REFERENCE SHEET

Random variables & moments

A distribution describes a random quantity; moments summarize selected features.

Before this Events, conditioning & Bayes, Integration & accumulation

Var(X) = E[X²] − E[X]²

Definitions

Random variable

A measurable function from a probability space to ℝ. Its distribution assigns probabilities to values or sets of values.

Expectation

For discrete X, E[X]=Σₓ xP(X=x); for a density f, E[X]=∫xf(x)dx, when absolutely integrable.

Variance and covariance

Var(X)=E[(X−E[X])²]; Cov(X,Y)=E[(X−E[X])(Y−E[Y])], assuming finite second moments.

Key theorems

Linearity of expectation

E[aX+bY]=aE[X]+bE[Y] for integrable X,Y, without requiring independence.

Variance of a sum

Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y). Independence with finite second moments implies zero covariance.

Total expectation

For integrable X and a random variable Y, E[E[X|Y]]=E[X].

Notable applications

Risk and variability

Means describe location while variance describes squared spread; neither alone identifies a general distribution.

Monte Carlo

Average function values at random samples to estimate expectations and integrals.

Work it out

Try before revealing

01 / Compute

Let X be Bernoulli(p). Find E[X] and Var(X).

Hint

X²=X.

Solution

E[X]=p and E[X²]=p, hence Var(X)=p−p²=p(1−p).

02 / Explain

For X uniform on [−1,1] and Y=X², show zero covariance but dependence.

Hint

Use symmetry for the odd moments.