Sum of continuous random variables

[EKW]

Consider:

```

>>> from sympy import *
>>> from sympy.stats import *
>>> t = Symbol('t')
>>> n = Uniform('n', 0, 1)
>>> print(density(n + n)(t))

Piecewise((1, (0 <= t/2) & (t/2 <= 1)), (0, True))/2

```

This is a uniform distribution from 0 to 2.

Am I misunderstanding what the sum of two random variables represents in sympy, or is this totally wrong? I would expect the result to be a triangular distribution.

[Francesco Bonazzi]

On Sunday, 8 October 2017 01:10:14 UTC-4, Francesco Bonazzi wrote:

This result is definitely wrong.

Sorry, ignore my previous answer, that result is correct. If you sum n + n, you are summing the same random value. To get the effect of summing two uniform distributions, just define a new uniform distribution:

In [8]: m = Uniform('m', 0, 1)

In [9]: density(n+m)
Out[9]:
      ⎛⎧                                 1     ⎞   ⎛⎧                        
      ⎜⎪            0               for ─── < 1⎟   ⎜⎪                        
      ⎜⎪                                │z│    ⎟   ⎜⎪                        
z ↦ - ⎜⎨                                       ⎟ + ⎜⎨                        
      ⎜⎪  ╭─╮0, 2 ⎛2, 1       │ 1⎞             ⎟   ⎜⎪  ╭─╮0, 2 ⎛2, 1       │  
      ⎜⎪z⋅│╶┐     ⎜           │ ─⎟   otherwise ⎟   ⎜⎪z⋅│╶┐     ⎜           │ ─
      ⎝⎩  ╰─╯2, 2 ⎝      1, 0 │ z⎠             ⎠   ⎝⎩  ╰─╯2, 2 ⎝      1, 0 │ p

                                                                     1       ⎞
                 0                                            for ─────── < 1⎟
                                                                  │z - 1│    ⎟
                                                                             ⎟
       1        ⎞   ╭─╮0, 2 ⎛2, 1       │         1        ⎞                 ⎟
────────────────⎟ - │╶┐     ⎜           │ ─────────────────⎟     otherwise   ⎟
olar_lift(z - 1)⎠   ╰─╯2, 2 ⎝      1, 0 │ polar_lift(z - 1)⎠                 ⎠

The integral is unable to simplify the expression, but that's another issue.