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Multivariate probability - Exam P formula sheet

Every Exam P formula in multivariate probability on one printable sheet, condensed from the 3 reference pages that teach them.

3 topics, 14 formula lines. Every line is condensed from the Exam P formula pages, which stay free to read in full. Print this page to get the sheet on paper; the site chrome drops away.

What is on this sheet

Multivariate probability

Multivariate probability

Joint distributions

Joint density
∬f(x,y) dx dy=1\iint f(x,y)\,dx\,dy = 1
Marginal
fX(x)=∫f(x,y) dyf_X(x) = \int f(x,y)\,dy
Conditional
fY∣X(y∣x)=f(x,y)fX(x)f_{Y\mid X}(y\mid x) = \dfrac{f(x,y)}{f_X(x)}
Independence
f(x,y)=fX(x)fY(y) on a rectangular supportf(x,y) = f_X(x)f_Y(y) \text{ on a rectangular support}
Probability of a region
P((X,Y)∈A)=∬Af(x,y) dx dyP((X,Y)\in A) = \iint_A f(x,y)\,dx\,dy

Covariance

Covariance
Cov⁡(X,Y)=E[XY]−E[X]E[Y]\operatorname{Cov}(X,Y) = E[XY] - E[X]E[Y]
Correlation
ρ=Cov⁡(X,Y)σXσY,−1≤ρ≤1\rho = \dfrac{\operatorname{Cov}(X,Y)}{\sigma_X\sigma_Y}, \quad -1 \le \rho \le 1
Variance of a sum
Var⁡(aX+bY)=a2σX2+b2σY2+2abCov⁡(X,Y)\operatorname{Var}(aX+bY) = a^{2}\sigma_X^{2} + b^{2}\sigma_Y^{2} + 2ab\operatorname{Cov}(X,Y)
Bilinearity
Cov⁡(aX+b, cY+d)=acCov⁡(X,Y)\operatorname{Cov}(aX+b,\,cY+d) = ac\operatorname{Cov}(X,Y)
Independence
X⊥Y⇒Cov⁡(X,Y)=0X \perp Y \Rightarrow \operatorname{Cov}(X,Y)=0the converse is false

Conditional expectation

Double expectation
E[X]=E[E[X∣Y]]E[X] = E\big[E[X\mid Y]\big]
Conditional variance formula
Var⁡(X)=E[Var⁡(X∣Y)]+Var⁡(E[X∣Y])\operatorname{Var}(X) = E\big[\operatorname{Var}(X\mid Y)\big] + \operatorname{Var}\big(E[X\mid Y]\big)
Compound mean
E[S]=E[N] E[X]E[S] = E[N]\,E[X]
Compound variance
Var⁡(S)=E[N]Var⁡(X)+Var⁡(N) E[X]2\operatorname{Var}(S) = E[N]\operatorname{Var}(X) + \operatorname{Var}(N)\,E[X]^{2}

Multivariate probability - Exam P formula sheet. Formulas condensed from the free reference pages on this site. Not affiliated with or endorsed by the Society of Actuaries.