# Complex Uniform Distribution between 1+0im and 2+0im

**URL:** <https://discourse.julialang.org/t/complex-uniform-distribution-between-1-0im-and-2-0im/32831>\
**Category:** Statistics\
**Created:** [December 31, 2019, 8:51pm UTC](https://discourse.julialang.org/t/complex-uniform-distribution-between-1-0im-and-2-0im/32831 "2019-12-31T20:51:57Z")\
**Posts on this page:** 8\
**Page:** 1

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**Author:** ![Amin\_Yahyaabadi](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/amin_yahyaabadi/32/9826_2.png) [@Amin\_Yahyaabadi](https://discourse.julialang.org/u/Amin_Yahyaabadi)\
**Post date:** [December 31, 2019, 8:51pm UTC](https://discourse.julialang.org/t/complex-uniform-distribution-between-1-0im-and-2-0im/32831/1 "2019-12-31T20:51:57Z")

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I am trying to extend Uniform so `rand` can generate random numbers for Complex types ([to test IntelVectorMath library for Complex types](https://github.com/JuliaMath/VML.jl/blob/94fadec653387d0aa7e5b7389167769c0c0eee3e/benchmark/benchmark.jl#L105))

```julia
# zeros like syntax
function Distributions.Uniform(::Type{T}, a, b) where {T <: Real}
    return Uniform(T(a), T(b))
end

# Complex support
function Distributions.Uniform(T::Type{Complex{P}}, a, b) where {P}
    aR, aI =reim(T(a))
    bR, bI =reim(T(b))
    if aR == bR
        outR = P(0) #?? Ask about this
    else
        outR = Uniform(P, min(aR,bR), max(aR,bR))
    end
    if aI == bI
        outI = P(0) #?? Ask about this
    else
        outI = Uniform(P, min(aI,bI), max(aI,bI))
    end
    return complex(outR, outI)
end

# doesn't work:
Uniform(Complex{Float64},1,2)

```

The problem arises when a and b are equal. In literature, the probability is 0. However, in Julia’s implementation, it is not apparent that when we define `Unifrom(a,b)`, is it a `[a, b]` or `(a,b)`.

If it is `[a, b]` then `Uniform(a, a)` should always give `a`  
if is is `(a, b)` then `Uniform(a, a)` should give a constant 0 distribution that rand always gives some null situation.

My implementation is faulty, but I can’t find anything in Distributions doc anything about `Complex` types. Probably it should be a multi-dimensional distribution.

---

<div class="post-metadata">

**Author:** ![Syx\_Pek](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/syx_pek/32/6364_2.png) [@Syx\_Pek](https://discourse.julialang.org/u/Syx_Pek)\
**Post date:** [December 31, 2019, 9:00pm UTC](https://discourse.julialang.org/t/complex-uniform-distribution-between-1-0im-and-2-0im/32831/2 "2019-12-31T21:00:02Z")

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Here is my opinion on the issue.

`Uniform(a, b)` as per the docs returns [a, b] but demands that a \< b. I feel like this should stay.

We should introduce a new distribution `d = Singleton{X}` that `rand(d)` returns the constant `X`. Then the product measure defined by OP will work as intended. This notation is used in [https://github.com/MikeInnes/Poirot.jl](https://github.com/MikeInnes/Poirot.jl) .

Last, we should resolve [https://github.com/JuliaStats/Distributions.jl/issues/142](https://github.com/JuliaStats/Distributions.jl/issues/142) and Ill like to hijack the issue and say that generic convolution should also work.

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<div class="post-metadata">

**Author:** ![Amin\_Yahyaabadi](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/amin_yahyaabadi/32/9826_2.png) [@Amin\_Yahyaabadi](https://discourse.julialang.org/u/Amin_Yahyaabadi)\
**Post date:** [December 31, 2019, 9:05pm UTC](https://discourse.julialang.org/t/complex-uniform-distribution-between-1-0im-and-2-0im/32831/3 "2019-12-31T21:05:31Z")

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I updated OP. We also need extend `complex()` so it can get `Distributions` types.

Another issue that is related is my problem with the Uniform definition and its API. I talked about it here:

[https://github.com/JuliaStats/Distributions.jl/issues/1041](https://github.com/JuliaStats/Distributions.jl/issues/1041)

I created a PR to solve it here:

[https://github.com/JuliaStats/Distributions.jl/pull/1045/files](https://github.com/JuliaStats/Distributions.jl/pull/1045/files)  
From the mathematical point of view, `a` and `b` bounds may not be part of the distribution and so their type shouldn’t be part of the Distribution definition. **Actually their probability is equal to 0!**

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<div class="post-metadata">

**Author:** ![Syx\_Pek](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/syx_pek/32/6364_2.png) [@Syx\_Pek](https://discourse.julialang.org/u/Syx_Pek)\
**Post date:** [December 31, 2019, 9:57pm UTC](https://discourse.julialang.org/t/complex-uniform-distribution-between-1-0im-and-2-0im/32831/4 "2019-12-31T21:57:23Z")

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Having probability 0 does not mean it does not occur. Either definition is self-consistent.

To see this more clearly, the probability for any value on an interval [a, b] gets returned by a uniform random distribution on it is 0.

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**Author:** ![Tamas\_Papp](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/tamas_papp/32/25949_2.png) [@Tamas\_Papp](https://discourse.julialang.org/u/Tamas_Papp)\
**Post date:** [January 1, 2020, 7:03am UTC](https://discourse.julialang.org/t/complex-uniform-distribution-between-1-0im-and-2-0im/32831/5 "2020-01-01T07:03:11Z")

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> [@Amin\_Yahyaabadi](#):
>
> The problem arises when a and b are equal.

I imagine that the problem arises before this: since [complex numbers have no canonical ordering](https://en.wikipedia.org/wiki/Complex_number#Ordering), it’s hard to make sense of a \< x \< b. Commonly, complex uniform distributions are defined over [unit circles](https://en.wikipedia.org/wiki/Complex_random_variable#Uniform_distribution).

As for the problem of singular/meaningless parameters: I think it is preferable if constructors simply error, instead of doing something clever. Currently

```julia
julia> Uniform(0,0)
ERROR: ArgumentError: Uniform: the condition a < b is not satisfied.

```

which is fine, but

```julia
julia> Normal(1, 0)
Normal{Float64}(μ=1.0, σ=0.0)

```

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<div class="post-metadata">

**Author:** ![Syx\_Pek](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/syx_pek/32/6364_2.png) [@Syx\_Pek](https://discourse.julialang.org/u/Syx_Pek)\
**Post date:** [January 1, 2020, 12:32pm UTC](https://discourse.julialang.org/t/complex-uniform-distribution-between-1-0im-and-2-0im/32831/6 "2020-01-01T12:32:51Z")

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I think the reason why `Uniform` does not permit `a = b` is due to the docs being written as

```julia
  Uniform(a,b)

  The continuous uniform distribution over an interval [a, b] has probability
  density function

```

f(x; a, b) = \frac{1}{b - a}, \quad a \le x \le b

which would imply that a \neq b due to the division by 0. But returning the `Singleton` as described in my first reply is perfectly sensible to, with some rewriting of the docs.

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<div class="post-metadata">

**Author:** ![Tamas\_Papp](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/tamas_papp/32/25949_2.png) [@Tamas\_Papp](https://discourse.julialang.org/u/Tamas_Papp)\
**Post date:** [January 1, 2020, 3:01pm UTC](https://discourse.julialang.org/t/complex-uniform-distribution-between-1-0im-and-2-0im/32831/7 "2020-01-01T15:01:48Z")

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> [@Syx\_Pek](#):
>
> returning the `Singleton` as described in my first reply is perfectly sensible

No, it would be [type unstable](https://docs.julialang.org/en/v1/manual/performance-tips/#Write-%22type-stable%22-functions-1). While the compiler can handle this to a limited extent, it is not good design without a compelling reason.

Also, it is a convention for `T` constructors to return `T` objects — this is not enforced, but it is not good style to violate this.

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<div class="post-metadata">

**Author:** ![Syx\_Pek](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/syx_pek/32/6364_2.png) [@Syx\_Pek](https://discourse.julialang.org/u/Syx_Pek)\
**Post date:** [January 2, 2020, 9:11am UTC](https://discourse.julialang.org/t/complex-uniform-distribution-between-1-0im-and-2-0im/32831/8 "2020-01-02T09:11:23Z")

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You’re right I didn’t think this through in that direction.