Actuarial Mathematics Unit 1 Review: Probability Theory & Distributions

Key Concepts and Terminology

• Probability measures the likelihood of an event occurring and ranges from 0 (impossible) to 1 (certain)
• Random variables assign numerical values to outcomes of a random experiment
  • Discrete random variables have countable outcomes (number of heads in 10 coin flips)
  • Continuous random variables have uncountable outcomes (time until next customer arrives)
• Probability distributions describe the probabilities of different outcomes for a random variable
• Expectation (mean) represents the average value of a random variable over many trials
• Variance and standard deviation measure the dispersion or spread of a probability distribution
• Independence implies that the occurrence of one event does not affect the probability of another event
• Conditional probability measures the probability of an event given that another event has occurred
• Bayes' theorem relates conditional probabilities and is used for updating probabilities based on new information

Probability Basics

• Probability axioms define the properties that probability measures must satisfy
  • Non-negativity: $P(A) \geq 0$ for any event $A$
  • Normalization: $P(\Omega) = 1$, where $\Omega$ is the sample space (set of all possible outcomes)
  • Countable additivity: For mutually exclusive events $A_1, A_2, \ldots$, $P(\bigcup_{i=1}^{\infty} A_i) = \sum_{i=1}^{\infty} P(A_i)$
• Probability of the complement: $P(A^c) = 1 - P(A)$, where $A^c$ is the complement of event $A$
• Probability of the union of two events: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
• Conditional probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$, where $P(B) > 0$
• Independence: Events $A$ and $B$ are independent if $P(A \cap B) = P(A)P(B)$
• Multiplication rule: $P(A \cap B) = P(A)P(B|A)$
• Law of total probability: For a partition $\{B_1, B_2, \ldots, B_n\}$ of the sample space, $P(A) = \sum_{i=1}^{n} P(A|B_i)P(B_i)$

Random Variables

• Random variables map outcomes of a random experiment to real numbers
• Probability mass function (PMF) for a discrete random variable $X$: $p_X(x) = P(X = x)$
  • Properties: $p_X(x) \geq 0$ and $\sum_x p_X(x) = 1$
• Cumulative distribution function (CDF) for a random variable $X$: $F_X(x) = P(X \leq x)$
  • Properties: $F_X(x)$ is non-decreasing, right-continuous, $\lim_{x \to -\infty} F_X(x) = 0$, and $\lim_{x \to \infty} F_X(x) = 1$
• Probability density function (PDF) for a continuous random variable $X$: $f_X(x) = F_X'(x)$
  • Properties: $f_X(x) \geq 0$ and $\int_{-\infty}^{\infty} f_X(x) dx = 1$
• Relationship between CDF and PDF: $F_X(x) = \int_{-\infty}^{x} f_X(t) dt$
• Quantile function (inverse CDF): $Q_X(p) = \inf\{x: F_X(x) \geq p\}$, where $0 < p < 1$

Probability Distributions

• Bernoulli distribution: Models a single trial with two possible outcomes (success or failure)
  • PMF: $p_X(x) = p^x(1-p)^{1-x}$, where $x \in \{0, 1\}$ and $0 < p < 1$
• Binomial distribution: Models the number of successes in a fixed number of independent Bernoulli trials
  • PMF: $p_X(x) = \binom{n}{x}p^x(1-p)^{n-x}$, where $x \in \{0, 1, \ldots, n\}$, $0 < p < 1$, and $\binom{n}{x} = \frac{n!}{x!(n-x)!}$
• Poisson distribution: Models the number of events occurring in a fixed interval of time or space
  • PMF: $p_X(x) = \frac{e^{-\lambda}\lambda^x}{x!}$, where $x \in \{0, 1, 2, \ldots\}$ and $\lambda > 0$
• Exponential distribution: Models the time between events in a Poisson process
  • PDF: $f_X(x) = \lambda e^{-\lambda x}$, where $x > 0$ and $\lambda > 0$
• Normal (Gaussian) distribution: Models many natural phenomena and is characterized by its bell-shaped curve
  • PDF: $f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$, where $x \in \mathbb{R}$, $\mu \in \mathbb{R}$, and $\sigma > 0$

Common Probability Distributions

• Uniform distribution: Models a random variable with equally likely outcomes over a specified interval
  • Discrete uniform: $p_X(x) = \frac{1}{n}$, where $x \in \{1, 2, \ldots, n\}$
  • Continuous uniform: $f_X(x) = \frac{1}{b-a}$, where $x \in [a, b]$
• Geometric distribution: Models the number of trials until the first success in a sequence of independent Bernoulli trials
  • PMF: $p_X(x) = (1-p)^{x-1}p$, where $x \in \{1, 2, \ldots\}$ and $0 < p < 1$
• Negative binomial distribution: Models the number of trials until a specified number of successes occur in a sequence of independent Bernoulli trials
  • PMF: $p_X(x) = \binom{x-1}{r-1}p^r(1-p)^{x-r}$, where $x \in \{r, r+1, \ldots\}$, $0 < p < 1$, and $r \in \{1, 2, \ldots\}$
• Gamma distribution: Generalizes the exponential distribution and models waiting times and lifetimes
  • PDF: $f_X(x) = \frac{1}{\Gamma(\alpha)\beta^\alpha}x^{\alpha-1}e^{-\frac{x}{\beta}}$, where $x > 0$, $\alpha > 0$, and $\beta > 0$
• Beta distribution: Models probabilities, proportions, and percentages
  • PDF: $f_X(x) = \frac{1}{B(\alpha, \beta)}x^{\alpha-1}(1-x)^{\beta-1}$, where $x \in (0, 1)$, $\alpha > 0$, and $\beta > 0$

Expectation and Variance

• Expectation (mean) of a discrete random variable $X$: $E[X] = \sum_x xp_X(x)$
• Expectation (mean) of a continuous random variable $X$: $E[X] = \int_{-\infty}^{\infty} xf_X(x) dx$
• Linearity of expectation: $E[aX + bY] = aE[X] + bE[Y]$ for constants $a$ and $b$ and random variables $X$ and $Y$
• Variance of a random variable $X$: $Var(X) = E[(X - E[X])^2] = E[X^2] - (E[X])^2$
• Standard deviation: $\sigma_X = \sqrt{Var(X)}$
• Covariance between random variables $X$ and $Y$: $Cov(X, Y) = E[(X - E[X])(Y - E[Y])]$
  • Properties: $Cov(X, X) = Var(X)$ and $Cov(aX + b, cY + d) = acCov(X, Y)$ for constants $a$, $b$, $c$, and $d$
• Correlation coefficient between random variables $X$ and $Y$: $\rho_{X,Y} = \frac{Cov(X, Y)}{\sigma_X\sigma_Y}$
  • Properties: $-1 \leq \rho_{X,Y} \leq 1$, $\rho_{X,Y} = 1$ for perfect positive linear relationship, and $\rho_{X,Y} = -1$ for perfect negative linear relationship

Multivariate Distributions

• Joint probability mass function (PMF) for discrete random variables $X$ and $Y$: $p_{X,Y}(x, y) = P(X = x, Y = y)$
• Joint probability density function (PDF) for continuous random variables $X$ and $Y$: $f_{X,Y}(x, y)$
  • Properties: $f_{X,Y}(x, y) \geq 0$ and $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} f_{X,Y}(x, y) dx dy = 1$
• Marginal PMF: $p_X(x) = \sum_y p_{X,Y}(x, y)$ and $p_Y(y) = \sum_x p_{X,Y}(x, y)$
• Marginal PDF: $f_X(x) = \int_{-\infty}^{\infty} f_{X,Y}(x, y) dy$ and $f_Y(y) = \int_{-\infty}^{\infty} f_{X,Y}(x, y) dx$
• Conditional PMF: $p_{Y|X}(y|x) = \frac{p_{X,Y}(x, y)}{p_X(x)}$, where $p_X(x) > 0$
• Conditional PDF: $f_{Y|X}(y|x) = \frac{f_{X,Y}(x, y)}{f_X(x)}$, where $f_X(x) > 0$
• Independence for discrete random variables: $p_{X,Y}(x, y) = p_X(x)p_Y(y)$ for all $x$ and $y$
• Independence for continuous random variables: $f_{X,Y}(x, y) = f_X(x)f_Y(y)$ for all $x$ and $y$

Applications in Actuarial Science

• Pricing insurance policies using probability distributions to model claim frequencies and severities
  • Poisson distribution for modeling claim counts
  • Exponential, gamma, and Pareto distributions for modeling claim sizes
• Calculating risk measures such as Value-at-Risk (VaR) and Tail Value-at-Risk (TVaR) using quantiles of loss distributions
• Estimating reserves for outstanding claims using stochastic models and simulation techniques
• Assessing the solvency and capital requirements of insurance companies using probabilistic approaches
  • Solvency II and Swiss Solvency Test frameworks
• Pricing and hedging financial derivatives using stochastic calculus and martingale methods
  • Black-Scholes model for pricing European options
  • Binomial and trinomial tree models for pricing American options
• Modeling dependence between risks using copulas and multivariate distributions
  • Gaussian copula for modeling linear dependence
  • t-copula and Archimedean copulas (Clayton, Gumbel, Frank) for modeling non-linear dependence
• Applying credibility theory to blend individual and collective risk information for experience rating
  • Bühlmann-Straub model and empirical Bayes estimators