# Summing Normal random variables

I would like to plot the graph of the sum of two random variables.So, I wrote

Plot[PDF[NormalDistribution[0, 1]+NormalDistribution[0, 1], x], {x, -10, 10}]


Why doesn't it work?

• Also, if you need exactly what you tried to execute you can use Plot[PDF[NormalDistribution[0, 1], x] + PDF[NormalDistribution[0, 1], x], {x, -10, 10}, Filling -> Axis, PlotRange -> All] Commented Feb 5, 2014 at 22:25
• And also thanks to Szabolcs for the MixtureDistribution command. ;) Commented Feb 5, 2014 at 22:31
• @Anoldmaninthesea. Actually I was very stupid and gave you an incorrect answer. I apologize. I corrected it now. Commented Feb 5, 2014 at 22:32
• No worries. Thanks for the help. =) Commented Feb 5, 2014 at 22:44

You'll need to use TransformedDistribution to achieve this:

d = TransformedDistribution[
a + b,
{a \[Distributed] NormalDistribution[0, 1], b \[Distributed] NormalDistribution[2, 2]}]

(* ==> NormalDistribution[2, Sqrt[5]] *)


This represents the distribution of a+b if a is distributed according to NormalDistribution[0, 1] and b is distributed according to NormalDistribution[2, 2].

In this case the result is automatically simplified to another NormalDistribution, but it can be used in cases when it can't be automatically simplified as well.

As usual, you can obtain the PDF using PDF[d, x], and then plot it.

• Many thanks Szabolcs. ;) Commented Feb 5, 2014 at 22:44

You can also try

Plot[PDF[NormalDistribution[0, 1], x] +
PDF[NormalDistribution[0, 1], x], {x, -10, 10}, PlotRange -> All]


so just make each Normal Distribution it's own PDF.

• This is incorrect. The OP wanted the PDF of the sum of two random variables, not the sum of two PDFs. Commented Sep 10, 2014 at 12:36