1.6 Continuous distributions (normal, exponential, gamma)

Continuous distributions are essential tools in actuarial mathematics for modeling random variables that can take on any value within a range. Unlike discrete distributions, they use probability density functions to describe the likelihood of specific values occurring.

Normal, exponential, and gamma distributions are key continuous distributions in actuarial science. Each has unique properties and applications, from modeling claim sizes to estimating lifetimes. Understanding these distributions helps actuaries analyze risks and price insurance products accurately.

Properties of continuous distributions

• Continuous distributions are used to model random variables that can take on any value within a specified range
• Unlike discrete distributions, the probability of a continuous random variable taking on a specific value is zero
• Key properties of continuous distributions include the probability density function (PDF), cumulative distribution function (CDF), and moments

Probability density functions

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• The PDF, denoted as $f(x)$, describes the relative likelihood of a continuous random variable taking on a specific value
• The PDF is non-negative for all values of $x$, i.e., $f(x) \geq 0$
• The total area under the PDF curve is equal to 1, i.e., $\int_{-\infty}^{\infty} f(x) dx = 1$
• The probability of a continuous random variable falling within a specific range $[a, b]$ is given by $P(a \leq X \leq b) = \int_{a}^{b} f(x) dx$

Cumulative distribution functions

• The CDF, denoted as $F(x)$, represents the probability that a continuous random variable $X$ takes on a value less than or equal to $x$
• The CDF is defined as $F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t) dt$
• The CDF is a non-decreasing function, i.e., if $x_1 < x_2$, then $F(x_1) \leq F(x_2)$
• The CDF ranges from 0 to 1, with $\lim_{x \to -\infty} F(x) = 0$ and $\lim_{x \to \infty} F(x) = 1$

Moments of continuous distributions

• Moments provide a way to characterize the properties of a continuous distribution, such as central tendency, dispersion, and shape
• The $n$-th moment of a continuous random variable $X$ is defined as $E[X^n] = \int_{-\infty}^{\infty} x^n f(x) dx$
• The first moment is the mean or expected value, given by $\mu = E[X] = \int_{-\infty}^{\infty} x f(x) dx$
• The second central moment is the variance, given by $\sigma^2 = E[(X - \mu)^2] = \int_{-\infty}^{\infty} (x - \mu)^2 f(x) dx$

Normal distribution

• The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric and bell-shaped
• It is widely used in various fields, including actuarial science, due to its well-defined properties and central limit theorem

Probability density function of normal distribution

• The PDF of a normal distribution with mean $\mu$ and standard deviation $\sigma$ is given by $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$
• The PDF is symmetric around the mean, with the peak at $x = \mu$
• The shape of the PDF is determined by the standard deviation $\sigma$, with smaller values of $\sigma$ resulting in a more concentrated distribution

Standard normal distribution

• The standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1
• The PDF of the standard normal distribution is given by $\phi(z) = \frac{1}{\sqrt{2\pi}} e^{-\frac{z^2}{2}}$
• The CDF of the standard normal distribution is denoted by $\Phi(z)$ and can be used to calculate probabilities for any normal distribution through standardization

Applications of normal distribution

• Modeling the distribution of heights, weights, or IQ scores in a population
• Analyzing the distribution of errors in measurements or observations
• Calculating the probability of an event occurring within a certain number of standard deviations from the mean

Exponential distribution

• The exponential distribution is a continuous probability distribution that models the time between events in a Poisson process
• It is often used to model the time until failure or the waiting time between events

Probability density function of exponential distribution

• The PDF of an exponential distribution with rate parameter $\lambda$ is given by $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$
• The mean and standard deviation of an exponential distribution are both equal to $\frac{1}{\lambda}$
• The CDF of an exponential distribution is given by $F(x) = 1 - e^{-\lambda x}$ for $x \geq 0$

Memoryless property

• The exponential distribution possesses the memoryless property, which means that the probability of an event occurring in the next time interval does not depend on how much time has already elapsed
• Mathematically, $P(X > s + t | X > s) = P(X > t)$ for all $s, t \geq 0$
• This property makes the exponential distribution suitable for modeling constant failure rates or inter-arrival times

Applications of exponential distribution

• Modeling the time between customer arrivals in a queue
• Analyzing the lifetime of electronic components or light bulbs
• Estimating the time until the next earthquake or volcanic eruption

Gamma distribution

• The gamma distribution is a continuous probability distribution that generalizes the exponential distribution by allowing for a shape parameter
• It is used to model waiting times, time until failure, and other positive, continuous random variables

Probability density function of gamma distribution

• The PDF of a gamma distribution with shape parameter $\alpha$ and rate parameter $\beta$ is given by $f(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}$ for $x \geq 0$
• The mean of a gamma distribution is $\frac{\alpha}{\beta}$, and the variance is $\frac{\alpha}{\beta^2}$
• The CDF of a gamma distribution is given by $F(x) = \frac{\gamma(\alpha, \beta x)}{\Gamma(\alpha)}$, where $\gamma(\alpha, \beta x)$ is the lower incomplete gamma function

Special cases of gamma distribution

• When $\alpha = 1$, the gamma distribution reduces to the exponential distribution with rate parameter $\beta$
• When $\alpha = n/2$ and $\beta = 1/2$, the gamma distribution becomes the chi-square distribution with $n$ degrees of freedom
• The Erlang distribution is a special case of the gamma distribution with integer shape parameter $\alpha$

Applications of gamma distribution

• Modeling the waiting time until the $\alpha$-th event in a Poisson process
• Analyzing the total amount of rainfall over a fixed period
• Estimating the time required to complete a complex task or project

Relationship between distributions

• Many continuous distributions are related to each other through limiting cases or special parameterizations
• Understanding these relationships can help in selecting appropriate models and simplifying calculations

Normal distribution as limiting case

• The normal distribution can be derived as a limiting case of the binomial distribution as the number of trials approaches infinity and the probability of success remains fixed
• The Poisson distribution also approaches the normal distribution when the rate parameter is large
• The central limit theorem states that the sum of a large number of independent and identically distributed random variables will be approximately normally distributed

Exponential distribution vs gamma distribution

• The exponential distribution is a special case of the gamma distribution with shape parameter $\alpha = 1$
• The sum of $n$ independent exponential random variables with rate parameter $\lambda$ follows a gamma distribution with shape parameter $n$ and rate parameter $\lambda$
• The gamma distribution can be used to model more flexible waiting times or time-to-failure scenarios compared to the exponential distribution

Transformations of continuous distributions

• Transforming a continuous random variable can lead to new distributions with different properties
• Linear and non-linear transformations are commonly used to modify the location, scale, or shape of a distribution

Linear transformations

• If $X$ is a continuous random variable and $Y = aX + b$ for constants $a \neq 0$ and $b$, then $Y$ follows a linearly transformed distribution
• The PDF of $Y$ is given by $f_Y(y) = \frac{1}{|a|} f_X(\frac{y-b}{a})$
• Linear transformations preserve the shape of the original distribution but change the location and scale parameters

Non-linear transformations

• Non-linear transformations can be used to create new distributions with different shapes or properties
• Examples of non-linear transformations include:
  • Exponential transformation: $Y = e^X$
  • Logarithmic transformation: $Y = \log(X)$
  • Power transformation: $Y = X^p$ for some constant $p$
• The PDF of the transformed variable can be derived using the change of variables technique, given by $f_Y(y) = f_X(g^{-1}(y)) \left| \frac{d}{dy} g^{-1}(y) \right|$, where $g$ is the transformation function

Estimation and inference

• Estimating the parameters of a continuous distribution from sample data is a crucial task in actuarial science
• Two common methods for parameter estimation are maximum likelihood estimation (MLE) and method of moments estimation (MME)

Maximum likelihood estimation

• MLE is a popular method for estimating the parameters of a distribution by maximizing the likelihood function
• The likelihood function is the joint probability density function of the observed data, viewed as a function of the parameters
• The MLE estimates are the parameter values that maximize the likelihood function or, equivalently, the log-likelihood function
• MLE is asymptotically efficient and consistent under certain regularity conditions

Method of moments estimation

• MME is a simple and intuitive method for estimating the parameters of a distribution by equating the sample moments to the corresponding population moments
• The $k$-th sample moment is given by $m_k = \frac{1}{n} \sum_{i=1}^n X_i^k$, where $X_1, X_2, \ldots, X_n$ are the observed data points
• The MME estimates are obtained by solving a system of equations that equate the sample moments to the corresponding theoretical moments, which are functions of the parameters
• MME is consistent but not always efficient compared to MLE

Confidence intervals for parameters

• Confidence intervals provide a range of plausible values for the true parameters based on the sample data
• A $(1-\alpha)100\%$ confidence interval for a parameter $\theta$ is an interval $[L, U]$ such that $P(L \leq \theta \leq U) = 1-\alpha$
• Confidence intervals can be constructed using various methods, such as:
  • Inverting a hypothesis test (e.g., t-interval for the mean)
  • Using the asymptotic normality of the MLE (Wald interval)
  • Bootstrapping or resampling techniques
• The width of the confidence interval decreases as the sample size increases, reflecting the increased precision of the estimates

Goodness-of-fit tests

• Goodness-of-fit tests are used to assess whether a given continuous distribution adequately describes the observed data
• These tests compare the observed frequencies or empirical distribution function (EDF) with the expected frequencies or theoretical CDF under the null hypothesis

Chi-square test

• The chi-square test is a goodness-of-fit test that compares the observed frequencies in bins with the expected frequencies under the hypothesized distribution
• The test statistic is given by $\chi^2 = \sum_{i=1}^k \frac{(O_i - E_i)^2}{E_i}$, where $O_i$ and $E_i$ are the observed and expected frequencies in the $i$-th bin, respectively
• The test statistic follows a chi-square distribution with $k-1-m$ degrees of freedom, where $m$ is the number of estimated parameters
• The chi-square test is sensitive to the choice of bins and may have low power for small sample sizes

Kolmogorov-Smirnov test

• The Kolmogorov-Smirnov (KS) test is a non-parametric goodness-of-fit test that compares the EDF with the theoretical CDF
• The test statistic is the maximum absolute difference between the EDF and the CDF, given by $D_n = \sup_x |F_n(x) - F(x)|$, where $F_n(x)$ is the EDF and $F(x)$ is the theoretical CDF
• The KS test is distribution-free and does not require binning, making it more powerful than the chi-square test for small sample sizes
• The critical values for the KS test are based on the Kolmogorov distribution and depend on the sample size and significance level

Applications in actuarial science

• Continuous distributions play a vital role in actuarial science, as they are used to model various types of risk and uncertainty
• Some common applications include modeling claim severity, lifetime distributions, and pricing insurance products

Modeling claim severity

• Claim severity refers to the size or amount of individual claims in an insurance portfolio
• Continuous distributions, such as the lognormal, gamma, or Pareto distribution, are often used to model claim severity
• The choice of distribution depends on factors such as the type of insurance, the characteristics of the policyholders, and the historical claim data
• Accurately modeling claim severity is essential for setting appropriate premiums, calculating reserves, and managing risk

Modeling lifetime distributions

• Lifetime distributions are used to model the time until death or failure in various contexts, such as life insurance, annuities, and reliability engineering
• The exponential distribution is a simple model for constant failure rates, while the Weibull distribution allows for increasing or decreasing failure rates over time
• Other distributions, such as the gamma, lognormal, or generalized gamma, can provide more flexibility in modeling lifetime data
• Estimating the parameters of lifetime distributions is crucial for pricing insurance products, calculating reserves, and assessing the financial stability of insurance companies

Pricing insurance products

• Pricing insurance products involves determining the premiums that policyholders must pay to cover the expected claims and expenses, while ensuring the profitability of the insurer
• Continuous distributions are used to model the frequency and severity of claims, as well as the time value of money and investment returns
• Actuaries use techniques such as risk classification, credibility theory, and experience rating to refine the pricing models based on the characteristics of the policyholders and the historical claims experience
• Stochastic simulation and scenario testing can be employed to assess the sensitivity of the pricing models to various assumptions and to quantify the uncertainty in the premium estimates