Tag Archives: Inverse Transformed

Parametric Distributions

Parametric distributions are functions in several dimensions.  Various parametric distributions are given in the exam tables.  Each input variable or dimension of the distribution function is called a parameter.  While studying, it is important to keep in mind that parameters are simply abstract devices built into a distribution function which allow us, through their manipulation, to tweak the shape of the distribution.  Ultimately, we are still only interested in things like $Pr(X\le x)$ and the distribution function parameters are used to help describe the distribution of $X$.

Transformations

1. Scaling:  If a random variable $X$ has a scaleable parametric distribution with parameters $(a_1, a_2, ..., a_n, \theta)$, then one of these parameters can be called a scale parameter and is denoted by $\theta$.  Having the scaleable property implies that $cX$ can be described with the same distribution function as $X$, except that the parameters of its distribution are $(a_1,a_2,..., a_n,c\theta)$ where $c$ is the scale factor.  In terms of probability, scaling a random variable has the following effect– if $Y = cX$ with $c >0$, then $Pr(Y \le y) = Pr(cX\le y) = Pr(X \le \frac{y}{c})$.
Caveat: The Inverse Gaussian as given in the exam tables has a $\theta$ in its set of parameters; however, this is not a scale distribution.  To scale a Lognormal distribution, adjust the parameters to $(\mu + \ln{c}, \sigma)$ where $c$ is the scale factor and $\mu$ and $\sigma$ are the usual parameters.  All the rest of the distributions given in appendix A are scale distributions.
2. Raising to a power:  A random variable raised to a positive power is called transformed.  If it is raised to -1 it is called inverse. If it is raised to a power less than -1, it is called inverse transformed.  When raising to a power, the scale parameter needs to be readjusted to remain a scale parameter in the new distribution.
3. Exponentiating:  An example is the lognormal distribution.  If $X$ is normal, then $Y = e^X$ is lognormal.  In terms of probability, $F_Y(y) = F_X(\ln{y})$.
Splicing
You can create a new distribution function by defining different distribution probability densities on different domain intervals.  As long as the piecewise integral of the spliced distribution is 1, it is a valid distribution.  Since total probability has to be exactly 1, scaling is an important tool that allows us to do this.
Tail Weight
Since a density function must integrate to 1, it must tend to 0 at the extremities of its domain.  If density function A tends towards zero at a slower rate than density function B, then density A is said to have a heavier tail than density B.  Some important measures of tail weight:
1. Tail weight decreases inversely with respect to the number of positive raw or central moments that exist.
2. The limit of the ratio of one density or survival function over another may tend to zero or infinity depending on which has the greater tail weight.
3. An increasing hazard rate function implies a lighter tail and vice versa.
4. An increasing mean residual life function means a heavier tail and vice versa.