formulating pdf from supports

Question

for a pdf which is triangular, with support over (b, 2b) and a peak when x = 5b/3, what would the pdf be? How do you determine a pdf from supports and a given shape?

Answer 1

In the case of the triangular, this is simple algebraic calculation using known facts.

Let us solve a similar problem; consider support on $(0,\theta)$ and then put the peak at $k\theta$. (Yours simply being a shifted version of this with a specific $k$)

Notice that $\theta$ is simply scale parameter, so let us solve the even simpler problem on $(0,1)$ with peak at $k$; the more general problem is simply a matter of introducing a scale parameter.

You can obtain $h$, the height at the peak by using the formula for the area of a triangle.

You can then write the equations of the line segments joining (0,0) to (k,h) and (k,h) to (1,0) by using the two-point form of the equation of a straight line.

So far this is nothing beyond basic Cartesian geometry that I presume you covered as a teenager.

You can then from those equations write down the density on four subintervals of $\mathbb{R}$, that to the left of 0, between 0 and k, between k and 1, and above 1. Done properly, this will give a complete definition of the pdf; it's a trivial matter to convert to my original $(0,\theta)$ case (scaling the width by $\theta$ and the height by $1/\theta$) and then to shift to your own example. (Or you could employ the same strategy on your original example directly -- but I am very much in the vein of converting already-solved problems when I can.)

More complicated pdf examples may in many cases follow broadly similar strategies (but employ the given facts differently since it will be a different problem).