Exam PRandom variablesFree to read

Expectation, variance and moments

E[X] is the centre, Var(X) the spread, and Var(X) = E[X²] − (E[X])² is the identity that turns most exam questions into two sums.

The formulas

Expectation

$$E[X] = \sum_x x\,p(x) \quad\text{or}\quad \int_{-\infty}^{\infty} x f(x)\,dx$$

Law of the unconscious statistician

$$E[g(X)] = \sum_x g(x)p(x) \quad\text{or}\quad \int g(x)f(x)\,dx$$

Variance

$$\operatorname{Var}(X) = E\!\left[(X-\mu)^2\right] = E[X^{2}] - \mu^{2}$$

Linear transform

$$E[aX+b] = aE[X]+b, \qquad \operatorname{Var}(aX+b) = a^{2}\operatorname{Var}(X)$$

Survival form

$$E[X] = \int_0^{\infty} S(x)\,dx$$

non-negative X

Where it comes from

1. Expanding $E[(X-\mu)^2] = E[X^2 - 2\mu X + \mu^2]$ and using linearity gives $E[X^2] - 2\mu^2 + \mu^2 = E[X^2] - \mu^2$.
2. The shift $b$ moves the whole distribution and cannot change spread, which is why it disappears from the variance.
3. The survival form comes from swapping the order of integration in $\int_0^\infty \int_0^x f(x)\,dt\,dx$.

Worked example

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

• Var = E[X²] − (E[X])². Compute the second moment, never the squared deviations, unless the question hands you deviations.
• Adding a constant never changes a variance; multiplying by a scales it by a².

Traps

• Reporting the standard deviation when the question asked for the variance (or the reverse).
• Applying E[g(X)] = g(E[X]) - true only for linear g.

Related

• Moment generating functions
• Covariance, correlation and sums
• Random variables, CDFs and survival functions

Drill this: the Exam P question bank has original questions on this topic, and today’s free round is open to everyone.