# formulating pdf from supports

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?

• – Tim
Commented Sep 22, 2016 at 14:04
• This question sounds like self-study so please add the tag and tell us what you have tried so far. Commented Sep 22, 2016 at 14:32

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).