# Cdf of multivariate normal in distributions.jl

**URL:** <https://discourse.julialang.org/t/cdf-of-multivariate-normal-in-distributions-jl/4773>\
**Category:** Statistics\
**Created:** [July 10, 2017, 9:21pm UTC](https://discourse.julialang.org/t/cdf-of-multivariate-normal-in-distributions-jl/4773 "2017-07-10T21:21:42Z")\
**Posts on this page:** 20\
**Page:** 1

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**Author:** ![newb\_gk](https://avatars.discourse-cdn.com/v4/letter/n/e19b73/32.png) [@newb\_gk](https://discourse.julialang.org/u/newb_gk)\
**Post date:** [July 10, 2017, 9:21pm UTC](https://discourse.julialang.org/t/cdf-of-multivariate-normal-in-distributions-jl/4773/1 "2017-07-10T21:21:42Z")

</div>

I’m trying to use the cdf of a multivariate normal distribution using Distributions.jl.  
While

```julia
d = Normal(0,1)
d2 = MvNormal([0.;0.], [1. 0.;0. 1.])
x = [1. ; 2.]
e1 = cdf(d,x[1])
e2 = cdf(d,x[2])

```

works (as long as the mean vector and the var-cov matrix for the multivariate normal are Floats ), I get an error whenever I write  
`e3 = cdf(d2,x)`  
Here is the error message:

```julia
ERROR: MethodError: no method matching cdf(::Distributions.MvNormal{Float64,PDMats.PDMat{Float64,Array{Float64,2}},Array{Float64,1}}, ::Array{Float64,1})
Closest candidates are:
  cdf(::Distributions.Distribution{Distributions.Univariate,S<:Distributions.ValueSupport}, ::AbstractArray{T,N}) at /Applications/JuliaPro-0.5.1.1.app/Contents/Resources/pkgs-0.5.1.1/v0.5/Distributions/src/univariates.jl:205

```

Does the cdf function support multivariate distributions? I couldn’t find anything on this in the documentation of Distributions.jl.

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<div class="post-metadata">

**Author:** ![andreasnoack](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/andreasnoack/32/27_2.png) [@andreasnoack](https://discourse.julialang.org/u/andreasnoack)\
**Post date:** [July 11, 2017, 12:53am UTC](https://discourse.julialang.org/t/cdf-of-multivariate-normal-in-distributions-jl/4773/2 "2017-07-11T00:53:03Z")

</div>

No, there is not a `cdf` for the multivariate normal. Generally, it is a slightly complicated computation. We have some code evaluating the bi- and trivariate case but it hasn’t been used for a long time. For some time, I’ve wanted a dedicated package for multivariate distributions that would be using `StaticArrays` for storage. The `cdf` code for fit well in such a package.

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**Author:** ![newb\_gk](https://avatars.discourse-cdn.com/v4/letter/n/e19b73/32.png) [@newb\_gk](https://discourse.julialang.org/u/newb_gk)\
**Post date:** [July 11, 2017, 2:53pm UTC](https://discourse.julialang.org/t/cdf-of-multivariate-normal-in-distributions-jl/4773/3 "2017-07-11T14:53:36Z")

</div>

That’s a shame. We really should have something in Julia for this. I’ll work around it using PyCall for the time being.

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<div class="post-metadata">

**Author:** ![andreasnoack](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/andreasnoack/32/27_2.png) [@andreasnoack](https://discourse.julialang.org/u/andreasnoack)\
**Post date:** [July 11, 2017, 4:24pm UTC](https://discourse.julialang.org/t/cdf-of-multivariate-normal-in-distributions-jl/4773/4 "2017-07-11T16:24:49Z")

</div>

Are you mainly interested in 2D and 3D or higher dimensions? Do you know how higher dimensions are handled in Python?

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<div class="post-metadata">

**Author:** ![newb\_gk](https://avatars.discourse-cdn.com/v4/letter/n/e19b73/32.png) [@newb\_gk](https://discourse.julialang.org/u/newb_gk)\
**Post date:** [July 11, 2017, 4:31pm UTC](https://discourse.julialang.org/t/cdf-of-multivariate-normal-in-distributions-jl/4773/5 "2017-07-11T16:31:28Z")

</div>

My interest is currently for 2D. I have no idea how higher dimensions are handled in Python right now but hopefully I’ll learn a little bit about it in the next few days.

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<div class="post-metadata">

**Author:** ![andreasnoack](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/andreasnoack/32/27_2.png) [@andreasnoack](https://discourse.julialang.org/u/andreasnoack)\
**Post date:** [July 11, 2017, 4:42pm UTC](https://discourse.julialang.org/t/cdf-of-multivariate-normal-in-distributions-jl/4773/6 "2017-07-11T16:42:16Z")

</div>

If you are interested in spending a little time on it, we could probably get the 2D Julia version working again. If we do it with StaticArrays it will also be extremely fast.

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**Author:** ![crsl4](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/crsl4/32/6647_2.png) [@crsl4](https://discourse.julialang.org/u/crsl4)\
**Post date:** [December 13, 2017, 6:20pm UTC](https://discourse.julialang.org/t/cdf-of-multivariate-normal-in-distributions-jl/4773/7 "2017-12-13T18:20:47Z")

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Hello,  
I am also interested in a julia version for the cdf for 2D normal. I might be able to spend a bit time on this, but where can I find the version that exists already? I am not familiar with StaticArrays though, but I can certainly give it a try.  
Thanks!

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<div class="post-metadata">

**Author:** ![andreasnoack](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/andreasnoack/32/27_2.png) [@andreasnoack](https://discourse.julialang.org/u/andreasnoack)\
**Post date:** [December 13, 2017, 7:10pm UTC](https://discourse.julialang.org/t/cdf-of-multivariate-normal-in-distributions-jl/4773/9 "2017-12-13T19:10:26Z")

</div>

It is here [https://github.com/JuliaStats/StatsFuns.jl/blob/e21bc26b1773aeb86bc8df4832db9c371d09ba2b/src/tvpack.jl#L338](https://github.com/JuliaStats/StatsFuns.jl/blob/e21bc26b1773aeb86bc8df4832db9c371d09ba2b/src/tvpack.jl#L338)

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<div class="post-metadata">

**Author:** ![oschulz](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/oschulz/32/2998_2.png) [@oschulz](https://discourse.julialang.org/u/oschulz)\
**Post date:** [November 18, 2019, 8:23am UTC](https://discourse.julialang.org/t/cdf-of-multivariate-normal-in-distributions-jl/4773/10 "2019-11-18T08:23:41Z")

</div>

I think it would be very helpful to have a CDF defined for `MvNormal`, analog to MATLAB’s [mvncdf](https://www.mathworks.com/help/stats/mvncdf.html). It would definitely help when dealing with truncated Gaussian priors, etc.

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**Author:** ![blackeneth](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/blackeneth/32/10353_2.png) [@blackeneth](https://discourse.julialang.org/u/blackeneth)\
**Post date:** [November 27, 2019, 12:10am UTC](https://discourse.julialang.org/t/cdf-of-multivariate-normal-in-distributions-jl/4773/11 "2019-11-27T00:10:46Z")

</div>

I’m working on it (shh…)

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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:** [November 27, 2019, 7:49am UTC](https://discourse.julialang.org/t/cdf-of-multivariate-normal-in-distributions-jl/4773/12 "2019-11-27T07:49:55Z")

</div>

I was surprised to find that there is a whole _book_ about this:

> **[Computation of Multivariate Normal and t Probabilities](https://link.springer.com/book/10.1007/978-3-642-01689-9)**

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<div class="post-metadata">

**Author:** ![mschauer](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/mschauer/32/13946_2.png) [@mschauer](https://discourse.julialang.org/u/mschauer)\
**Post date:** [November 27, 2019, 3:10pm UTC](https://discourse.julialang.org/t/cdf-of-multivariate-normal-in-distributions-jl/4773/13 "2019-11-27T15:10:31Z")

</div>

I have the bivariate case in [https://github.com/mschauer/GaussianDistributions.jl:](https://github.com/mschauer/GaussianDistributions.jl:)

[https://github.com/mschauer/GaussianDistributions.jl/blob/master/src/bivariate.jl](https://github.com/mschauer/GaussianDistributions.jl/blob/master/src/bivariate.jl)

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**Author:** ![mschauer](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/mschauer/32/13946_2.png) [@mschauer](https://discourse.julialang.org/u/mschauer)\
**Post date:** [November 27, 2019, 3:13pm UTC](https://discourse.julialang.org/t/cdf-of-multivariate-normal-in-distributions-jl/4773/14 "2019-11-27T15:13:45Z")

</div>

Ah, I did not know we had that!

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<div class="post-metadata">

**Author:** ![blackeneth](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/blackeneth/32/10353_2.png) [@blackeneth](https://discourse.julialang.org/u/blackeneth)\
**Post date:** [November 28, 2019, 3:44pm UTC](https://discourse.julialang.org/t/cdf-of-multivariate-normal-in-distributions-jl/4773/15 "2019-11-28T15:44:26Z")

</div>

Yes, Prof. Genz literally “wrote the book on it”!

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<div class="post-metadata">

**Author:** ![acxz](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/acxz/32/16759_2.png) [@acxz](https://discourse.julialang.org/u/acxz)\
**Post date:** [September 28, 2022, 4:58pm UTC](https://discourse.julialang.org/t/cdf-of-multivariate-normal-in-distributions-jl/4773/16 "2022-09-28T16:58:33Z")

</div>

I apologize if this is a necro-bump, but came across this thread as the first Google Search entry.

Has the CDF for Multivariate normals been implemented in Julia now? I essentially looking for an analogous to just sample the CDF of a normal distribution, maybe something akin to R’s `pnorm`.

Edit: Found what I wanted with the combination of [Univariate Distributions · Distributions.jl](https://juliastats.org/Distributions.jl/stable/univariate/#Distributions.Normal) and [Univariate Distributions · Distributions.jl](https://juliastats.org/Distributions.jl/stable/univariate/#Distributions.cdf-Tuple%7BUnivariateDistribution,%20Real%7D)

E.g.:

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

julia> Distributions.cdf(dist, 3/640) - Distributions.cdf(dist, -1/128)
0.00498673995194876

```

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<div class="post-metadata">

**Author:** ![Paul\_Soderlind](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/paul_soderlind/32/1753_2.png) [@Paul\_Soderlind](https://discourse.julialang.org/u/Paul_Soderlind)\
**Post date:** [September 28, 2022, 9:27pm UTC](https://discourse.julialang.org/t/cdf-of-multivariate-normal-in-distributions-jl/4773/17 "2022-09-28T21:27:47Z")

</div>

MvNormalCDF.jl at [GitHub - PharmCat/MvNormalCDF.jl: Quasi-Monte-Carlo numerical computation of multivariate normal probabilities](https://github.com/PharmCat/MvNormalCDF.jl)

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**Author:** ![acxz](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/acxz/32/16759_2.png) [@acxz](https://discourse.julialang.org/u/acxz)\
**Post date:** [September 29, 2022, 4:02pm UTC](https://discourse.julialang.org/t/cdf-of-multivariate-normal-in-distributions-jl/4773/18 "2022-09-29T16:02:44Z")

</div>

Thanks for the link!

Looks like here is the open issue: [CDF of Multivariate Normal Distribution · Issue #260 · JuliaStats/Distributions.jl · GitHub](https://github.com/JuliaStats/Distributions.jl/issues/260)

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<div class="post-metadata">

**Author:** ![Fuzeq](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/fuzeq/32/49148_2.png) [@Fuzeq](https://discourse.julialang.org/u/Fuzeq)\
**Post date:** [April 19, 2023, 12:13pm UTC](https://discourse.julialang.org/t/cdf-of-multivariate-normal-in-distributions-jl/4773/19 "2023-04-19T12:13:19Z")

</div>

I used HCubature to calculate integral of PDF. Maybe not the fastest, but it works

```julia
using HCubature, Distributions, Plots, LinearAlgebra
plotly()
μ = [0.0, 0.0]
Σ = [1.0 0.0; 0.0 1.0]

my_pdf(x) = ((2*pi)^2*det(Σ))^(-0.5)*exp(-0.5*transpose(x-μ)*inv(Σ)*(x-μ))
mv_norm_cdf(x,y)= hcubature(x -> my_pdf(x), [-10.0, -10.0], [x, y])[1]

xs = -5:0.5:5
ys = -5:0.5:5

surface(xs,ys, mv_norm_cdf, xlabel="X", ylabel="Y", zlabel="CDF", color=:viridis, zlim=[0,1])

```

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<div class="post-metadata">

**Author:** ![Fuzeq](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/fuzeq/32/49148_2.png) [@Fuzeq](https://discourse.julialang.org/u/Fuzeq)\
**Post date:** [April 20, 2023, 1:36pm UTC](https://discourse.julialang.org/t/cdf-of-multivariate-normal-in-distributions-jl/4773/20 "2023-04-20T13:36:27Z")

</div>

I optimalized my code so it works faster. I used 99.7 rule to define the interval of calculation.

```julia

using HCubature, Distributions, Plots, BenchmarkTools,Statistics, StatsBase,Random

function mv_norm_cdf(x,y , μ, Σ)
    x_low = μ[1]-3*Σ[1,1]
    y_low = μ[2]-3*Σ[2,2]
    dist = MvNormal(μ,Σ)
    return hcubature(x -> pdf(dist,x), [x_low, y_low], [x, y])[1]
end

function mv_norm_cdf_plot(μ, Σ)
    x_low = μ[1]-3*Σ[1,1]
    y_low = μ[2]-3*Σ[2,2]
    x_high = μ[1]+3*Σ[1,1]
    y_high = μ[2]+3*Σ[2,2]
    dist = MvNormal(μ,Σ)
    _mv_norm_cdf(x,y) = hcubature(x -> pdf(dist,x), [x_low, y_low], [x, y])[1]

    xs = x_low:0.5:x_high
    ys = y_low:0.5:y_high

    surface(xs,ys, _mv_norm_cdf, xlabel="X", ylabel="Y", zlabel="CDF", color=:viridis, zlim=[0,1])
end

```

Here is example use

```julia
μ = [0.0, 0.0]
Σ = [1.0 0.0; 0.0 1.0]

mv_norm_cdf(0,0,μ,Σ)
mv_norm_cdf_plot(μ,Σ)

```

![image](https://global.discourse-cdn.com/julialang/original/3X/2/a/2a7130d993bd245de84a03dd551dbc5fdf046796.png)

Also to compare example of empirical cdf

```julia
function mv_normal_ecdf(μ,Σ)
    x_low = μ[1]-3*Σ[1,1]
    y_low = μ[2]-3*Σ[2,2]
    x_high = μ[1]+3*Σ[1,1]
    y_high = μ[2]+3*Σ[2,2]
    dist = MultivariateNormal(μ, Σ)
    n = 1000
    sample = rand(dist, n)
    
    ecdf(x,y) = sum((sample[1,i]<x && sample[2,i]<y) for i in 1:n)/n
    
    xs = x_low:0.25:x_high
    ys = y_low:0.25:y_high
    
    surface(xs,ys,ecdf, xlabel="X", ylabel="Y", zlabel="ECDF", color=:viridis, zlim=[0,1])
end

```

![image](https://global.discourse-cdn.com/julialang/original/3X/6/e/6e60e6ee376f5b5825204a80c2c9075734bb4910.png)

---

<div class="post-metadata">

**Author:** ![Dan](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/dan/32/42581_2.png) [@Dan](https://discourse.julialang.org/u/Dan)\
**Post date:** [April 20, 2023, 2:14pm UTC](https://discourse.julialang.org/t/cdf-of-multivariate-normal-in-distributions-jl/4773/21 "2023-04-20T14:14:36Z")

</div>

> [@Fuzeq](#):
>
> Also to compare example of empirical cdf

Can you add a plot of the difference also?

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