Random variables - Exam P formula sheet

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

3 topics, 16 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

Random variables

CDF: $F (x) = P (X \leq x)$
Interval probability: $P (a < X \leq b) = F (b) - F (a)$
Survival function: $S (x) = 1 - F (x) = P (X > x)$
Density: $f (x) = F^{'} (x)$ continuous case
Percentile: $F (π_{p}) = p$
Hazard rate: $h (x) = \frac{f ( x )}{S ( x )} = - \frac{d}{d x} ln S (x)$

Expectation and variance

Expectation: $E [X] = \sum_{x} x p (x) or \int_{- \infty}^{\infty} x f (x) d x$
Law of the unconscious statistician: $E [g (X)] = \sum_{x} g (x) p (x) or \int g (x) f (x) d x$
Variance: $Var (X) = E [(X - μ)^{2}] = E [X^{2}] - μ^{2}$
Linear transform: $E [a X + b] = a E [X] + b, Var (a X + b) = a^{2} Var (X)$
Survival form: $E [X] = \int_{0}^{\infty} S (x) d x$ non-negative X

MGFs

Definition: $M_{X} (t) = E [e^{tX}]$
Moments: $M_{X}^{(k)} (0) = E [X^{k}]$
Sum of independents: $M_{X + Y} (t) = M_{X} (t) M_{Y} (t)$
Linear transform: $M_{a X + b} (t) = e^{b t} M_{X} (a t)$
Uniqueness: $M_{X} = M_{Y} \Rightarrow X = d Y$

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