Continuous random variables

Visualizing PDFs

The plots I’ve included are the typical way to visualize PDFs, but I much prefer to visualize them as the number lines below. Implementation-wise, I’ve just sampled 300 values from each distribution and plotted each as a red dot with opacity 0.12.

Transforming variables

If I know the PDF of , how can I find the PDF of ?

Using our number line visual, let’s animate the transform from some random variable to :

Observe two things:

1. A point originally at moves to .
2. The density at every point is halved.

Therefore,

(Typically, we rearrange this as , but I find the first expression more enlightening.)

Let’s tackle a slightly more challenging transform, .

Note that the density is no longer uniformly stretched by a factor of ; instead, the density at point will get stretched by . Therefore for ,

Exercise

In general, what is the PDF of in terms of and ?