Find the PDF of W = X + Y + Z on a Uniform Distribution

[Dwolfson]

I am stumped.

I have that W=X+Y+Z and that S=X+Y

These are all X, Y, & Z and Independent and Uniformly Distributed on (0,1)

I found the pdf of S to be (Assume all these < rep. less than or equal to):

S when 0<S<1
2-S when 0<S<1

So I continued:

To do pdf of S+Z=W

I figured there will be 3 intervals:

when 0<W<1, 1<W<2, and 2<W<3:

I Have figured out the one from 0<W<1

to be integral from 0 to W pdf(w)=S(pdf(W-S))ds

= W^2/2

For the other two intervals I am struggling on which pdf of S to use and what is the interval of integration..

Thank you in advance for your help,
--Derek

 

[lanedance]

so the sum of 2 RVs is given by their convolution, in particular the square pulse integral
http://en.wikipedia.org/wiki/Convolution

so for S = X+Y, with $p_X(X=x), \ p_Y(Y=y)$
$$p_S(s) = \int dx p_X(x) p_Y(s-x)$$

similarly, it should just follow that for W = S + Z
$$p_W(w) = \int dz p_Z(z) p_S(w-z)$$

 

[lanedance]

I am stumped.

I have that W=X+Y+Z and that S=X+Y

These are all X, Y, & Z and Independent and Uniformly Distributed on (0,1)

I found the pdf of S to be (Assume all these < rep. less than or equal to):

S when 0<S<1
2-S when 0<S<1

and i assume you mean
2-S when 1<S<2

 

[Dwolfson]

Please look here -- I did some further work and plugged this thread into the proper message board:

https://www.physicsforums.com/showthread.php?p=3019646#post3019646