# 'Triangular Distributions' Probability Density Function

1. Oct 9, 2008

### pone

(\Triangular" distributions.) Let X be a continuous random variable with prob-
ability density function f(x). Suppose that all we know about f is that a </= X </= b,
f(a) = f(b) = 0, and that there exists a value c between a and b where f is at a maxi-
mum. A natural density function to consider in this case is a piece-wise linear function,
corresponding to lines connecting (a; 0) with (c; f(c)), and (c; f(c)) with (b; 0).
a) What is the value of f(c)?
b) Sketch a graph of f(x).
c) Compute the expected value E(X) and the variance Var(X).

I have not been given any numbers and am very confused as to how there could be a numerical answer to this question. I know the probability density function looks like a triangle, with f(a) and f(b) on the x-axis, but am not sure where to go with this. Anyone have a suggestion?

2. Oct 9, 2008

### mathman

The density function will look like:

f(x)=K(x-a) a<x<c
f(x)=L(b-x) c<x<b

where L and K are determined by:
L(b-c)=K(c-a)
integral from a to b of f(x)=1.

Your results will be functions of a, b and c, so don't expect to get numbers unless a, b, and c are specified.

Last edited: Oct 10, 2008