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

$$E[X]=\sum _{x}xp(x)\text{or}{\int }_{-\infty }^{\infty }xf(x)dx$$

Law of the unconscious statistician

$$E[g(X)]=\sum _{x}g(x)p(x)\text{or}\int g(x)f(x)dx$$

Variance

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

Linear transform

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

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 ${\int }_0^{\infty }{\int }_0^{x}f(x)dtdx$.

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.