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

\int_a^b f(x)\,dx

. For discrete variables, the analogous function is the probability mass function (PMF).

Cumulative Distribution Function (CDF)

F(x) = P(X \le 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] = \int x \cdot f(x)\,dx

for continuous variables, or

\sum x \cdot P(X=x)

for discrete variables. It represents the long-run average of the random variable.

Variance

\text{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) \cdot 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

\text{Cov}(X,Y) = E[XY] - E[X] \cdot 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[\min(X, d)] / E[X]

, where

E[\min(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

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