Shape, Rate, Scale, and Location Parameters

Alec James van Rassel

library(Distributacalcul)

Context

When I started learning about probability distributions, the parameters of distributions were simply “alpha”, “beta”, “mu”, etc. to me.

It is only when I started using R that I realized they were called “shape”, “rate”, and “scale” parameters. Sometimes, there was even a “location” or a “dispersion” parameter ! But what do these mean ? That’s what this vignette explains.

The location parameter

To get a feel for this, see what happens to the density of the Normal distribution when you change \(\mu\) :

distributacalculVis(law = "Norm", mod = "functions")

If you notice, changing \(\mu\) only changes where on the x-axis the density is centered or, located. We call this a location parameter.

In brief, the location parameter is exactly what it sounds like.

The scale parameter

The scale parameter changes the scale of the distribution.

To get a feel for this, try changing the scale parameter of the Gamma distribution \(\beta\) below from 1 to 2 to 3 :

distributacalculVis(law = "Gamma", mod = "functions")

As you increase the scale parameter, the distribution becomes increasingly compressed.

To understand why this happens mathematically, suppose we scale a random variable \(X\) by 3. That is to say, we multiply \(X\) by 3: \[ \Pr(3 \times X \leq x) = \Pr\left(X \leq \frac{x}{3}\right) \]

We see that scaling up a random variable by a constant scales down the values it takes.

Converting between centimeters and meters doesn’t change a number, only its scale. Similarly, the scale parameter doesn’t change the ‘shape’ of a distribution, only its scale.

The rate parameter

Mathematically, the rate parameter is one over the scale parameter : \(\text{rate} = \frac{1}{\text{scale}}\).

To understand the logic behind this, suppose that we scale a random variable \(X\) by \(\frac{1}{3}\). That is to say, we multiply \(X\) by \(\frac{1}{3}\): \[ \Pr\left(\frac{1}{3} \times X \leq x\right) = \Pr\left(X \leq 3x\right) \]

We see that scaling down a random variable by a fraction scales up the values it takes.

In practice, the Poisson distribution is often used to model the frequency of events. Particularly, it is used in actuarial science to model the number of accidents which occur.
The \(\lambda\) parameter of a discrete Poisson random variable is a rate parameter. Enter the \(\beta\) parameter as a proportion below (0.20, 0.30, etc.) and see what changing it does :

distributacalculVis(law = "Gamma", mod = "functions")

You can observe it has the opposite effect of the scale parameter.

In brief, mathematically the rate parameter is just the inverse of the scale parameter. In practice, however, it has very useful applications.

The shape parameter