Home/Formula sheets/Exam P

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

Joint distributions

Joint density: $\iint f (x, y) d x d y = 1$
Marginal: $f_{X} (x) = \int f (x, y) d y$
Conditional: $f_{Y ∣ X} (y ∣ x) = \frac{f ( x , y )}{f _{X} ( x )}$
Independence: $f (x, y) = f_{X} (x) f_{Y} (y) on a rectangular support$
Probability of a region: $P ((X, Y) \in A) = \iint_{A} f (x, y) d x d y$

Covariance

Covariance: $Cov (X, Y) = E [X Y] - E [X] E [Y]$
Correlation: $ρ = \frac{Cov ( X , Y )}{σ _{X} σ _{Y}}, - 1 \leq ρ \leq 1$
Variance of a sum: $Var (a X + bY) = a^{2} σ_{X}^{2} + b^{2} σ_{Y}^{2} + 2 ab Cov (X, Y)$
Bilinearity: $Cov (a X + b, c Y + d) = a c Cov (X, Y)$
Independence: $X ⊥ Y \Rightarrow Cov (X, Y) = 0$ the converse is false

Conditional expectation

Double expectation: $E [X] = E [E [X ∣ Y]]$
Conditional variance formula: $Var (X) = E [Var (X ∣ Y)] + Var (E [X ∣ Y])$
Compound mean: $E [S] = E [N] E [X]$
Compound variance: $Var (S) = E [N] Var (X) + Var (N) E [X]^{2}$

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.