Expectations and conditional expectations: formula sheet

Key formulas

Expectation, discrete

E[X] = Σ x · P(X = x)

Sum over all values x that X can take. The probabilities must add to 1.

Expectation, continuous

E[X] = ∫ x · f(x) dx

Integrate over the range where f(x) > 0. Check that ∫ f(x) dx = 1.

Expectation of a function

E[g(X)] = Σ g(x) · P(X = x) or E[g(X)] = ∫ g(x) · f(x) dx

Use the original probabilities or density of X, not those of g(X).

Linearity

E[aX + b] = a·E[X] + b

Holds for any constants a and b. Also E[X + Y] = E[X] + E[Y] for any X and Y.

Variance from expectations

Var(X) = E[X²] − (E[X])²

Needs E[X²], which is E[g(X)] with g(x) = x².

Expectation from survival function

E[X] = ∫ from 0 to ∞ of P(X > x) dx

For a non-negative continuous variable. Useful when the survival function is simple.

Linearity of expectation

E[aX + bY + c] = aE[X] + bE[Y] + c

Always true. No independence needed. Extends to any number of variables.

Variance (computational form)

Var(X) = E[X²] − (E[X])²

Usually the fastest route. Variance is never negative.

Variance of a linear function

Var(aX + c) = a²Var(X)

The constant c drops out. The multiplier a is squared, so a sign change does not matter.

Covariance

Cov(X, Y) = E[XY] − E[X]E[Y]

Also Cov(X, Y) = E[(X − μX)(Y − μY)]. Cov(X, X) = Var(X).

Variance of a linear combination

Var(aX + bY) = a²Var(X) + b²Var(Y) + 2ab Cov(X, Y)

For independent variables, Cov = 0 and the last term vanishes.

Covariance with constants

Cov(aX + b, cY + d) = ac Cov(X, Y)

Bilinear in both arguments. Constants added do not matter.

Correlation

ρ(X, Y) = Cov(X, Y) ÷ (σX σY)

Always between −1 and 1, when both standard deviations are non-zero.

Variance of a sum of n variables

Var(ΣXᵢ) = ΣVar(Xᵢ) + 2 Σ(i<j) Cov(Xᵢ, Xⱼ)

For independent variables, Var(ΣXᵢ) = ΣVar(Xᵢ).

Moments

k-th raw moment = E[Xᵏ]; k-th central moment = E[(X − μ)ᵏ]

Skewness = E[(X − μ)³] ÷ σ³. Kurtosis = E[(X − μ)⁴] ÷ σ⁴.

Conditional pmf

p(x | y) = P(X = x, Y = y) ÷ P(Y = y)

Needs P(Y = y) > 0. For fixed y, the values p(x | y) add up to 1 over x.

Conditional pdf

f(x | y) = f(x, y) ÷ f_Y(y)

Needs f_Y(y) > 0. Get f_Y(y) by integrating the joint pdf over x. For fixed y, f(x | y) integrates to 1 over x.

Conditional expectation (discrete)

E[X | Y = y] = Σ x · p(x | y)

Sum over all x with p(x | y) > 0.

Conditional expectation (continuous)

E[X | Y = y] = ∫ x · f(x | y) dx

Integrate over the range of x given y. The limits can depend on y.

Conditional expectation of a function

E[h(X) | Y = y] = ∫ h(x) f(x | y) dx

Use this for E[X² | Y = y] when you need a conditional variance.

Conditional variance

Var(X | Y = y) = E[X² | Y = y] − (E[X | Y = y])²

Same as the usual variance formula, using conditional moments.

Tower property

E[E[X | Y]] = E[X]

Use it to check your answer. Average E[X | Y = y] over the marginal of Y.

Independence

If X and Y are independent, f(x | y) = f_X(x) and E[X | Y] = E[X]

Independence is sufficient. A constant E[X | Y] alone does not prove independence.

Law of total expectation (tower property)

E[X] = E[E[X | Y]]

Needs E[X] to exist. Y can be discrete or continuous.

Law of total variance

Var(X) = E[Var(X | Y)] + Var(E[X | Y])

Needs Var(X) to be finite. First term is the average conditional variance. Second is the variance of the conditional mean.

Compound sum mean

E[S] = E[N] × E[X]

S = X₁ + ... + X_N, N independent of the Xᵢ, Xᵢ identically distributed.

Compound sum variance

Var(S) = E[N] × Var(X) + Var(N) × (E[X])²

Same conditions. It comes from E[Var(S | N)] = E[N]Var(X) and Var(E[S | N]) = Var(N)(E[X])², using independent Xᵢ.

Conditional moments of a compound sum

E[S | N] = N × E[X], Var(S | N) = N × Var(X)

Valid when the Xᵢ are independent and identically distributed and independent of N.

Variance from conditional variance definition

Var(X | Y) = E[X² | Y] − (E[X | Y])²

Use it to find conditional variance when only conditional moments are given.

Quick revision

• E[X] = Σ x P(X = x) for discrete; E[X] = ∫ x f(x) dx for continuous.
• E[g(X)] = Σ g(x) P(X = x) or ∫ g(x) f(x) dx. In general, E[g(X)] ≠ g(E[X]).
• Linearity: E[aX + bY] = aE[X] + bE[Y], with no independence needed.
• Var(X) = E[X²] − (E[X])²; Var(aX + b) = a²Var(X).
• Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y); if independent, Cov = 0.
• Zero covariance does not imply independence.
• If X and Y are independent, E[XY] = E[X]E[Y].
• Conditional pdf: f(x | y) = f(x, y) ÷ f_Y(y), where f_Y(y) > 0.
• E[X | Y] is a random variable, a function of Y.
• Law of total expectation: E[X] = E[E[X | Y]].
• Law of total variance: Var(X) = E[Var(X | Y)] + Var(E[X | Y]).
• For compound sums S = X₁ + … + X_N with i.i.d. claims independent of N: E[S] = E[N]E[X] and Var(S) = E[N]Var(X) + Var(N)(E[X])².

Common mistakes

• Writing E[g(X)] = g(E[X]) Fix: Apply g to each value inside the sum or integral. For X², compute E[X²] directly or use Var(X) + (E[X])².
• Forgetting to multiply by x (or g(x)) and integrating only f(x) Fix: Write the integrand explicitly as g(x)·f(x) before integrating.
• Writing Var(aX) = a Var(X). Fix: Remember the multiplier is squared: Var(aX) = a²Var(X).
• Writing Var(X − Y) = Var(X) − Var(Y). Fix: Use Var(X − Y) = Var(X) + Var(Y) − 2Cov(X, Y). For independent variables the variances add.
• Using the joint pmf or pdf directly as the conditional distribution. Fix: Always divide by the marginal of Y. Check that the conditional sums or integrates to 1.
• Dividing by the wrong marginal, for example f_X(x) instead of f_Y(y). Fix: The variable after the bar is the one you divide by. For X given Y, divide by f_Y(y).
• Writing Var(X) = E[Var(X | Y)] and forgetting the second term. Fix: Always write both terms first, then compute each. The second term is zero only if E[X | Y] is constant.
• Using Var(N)E[X] instead of Var(N)(E[X])². Fix: Var(aN) = a² Var(N). Here a = E[X], so square it.

Exam tips

• In MCQs, check whether the question wants E[X], E[X²] or Var(X). Examiners often offer E[X²] as a wrong option for the variance.
• For deductible or cap questions, write the payment as a piecewise function before integrating. Marks are given for the setup.
• In written answers, state the formula, the limits and the working. A correct setup earns marks even if the arithmetic slips.
• Check the result lies within the range of the variable. A mean of 1.5 for a variable on (0, 1) means an error.
• If the question gives a survival function, try ∫ P(X > x) dx first. It is often shorter.
• In written answers, state which rule you use, for example linearity, and whether independence is needed.
• For MCQs, check the variance sign trap: Var(X − Y) with independent X and Y adds the variances.
• When a question gives ρ instead of covariance, convert first using Cov = ρ σX σY. Remember σ is the square root of the variance.