1.5 Discrete distributions (Bernoulli, binomial, Poisson)

Discrete distributions like Bernoulli, binomial, and Poisson are key tools in actuarial math. They model events with specific outcomes, like insurance claims or defective products. Understanding these distributions helps predict probabilities and analyze risk.

Each distribution has unique properties and applications. Bernoulli models single trials, binomial handles multiple trials, and Poisson deals with rare events. Mastering these concepts is crucial for actuaries to make accurate predictions and informed decisions in various scenarios.

Bernoulli distribution

• The Bernoulli distribution is a discrete probability distribution that models a single trial with two possible outcomes (success or failure)
• It serves as the foundation for other discrete distributions like the binomial distribution
• The Bernoulli distribution is essential in actuarial mathematics for modeling events with binary outcomes, such as whether a claim is made or not

Probability mass function

• The probability mass function (PMF) of a Bernoulli random variable X is given by: $P(X=x) = p^x(1-p)^{1-x}$ for $x \in {0,1}$
  • $p$ represents the probability of success
  • $1-p$ represents the probability of failure
• The PMF assigns probabilities to the two possible outcomes of a Bernoulli trial (success: $x=1$, failure: $x=0$)

Mean and variance

• The mean (expected value) of a Bernoulli random variable X is given by: $E(X) = p$
• The variance of a Bernoulli random variable X is given by: $Var(X) = p(1-p)$
  • The variance measures the spread or dispersion of the distribution around its mean
• The standard deviation is the square root of the variance: $\sigma = \sqrt{p(1-p)}$

Applications of Bernoulli distribution

• Modeling the outcome of a single insurance claim (claim or no claim)
• Modeling the success or failure of a single medical treatment
• Analyzing the probability of a defective item in quality control (defective or non-defective)
• Serving as a building block for more complex distributions like the binomial distribution

Binomial distribution

• The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials
• It is a discrete probability distribution that describes the probability of obtaining a specific number of successes in n trials, each with a constant probability of success p
• The binomial distribution is crucial in actuarial mathematics for modeling events that have a fixed number of trials and two possible outcomes

Probability mass function

• The probability mass function (PMF) of a binomial random variable X is given by: $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$ for $k = 0,1,2,...,n$
  • $n$ represents the total number of trials
  • $p$ represents the probability of success in each trial
  • $\binom{n}{k}$ is the binomial coefficient, which calculates the number of ways to choose $k$ successes from $n$ trials
• The PMF gives the probability of observing exactly $k$ successes in $n$ trials

Cumulative distribution function

• The cumulative distribution function (CDF) of a binomial random variable X is given by: $F(x) = P(X \leq x) = \sum_{k=0}^{\lfloor x \rfloor} \binom{n}{k}p^k(1-p)^{n-k}$
  • The CDF gives the probability that the random variable X takes a value less than or equal to x
• The CDF can be used to calculate probabilities for ranges of values (e.g., $P(a \leq X \leq b) = F(b) - F(a-1)$)

Mean and variance

• The mean (expected value) of a binomial random variable X is given by: $E(X) = np$
• The variance of a binomial random variable X is given by: $Var(X) = np(1-p)$
  • The variance measures the spread or dispersion of the distribution around its mean
• The standard deviation is the square root of the variance: $\sigma = \sqrt{np(1-p)}$

Moment generating function

• The moment generating function (MGF) of a binomial random variable X is given by: $M_X(t) = (pe^t + 1 - p)^n$
• The MGF uniquely determines the distribution and can be used to calculate moments and other properties of the distribution
  • The $k$-th moment can be found by evaluating the $k$-th derivative of the MGF at $t=0$

Properties of binomial distribution

• The sum of two independent binomial random variables with the same success probability $p$ follows a binomial distribution with parameters $n_1 + n_2$ and $p$
• The binomial distribution approaches a normal distribution as $n$ increases and $p$ is not too close to 0 or 1 (usually when $np \geq 5$ and $n(1-p) \geq 5$)
  • This property allows for the use of the normal approximation to the binomial distribution in certain situations

Binomial approximation to hypergeometric

• The binomial distribution can be used to approximate the hypergeometric distribution when the population size is large relative to the sample size
• The approximation is appropriate when the sample size $n$ is less than 5% of the population size $N$
  • In this case, the binomial distribution with parameters $n$ and $p = K/N$ (where $K$ is the number of successes in the population) closely approximates the hypergeometric distribution

Applications of binomial distribution

• Modeling the number of insurance claims in a fixed number of policies
• Analyzing the number of defective items in a batch of products
• Predicting the number of successful treatments in a group of patients
• Calculating the probability of winning a certain number of games in a sports series

Poisson distribution

• The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space, given a known average rate of occurrence
• It is characterized by a single parameter $\lambda$, which represents the average number of events per interval
• The Poisson distribution is widely used in actuarial mathematics for modeling rare events and claim frequencies

Probability mass function

• The probability mass function (PMF) of a Poisson random variable X is given by: $P(X=k) = \frac{e^{-\lambda}\lambda^k}{k!}$ for $k = 0,1,2,...$
  • $\lambda$ represents the average number of events per interval
  • $e$ is the mathematical constant (Euler's number) approximately equal to 2.71828
• The PMF gives the probability of observing exactly $k$ events in the given interval

Cumulative distribution function

• The cumulative distribution function (CDF) of a Poisson random variable X is given by: $F(x) = P(X \leq x) = \sum_{k=0}^{\lfloor x \rfloor} \frac{e^{-\lambda}\lambda^k}{k!}$
  • The CDF gives the probability that the random variable X takes a value less than or equal to x
• The CDF can be used to calculate probabilities for ranges of values (e.g., $P(a \leq X \leq b) = F(b) - F(a-1)$)

Mean and variance

• The mean (expected value) and variance of a Poisson random variable X are both equal to the parameter $\lambda$: $E(X) = Var(X) = \lambda$
  • This property is unique to the Poisson distribution
• The standard deviation is the square root of the variance: $\sigma = \sqrt{\lambda}$

Moment generating function

• The moment generating function (MGF) of a Poisson random variable X is given by: $M_X(t) = e^{\lambda(e^t-1)}$
• The MGF uniquely determines the distribution and can be used to calculate moments and other properties of the distribution
  • The $k$-th moment can be found by evaluating the $k$-th derivative of the MGF at $t=0$

Properties of Poisson distribution

• The sum of two independent Poisson random variables with parameters $\lambda_1$ and $\lambda_2$ follows a Poisson distribution with parameter $\lambda_1 + \lambda_2$
• The Poisson distribution is the limiting case of the binomial distribution when $n$ is large, $p$ is small, and $np = \lambda$ remains constant
  • This property allows for the Poisson approximation to the binomial distribution in certain situations

Poisson approximation to binomial

• The Poisson distribution can be used to approximate the binomial distribution when the number of trials $n$ is large and the success probability $p$ is small, such that $np = \lambda$ remains constant
• The approximation is appropriate when $n \geq 20$ and $p \leq 0.05$, or when $n \geq 100$ and $np \leq 10$
  • In these cases, the Poisson distribution with parameter $\lambda = np$ closely approximates the binomial distribution

Applications of Poisson distribution

• Modeling the number of insurance claims in a fixed time period
• Analyzing the number of rare events (e.g., accidents, defects) in a given interval
• Predicting the number of customers arriving at a store within a specific time frame
• Calculating the probability of a certain number of events occurring in a fixed area or volume (e.g., number of bacteria in a petri dish)

Relationships between distributions

Binomial as sum of Bernoullis

• The binomial distribution can be derived as the sum of $n$ independent and identically distributed Bernoulli random variables
• Let $X_1, X_2, ..., X_n$ be independent Bernoulli random variables with success probability $p$, then $Y = \sum_{i=1}^n X_i$ follows a binomial distribution with parameters $n$ and $p$
  • This relationship demonstrates that the binomial distribution is a generalization of the Bernoulli distribution for multiple trials

Poisson as limit of binomial

• The Poisson distribution can be derived as the limiting case of the binomial distribution when the number of trials $n$ approaches infinity and the success probability $p$ approaches zero, while their product $np = \lambda$ remains constant
• Mathematically, if $X_n \sim Bin(n, \lambda/n)$, then $\lim_{n \to \infty} P(X_n = k) = \frac{e^{-\lambda}\lambda^k}{k!}$, which is the probability mass function of a Poisson distribution with parameter $\lambda$
  • This relationship allows for the use of the Poisson distribution as an approximation to the binomial distribution in certain situations

Fitting discrete distributions

Method of moments

• The method of moments is a technique for estimating the parameters of a distribution by equating the sample moments to the corresponding theoretical moments
• For a discrete distribution with parameter $\theta$, the $k$-th moment is given by: $E(X^k) = \sum_{x} x^k P(X=x; \theta)$
  • The first moment (mean) and second central moment (variance) are most commonly used
• To estimate the parameter, set the sample moments equal to the theoretical moments and solve for $\theta$
  • For example, to estimate the parameter $p$ of a Bernoulli distribution, set $\bar{x} = p$ and solve for $p$, giving $\hat{p} = \bar{x}$

Maximum likelihood estimation

• Maximum likelihood estimation (MLE) is a method for estimating the parameters of a distribution by maximizing the likelihood function
• The likelihood function $L(\theta; x_1, x_2, ..., x_n)$ is the joint probability mass function of the observed data, viewed as a function of the parameter $\theta$
  • For independent and identically distributed observations, the likelihood function is the product of the individual probability mass functions: $L(\theta; x_1, x_2, ..., x_n) = \prod_{i=1}^n P(X=x_i; \theta)$
• To find the MLE, take the logarithm of the likelihood function (log-likelihood), differentiate with respect to $\theta$, set the derivative equal to zero, and solve for $\theta$
  • The MLE has desirable properties such as consistency, asymptotic normality, and asymptotic efficiency

Discrete distribution examples

Modeling claim frequency

• The Poisson distribution is often used to model the number of insurance claims in a fixed time period
• The parameter $\lambda$ represents the average number of claims per unit time
  • $\lambda$ can be estimated from historical claim data using the method of moments or MLE
• Once the parameter is estimated, the Poisson distribution can be used to calculate probabilities and make predictions about future claim frequencies

Modeling rare events

• The Poisson distribution is well-suited for modeling rare events, such as accidents, defects, or natural disasters
• The rarity of the event is captured by the small value of the parameter $\lambda$, which represents the average number of occurrences per unit of time or space
  • For example, the number of earthquakes in a given region per year may follow a Poisson distribution with a small $\lambda$ value

Modeling success/failure experiments

• The binomial distribution can be used to model the number of successes in a fixed number of trials, where each trial has two possible outcomes (success or failure)
• Examples include modeling the number of defective items in a batch of products, the number of successful treatments in a group of patients, or the number of correct answers on a multiple-choice exam
  • The parameters $n$ and $p$ represent the number of trials and the probability of success in each trial, respectively
• The binomial distribution can help calculate probabilities, make predictions, and assess the likelihood of various outcomes in such experiments