Find the Density Function

Date: 01/24/2002 at 10:07:31
From: Tanya
Subject: Probability

Suppose X1, X2, X3 are independent  and uniform on(0,1). find the 
density function of S = X1 + X2 + X3.

Date: 01/24/2002 at 19:15:11
From: Doctor Anthony
Subject: Re: Probability

I shall use x, y, z rather than x1, x2, x3.

You can obtain the distribution function of S by finding the volume of 
the unit cube cut off by the plane x + y + z = s

The plane will be symmetrically positioned cutting the coordinate axes 
at (s,0,0), (0,s,0), (0,0,s)

The volume of the tetrahedron cut off by the plane x + y + z = s  is  

 Vol = (1/3) area of base x perpendicular height

  (1/3)(1/2 x s^2) (s)  =  (1/6)s^3

So   F(s) =  s^3/6    for 0 < s < 1

For 1 < s < 2 the volume is calculated as above but with the 3 small 
tetrahedra outside the unit cube subtracted. The volume of each of 
these exterior tetrahedra is  (1/6)(s-1)^3, so together they have a 
volume of (1/2)(s-1)^3, and the volume of the cube cut off by the 
plane is

      (1/6)s^3 - (1/2)(s-1)^3

and so

     F(s) = (1/6)s^3 - (1/2)(s-1)^3   for 1 < s < 2

For 2 < s < 3  we get the volume cut off as

         1 - (1/6)(3-s)^3     and so  

     F(s) =  1 - (1/6)(3-s)^3   for  2 < s < 3

To obtain the pdf we must differentiate F(s).  So we have

  f(s) =  (1/2)s^2                  for  0 < s < 1

       =  (1/2)s^2 - (3/2)(s-1)^2   for  1 < s < 2

       =  (1/2)(3-s)^2              for  2 < s < 3


- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/