Sum of uniform distributions

Question

Let’s say

$X_1$ ~ $uni(0,1)$

$X_2$ ~ $uni(0,1)$

$X_3$ ~ $uni(0,1)$

And

$Y=0.1X_1+0.3X_2+0.6X_3$

What’s the $F(Y)$ (i.e., CDF)?

This seems related to [your other recent question](http://stats.stackexchange.com/q/31771/2970). Maybe you should consider updating your other one to clarify your interests, instead. :) — cardinal, Jul 06 '12 at 19:45
You didn't mention independence between X1, X2 and X3. I am sure that Greg had to asswume it to get his asnwer. — Michael R. Chernick, Jul 06 '12 at 21:01

score 6 · Accepted Answer · answered Jul 06 '12 at 20:06

6

It is not one of the common named distributions, but the distr package in R can help you explore it:

library(distr)

d <- 0.1*Unif() + 0.3*Unif() + 0.6*Unif()
plot(d)

enter image description here

answered Jul 06 '12 at 20:06

Greg Snow

Is it possible to make the figures a little less tall so that it's a little easier to see what's going on? Stacking them vertically might work better for this particular interface. :) – cardinal Jul 06 '12 at 21:14
1

This is the default plot, you can also access the information to plot each individually and therefore arrange them however you want (or only plot a subset). You can do additional manipulations, or generate random observations from the distribution as well. – Greg Snow Jul 06 '12 at 21:43
(+1) Good information and interesting package. As Michael notes, it might help to make the independence assumption explicit. – cardinal Jul 06 '12 at 21:46
@cardinal, good point, my solution assumes independence. – Greg Snow Jul 06 '12 at 21:51
Thanks! this is very helpful, but can we find a closed form of $F(Y)$? – user9292 Jul 07 '12 at 16:41
Yes, you can find a closed form: but it has to be written as seven formulas for the intervals demarcated by $0,.1,.3,.4,.6,.7,.9,$ and $1$. For instance, for $t\in [0,0.1]$ the PDF is $250t^2/9$ while for $t\in [0.1,0.3]$ it equals $5(20t-1)/18,$ *etc.* That's because uniform PDFs are discontinuous at $0$ and $1$, leading to functional breaks in the PDFs and CDFs of their linear combinations at various places. See http://stats.stackexchange.com/questions/41467. – whuber Apr 21 '14 at 18:17

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