. Hence the σ/√n that will chase us through the whole course.

Where it shows up

A stock portfolio: expected return is E[X], risk is σ. Diversification works precisely through variance addition: uncorrelated assets damp the total spread.

Reliability engineering and queueing: the expected number of events and its variance (Poisson) let you size capacity and price the risk of overload.

Definitions

Random variable

a number that takes one value or another depending on the random outcome.

Expectation E[X]

E[X] = \sum x_i\, p_i

the variable's average over a long series of repeats: multiply each value xᵢ by its probability pᵢ and sum. It is the center of the distribution.

Variance Var[X]

\mathrm{Var}(X) = E\big[(X - E[X])^2\big]

the average squared deviation from the expectation; σ = √Var is the spread in original units.

Adding variances

\mathrm{Var}(X+Y) = \mathrm{Var}(X) + \mathrm{Var}(Y)

the variances of independent variables add up. The foundation of σ/√n, the CLT and ANOVA.

Independence

the outcome of one variable does not affect the distribution of the other. Without it, variance addition breaks.

Deep dive: the math and the mechanism (optional)

Where σ/√n comes from — the precision of the mean. Add n independent observations: their variances add, so the sum has variance n·σ². The mean is that same sum, divided by n. And when you divide a quantity by a constant c, its variance is divided by c² (the spread shrinks faster than the value itself). So the variance of the mean = n·σ² / n² = σ²/n, and the standard deviation of the mean (called the standard error) = σ/√n. That is why a larger sample gives a more precise mean — and why precision grows not like n but like √n.

An important asymmetry between the two rules. Expectation is linear ALWAYS: E[aX + b] = a·E[X] + b, and E[X + Y] = E[X] + E[Y] even for dependent quantities. Variances, however, add ONLY for independent ones. In general Var(X + Y) = Var(X) + Var(Y) + 2·Cov(X, Y), where Cov(X, Y) is the covariance, a measure of how they deviate together. It is exactly this covariance that variance-reduction methods exploit: pick an auxiliary quantity closely tied to the metric, and the surplus spread is subtracted from the variance of the estimate.