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: $F(x) = P(X \le x)$
Interval probability: $P(a < X \le b) = F(b) - F(a)$
Survival function: $S(x) = 1 - F(x) = P(X > x)$
Density: $f(x) = F'(x)$ continuous case
Percentile: $F(\pi_p) = p$
Hazard rate: $h(x) = \dfrac{f(x)}{S(x)} = -\dfrac{d}{dx}\ln S(x)$

Expectation and variance

Expectation: $E[X] = \sum_x x\,p(x) \quad\text{or}\quad \int_{-\infty}^{\infty} x f(x)\,dx$
Law of the unconscious statistician: $E[g(X)] = \sum_x g(x)p(x) \quad\text{or}\quad \int g(x)f(x)\,dx$
Variance: $\operatorname{Var}(X) = E\!\left[(X-\mu)^2\right] = E[X^{2}] - \mu^{2}$
Linear transform: $E[aX+b] = aE[X]+b, \qquad \operatorname{Var}(aX+b) = a^{2}\operatorname{Var}(X)$
Survival form: $E[X] = \int_0^{\infty} S(x)\,dx$ non-negative X

MGFs

Definition: $M_X(t) = E\!\left[e^{tX}\right]$
Moments: $M^{(k)}_X(0) = E[X^{k}]$
Sum of independents: $M_{X+Y}(t) = M_X(t)M_Y(t)$
Linear transform: $M_{aX+b}(t) = e^{bt}M_X(at)$
Uniqueness: $M_X = M_Y \Rightarrow X \stackrel{d}{=} Y$

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.