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### 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/
```
Associated Topics:
High School Calculus
High School Probability

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