Random Variable

A function that assigns a numerical value to each outcome in a sample space. Discrete random variables take countable values; continuous random variables take values in an interval.

Probability Density Function (PDF)

For a continuous random variable X, the function f(x) such that P(a < X < b) equals the integral of f(x) from a to b. For discrete variables, the analogous function is the probability mass function (PMF).

Cumulative Distribution Function (CDF)

F(x) = P(X <= x). A non-decreasing, right-continuous function that goes from 0 to 1 and completely characterizes the distribution of a random variable.

Expected Value

E[X] = integral of x*f(x) dx for continuous variables, or the sum of x*P(X=x) for discrete variables. It represents the long-run average of the random variable.

Variance

Var(X) = E[(X - E[X])^2] = E[X^2] - (E[X])^2. A measure of the spread of a distribution around its mean. The standard deviation is the square root of the variance.

Moment-Generating Function (MGF)

M(t) = E[e^(tX)]. If it exists in a neighborhood of t=0, it uniquely determines the distribution. The n-th derivative evaluated at t=0 gives the n-th moment.

Conditional Expectation

E[X|Y=y] is the expected value of X given that Y equals y. The law of total expectation states E[X] = E[E[X|Y]], which is useful for computing means of mixture distributions.

Bayes Theorem

P(A|B) = P(B|A)*P(A) / P(B). Used to update the probability of an event based on new information. In insurance, often applied to classify risks after observing claims.

Covariance

Cov(X,Y) = E[XY] - E[X]*E[Y]. Measures the linear association between two random variables. If X and Y are independent, their covariance is zero (but the converse is not necessarily true).

Loss Elimination Ratio

The proportion of expected losses eliminated by a deductible d: LER(d) = E[X ^ d] / E[X], where E[X ^ d] is the limited expected value at d. Used in insurance to quantify the effect of deductibles.

Survival Function

S(x) = P(X > x) = 1 - F(x). In insurance contexts, it gives the probability that a loss exceeds a given threshold. Essential for computing excess loss and stop-loss premiums.

Deductible

The amount of loss retained by the policyholder before insurance payments begin. An ordinary deductible reduces each payment by d; a franchise deductible pays the full loss once it exceeds d.

P Glossary