. and
The following is an equivalent calculation of that may be easier to use in some circumstances. ashas
In , we assume that the deductible is a positive integer.
functions. The random variable
In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. . The mean excess loss variable is the conditional variable and the mean excess loss function is defined by: In an insurance context, the mean excess loss function is the average payment in excess of a threshold given that the loss exceeds the threshold. . has a Gamma distribution with parameters
The formulation is obtained by using integration by parts (also see theorem 3.1 in [1]). distribution do they have?
increasing the number of degrees of freedom from
degrees of freedom respectively. The following is how the limited expected value is calculated depending on whether the loss is continuous or discrete. we
Thus, the Chi-square distribution is a special case of the Gamma distribution
The Gamma distribution can be thought of as a generalization of the Chi-square distribution. Example 2 constant:and
Therefore, a Gamma random variable with parameters
probability density
degrees of freedom and
. and
has
independent normal random variables having mean
has a Chi-square distribution with
and
variables. variables: What distribution do these variables have? Gamma Distribution Variance. "Gamma distribution", Lectures on probability theory and mathematical statistics, Third edition. is a Gamma random variable with parameters
and
Define the following random
Example 3 when a loss is less than , no payment is made to the insured entity, and when the loss exceeds , the insured entity is reimbursed for the amount of the loss in excess of the deductible . and
The random variable
In , the deductible is free to be any positive number and is the largest integer such that . is a strictly increasing function of
having mean
).
Let the loss random variable be exponential with pdf . We start with the following example: Example by Marco Taboga, PhD. It can be derived by using the definition of
In the lecture entitled Chi-square distribution we
degrees of freedom (remember that a Gamma random variable with parameters
independent normal random variables
has a Gamma distribution with parameters
where
However, the two distributions have the same number of degrees of freedom
Therefore, they have the same shape (one is the "stretched version of the
distribution changes when its parameters are changed.
degrees of freedom. obtains another Gamma random variable. The following plot contains the graphs of two Gamma probability density
has a Gamma distribution with parameters
. and variance
… the variables
aswhere
This page collects some plots of the Gamma
So the second question is about a per loss average.
Therefore
increased the more the distribution resembles a normal distribution).
Taboga, Marco (2017). The expected payment for large losses is always the unmodified expected plus a component that is increasing in .
has a Gamma distribution with parameters
Let its
degrees of freedom. defined as
iswhere
It
be two independent Chi-square random variables having
Gamma distribution and Poisson distribution, Derive some facts of the negative binomial distribution.
Online appendix.
in both cases, the two distributions have the same mean. are mutually independent standard normal random
degrees of freedom. iswhere
Chi-square distribution). other words,
The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. degrees of freedom and the random variable
Therefore,In
and variance
The gamma distribution is another widely used distribution. Thus the expected amount of loss eliminated (from the insurer’s point of view) is . and
By having a deductible provision in the policy, the insurer is now only liable for the amount and the amount the insurer is expected to pay per loss is . Therefore
density of a function of a continuous
degrees of freedom. Gamma Distribution Mean. The Gamma distribution is a scaled Chi-square distribution, A Gamma random variable times a strictly positive constant is a Gamma random variable, A Gamma random variable is a sum of squared normal random variables, Plot 1 - Same mean but different degrees of freedom, Plot 2 - Different means but same number of degrees of freedom. The gamma distribution represents continuous probability distributions of two-parameter family. is strictly
variable. is a strictly positive constant, then the random variable
It can be shown as follows: So, Variance = E[x 2] – [E(x 2)], where p = (E(x)) (Mean and Variance p(p+1) – p 2 = p These plots help us to understand how the shape of the Gamma
The mean excess loss function is computed by the following depending on whether the loss variable is continuous or discrete. Kindle Direct Publishing. The mean excess loss function of the Pareto distribution has a linear form that is increasing (see the previous post The Pareto distribution).
course, the above integrals converge only if
characteristic function and a Taylor series
Therefore, the moment generating function of a Gamma random variable exists
The following summarizes this observation. In the first question, the average is computed over all losses that are eligible for reimbursement (i.e., the loss exceeds the deductible). The following is how this expected value is calculated depending on whether the loss is continuous or discrete.
since
is the Gamma function.
,
and variance
and
and
If \(\alpha = 1\), then the corresponding gamma distribution is given by the exponential distribution, i.e., \(\text{gamma}(1,\lambda) = \text{exponential}(\lambda)\). degrees of freedom and mean
and
and
called lower incomplete Gamma function and is
Of all the losses that are eligible to be reimbursed by the insurer, what is the average payment made by the insurer to the insured? ; the second graph (blue line) is the probability density function of a Gamma
obtainwhere
can be written
https://www.statlect.com/probability-distributions/gamma-distribution.
The pdf is .
The expected value of a Gamma random variable
Suppose is the random loss and . The insurer’s expected payment without the deductible is . If a random variable
(). for all
To better understand the Gamma distribution, you can have a look at its
usually evaluated using specialized computer algorithms. from the previous
This can be easily proved using the formula
If
and
are independent (see the lecture entitled
and
Also see Example 3 below.
The exponential distribution has a constant mean excess loss function and is considered a medium tailed distribution. have explained that a Chi-square random variable
we have
is just a Chi square distribution with
Note that is the function that assigns the value of whenever and otherwise assigns the value of zero. can be derived thanks to the usual
has a Chi-square distribution with
have? has a Gamma distribution with parameters
Below you can find some exercises with explained solutions. . is also a Chi-square random variable when
all have a Gamma distribution. Therefore
: By
and multiplied by
The random variable
expansion: The distribution function
degrees of freedom, divided by
distribution.
with
In general, an increasing mean excess loss function is an indication of a heavy tailed distribution. Gamma distributions are devised with generally three kind of parameter combinations.
Bowers N. L., Gerber H. U., Hickman J. C., Jones D. A., Nesbit C. J. other" - it would look exactly the same on a different scale). for the density of a function of a continuous
When the loss does not reach the deductible, the payment is considered zero and when the loss is in excess of the deductible, the payment is . The survival function is: For the losses that exceed the deductible, the insurer’s expected payment is: Then the insurer’s expected payment per loss is : With a deductible in the policy, the following is the expected amount of loss eliminated (from the insurer’s point of view).