Moments

Let $X$ be a continuous random variable with probability density function $f_{X}left( x right)$. The k-th raw moment of $X$, denoted by $mu_{k}^{prime}$, is the expected value of the k-th power of $X$, provided it exists. The first raw moment $mu_{1}^{prime}$ is the mean of $X$ usually denoted by $mu$. The formula for $mu_{k}^{prime}$ is given as
$$mu_{k}^{prime} = Eleft( X^{k} right) = int_{0}^{infty}{x^{k}f_{X}left( x right)dx } .$$ The support of the random variable $X$ is assumed to be nonnegative since actuarial phenomena are rarely negative.

The k-th central moment of $X$, denoted by $mu_{k}$, is the expected value of the k-th power of the deviation of $X$ from its mean $mu$. The formula for $mu_{k}$ is given as
$$mu_{k} = Eleftlbrack {(X – mu)}^{k} rightrbrack = int_{0}^{infty}{left( x – mu right)^{k}f_{X}left( x right) dx }.$$
The second central moment $mu_{2}$ defines the variance of $X$, denoted by $sigma^{2}$. The square root of the variance is the standard deviation $sigma$. A further characterization of the shape of the distribution includes its degree of symmetry as well as its flatness compared to the normal distribution. The ratio of the third central moment to the cube of the standard deviation $left( mu_{3} / sigma^{3} right)$ defines the coefficient of skewness which is a measure of symmetry. A positive coefficient of skewness indicates that the distribution is skewed to the right (positively skewed). The ratio of the fourth central moment to the fourth power of the standard deviation $left(mu_{4} / sigma^{4} right)$ defines the coefficient of kurtosis. The normal distribution has a coefficient of kurtosis of 3. Distributions with a coefficient of kurtosis greater than 3 have heavier tails and higher peak than the normal, whereas distributions with a coefficient of kurtosis less than 3 have lighter tails and are flatter.

Example 3.1 (SOA) $X$ has a gamma distribution with mean 8 and skewness 1. Find the variance of $X$.

Solution

Quantiles

Percentiles can also be used to describe the characteristics of the distribution of $X$. The 100pth percentile of the distribution of $X$, denoted by $pi_{p}$, is the value of $X$ which satisfies
$$F_{X}left( {pi_{p}}^{-} right) leq p leq Fleft( pi_{p} right) ,$$ for $0 leq p leq 1$.

The 50-th percentile or the middle point of the distribution, $pi_{0.5}$, is the median. Unlike discrete random variables, percentiles of continuous variables are distinct.

Example 3.2 (SOA) Let $X$ be a continuous random variable with density function $f_{X}left( x right) = theta e^{- theta x}$, for $x gt 0$ and 0 elsewhere. If the median of this distribution is $frac{1}{3}$, find $theta$.
Solution

The Moment Generating Function

The moment generating function, denoted by $M_{X}left( t right)$ uniquely characterizes the distribution of $X$. While it is possible for two different distributions to have the same moments and yet still differ, this is not the case with the moment generating function. That is, if two random variables have the same moment generating function, then they have the same distribution. The moment generating is a real function whose k-th derivative at zero is equal to the k-th raw moment of $X$. The moment generating function is given by
$$M_{X}left( t right) = Eleft( e^{text{tX}} right) = int_{0}^{infty}{e^{text{tx}}f_{X}left( x right) dx }$$ for all $t$ for which the expected value exists.

Example 3.3 (SOA) The random variable $X$ has an exponential distribution with mean $frac{1}{b}$. It is found that $M_{X}left( – b^{2} right) = 0.2$.
Find $b$.
Solution

Example 3.4 Let $X_{1}$, $X_{2}$, …, $X_{n}$ be independent $text{Ga}left( alpha_{i},theta right)$ random variables. Find the distribution of $S = sum_{i = 1}^{n}X_{i}$.
Solution

Probability Generating Function

The probability generating function, denoted by $P_{X}left( z right)$, also uniquely characterizes the distribution of $X$. It is defined as
$$P_{X}left( z right) = Eleft( z^{X} right) = int_{0}^{infty}{z^{x}f_{X}left( x right) dx}$$ for all $z$ for which the expected value exists.

We can also use the probability generating function to generate moments of $X$. By taking the k-th derivative of $P_{X}left( z right)$ with respect to $z$ and evaluate it at $z = 1$, we get
$$Eleftlbrack Xleft( X – 1 right)ldotsleft( X – k + 1 right) rightrbrack .$$