# A comparison of common distributions in Julia, Python and R

**URL:** <https://discourse.julialang.org/t/a-comparison-of-common-distributions-in-julia-python-and-r/61604>\
**Category:** Statistics\
**Tags:** distributions\
**Created:** [May 21, 2021, 8:35pm UTC](https://discourse.julialang.org/t/a-comparison-of-common-distributions-in-julia-python-and-r/61604 "2021-05-21T20:35:13Z")\
**Posts on this page:** 2\
**Page:** 1

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**Author:** ![sylvaticus](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/sylvaticus/32/203883_2.png) [@sylvaticus](https://discourse.julialang.org/u/sylvaticus)\
**Post date:** [May 21, 2021, 8:35pm UTC](https://discourse.julialang.org/t/a-comparison-of-common-distributions-in-julia-python-and-r/61604/1 "2021-05-21T20:35:13Z")

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> **[GitHub - sylvaticus/commonDistributionsInJuliaPythonR: Common probability distributions in Julia, Python...](https://github.com/sylvaticus/commonDistributionsInJuliaPythonR)**
>
> Common probability distributions in Julia, Python and R

# Common distributions in Julia, Python and R

Please report errors on [GitHub - sylvaticus/commonDistributionsInJuliaPythonR: Common probability distributions in Julia, Python and R](https://github.com/sylvaticus/commonDistributionsInJuliaPythonR)

## Loading packages

- Julia: `using Distributions`
- Python: `from scipy import stats`
- R: `library(extraDistr)`

## Discrete distributions

- **Discrete Uniform** : Complete ignorance
- **Bernoulli** : Single binary trial
- **Binomial** : Number of successes in independent binary trials
- **Categorical** : Individual categorical trial
- **Multinomial** : Number of successes of the various categories in independent multinomial trials
- **Geometric** : Number of independent binary trials until (and including) the first success (discrete time to first success)
- **Hypergeometric** : Number of successes sampling without replacement from a bin with given initial number of items representing successes
- **Multivariate hypergeometric** : Number of elements sampled in the various categories from a bin without replacement
- **Poisson** : Number of independent arrivals in a given period given their average rate per that period length
- **Pascal** : Number of independent binary trials until (and including) the n-th success (discrete time to n-th success).

| Name | Parameters | Support | PMF | Expectations | Variance | CDF |
| --- | --- | --- | --- | --- | --- | --- |
| **D. Unif** | a,b ∈ Z with b ≧ a | x \in \{a,a+1,...,b\} | \frac{1}{b-a+1} | \frac{a+b}{2} | \frac{(b-a)(b-a+2)}{12} | \frac{x-a+1}{b-a+1} |
| **Bern** | p ∈ [0,1] | x ∈ {0,1} | p^x(1-p)^{1-x} | p | p(1-p) | \sum\_{i=0}^x p^i(1-p)^{1-i} |
| **Bin** | p ∈ [0,1], n in N⁺ | x \in \{0,...,n\} | {{n} \choose {x}} p^x(1-p)^{1-x} | np | n p(1-p) | \sum\_{i=0}^{x} {{n} \choose {i}} p^i(1-p)^{1-i} |
| **Cat** | p\_1,p\_2,...,p\_K with p\_k \in [0,1] and \sum\_{k=1}^K p\_k =1 | x ∈ {1,2,…,K} | \prod\_{k=1}^K p\_k^{\mathbb{1}(k=x)} | | | |
| **Multin** | n, p\_1,p\_2,...,p\_K with p\_k \in [0,1], \sum\_{k=1}^K p\_k =1 and n \in N^+ | x \in \mathbb{N}\_{0}^K | {{n} \choose {x\_1, x\_2,...,x\_K}} \prod\_{k=1}^K p\_k^{x\_K} | | | |
| **Geom** | p ∈ [0,1] | x ∈ N⁺ | (1-p)^{x-1}p | \frac{1}{p} | \frac{1-p}{p^2} | 1-(1-p)^x |
| **Hyperg** | n\_s,n\_f, n \in \mathbb{N}\_{0} | x \in \mathbb{N}\_{0} with x \leq n\_s | \frac{{n\_s \choose x} {n\_f \choose n-x} }{ (n\_s + n\_f) \choose n } | n \frac{n\_s}{n\_s+n\_f} | n\frac{n\_s}{n\_s+n\_f}\frac{n\_f}{n\_s+n\_f}\frac{n\_s+n\_f+n}{n\_s+n\_f+1} | |
| **Multiv hyperg** | n\_1,n\_2,...,n\_K, n with n \in \mathbb{N}\_{+}, n\_i \in \mathbb{N}\_{0} | x \in \mathbb{N}\_{0}^K with x\_i \leq n\_i ~ \forall i, \sum\_{i=1}^K x\_i = n | \frac{\prod\_{i=1}^K {n\_i \choose x\_i} }{ \sum\_{i=1}^K n\_i \choose n } | n\frac{n\_i}{\sum\_{i=1}^K n\_i} | n\frac{\sum\_{j=1}^K n\_j - n}{\sum\_{j=1}^K n\_j - 1} \frac{n\_i}{\sum\_{j=1}^K n\_j} \left(1 - \frac{n\_i}{\sum\_{j=1}^K n\_j} \right) | |
| **Pois** | λ in R⁺ | x ∈ N₀ | \frac{\lambda^xe^{-\lambda}}{x!} | \lambda | \lambda | |
| **Pasc** | n ∈ N⁺, p in [0,1] | x in N⁺ | {x-1 \choose n-1} p^n (1-p)^{x-n} | \frac{n}{p} | \frac{n(1-p)}{p^2} | |

| Distribution | Julia | Python (stats.[distributionName]) | R |
| --- | --- | --- | --- |
| Discrete uniform | `DiscreteUniform(lRange,uRange)` | `randint(lRange,uRange)` | `dunif(lRange,uRange)` |
| Bernoulli | `Bernoulli(p)` | `bernoulli(p)` | `bern(p)` |
| Binomial | `Binomial(n,p)` | `binom(n,p)` | `binom(n,p)` |
| Categorical | `Categorical(ps)` | Not Av. | `cat(ps)` |
| Multinomial | `Multinomial(n, ps)` | `multinomial(n, ps)` | `mnom(n,ps)` |
| Geometric | `Geometric(p)` | `geom(p)` | `geom(p)` |
| Hypergeometric | `Hypergeometric(nS, nF, nTrials)` | `hypergeom(nS+nF,nS,nTrials)` | `hyper(nS, nF, nTrias)` |
| Mv hypergeometric | Not Av. | `multivariate_hypergeom(initialNByCat,nTrials)` | `mvhyper(initialNByCat,nTrials)` |
| Poisson | `Poisson(rate)` | `poisson(rate)` | `pois(rate)` |
| Negative Binomial | `NegativeBinomial(nSucc,p)` | `nbinom(nSucc,p)` | `nbinom(nSucc,p)` |

## Continuous distributions

- **Uniform** complete ignorance, pick at random, all equally likely outcomes
- **Exponential** waiting time to first event whose rate is λ (continuous time to first success)
- **Normal** The asymptotic distribution of a sample means
- **Erlang** Time of the n-th arrival
- **Cauchy** The ratio of two independent zero-means normal r.v.
- **Chi-squared** The sum of the squared of iid standard normal r.v.
- **T distribution** The distribution of a sample means
- **F distribution** : The ratio of the ratio of two indep Χ² r.v. with their relative parameter
- **Beta distribution** The Beta distribution
- **Gamma distribution** Generalisation of the exponential, Erlang and chi-square distributions

| Name | Parameters | Support | PMF | Expectations | Variance | CDF |
| --- | --- | --- | --- | --- | --- | --- |
| **Unif** | a,b ∈ R with b ≧ a | x ∈ [a,b] | \frac{1}{b-a} | \frac{a+b}{2} | \frac{(b-a)^2}{12} | \frac{x-a}{b-a} |
| **Expo** | λ ∈ R⁺ | x ∈ R⁺ | \lambda e^{-\lambda x} | \frac{1}{\lambda} | \frac{1}{\lambda^2} | 1-e^{-\lambda x} |
| **Normal** | μ ∈R, σ² ∈ R⁺ | x ∈ R | \frac{1}{\sigma \sqrt{2 \pi}}e^\frac{-(x-\mu)^2}{2\sigma^2} | \mu | \sigma^2 | |
| **Erlang** | n ∈ N⁺, λ ∈ R⁺ | x ∈ Rₒ | \frac{\lambda^n x^{n-1} e^{-\lambda x} }{(n - 1) !} | \frac{n}{\lambda} | \frac{n}{\lambda^2} | |
| **Cauchy** | x₀ ∈ R (_location_), γ ∈ R⁺ (_scale_) | \frac{1}{\pi \gamma (1+(\frac{x-x\_0}{\gamma})^2) } | | | | |
| **Chi-sq** | d ∈ N⁺ | x ∈ R⁺ | \frac{1}{2^{}\frac{d}{2}\Gamma(\frac{d}{2})} x^{\frac{d}{2})-1}e^{-\frac{x}{2}} | d | 2d | |
| **T** | ν ∈ R⁺ | x ∈ R | \frac{ \Gamma(\frac{\nu +1}{2})}{\sqrt{\nu \pi} \Gamma(\frac{\nu}{2})} \left( 1 + \frac{x^2}{\nu} \right)^{- \frac{\nu + 1}{2}} | | | |
| **F** | d₁ ∈ N⁺ d₂ ∈ N⁺ | x ∈ R⁺ | \frac {\sqrt {\frac {(d\_1 x)^{d\_1} d\_2^{d\_2} } {(d\_1 x + d\_2)^{d\_1 + d\_2} } }} {x \mathrm {B} \left( \frac{d\_1}{2},\frac {d\_2}{2} \right) } | \frac{d\_2}{d\_2 -2} for d\_2 \> 2 | \frac{2 d\_2^2 (d\_1 + d\_2 -2)}{d\_1 (d\_2 -2)^2 (d\_2 -4)} for d\_2 \> 4 | |
| **Beta** | α, β ∈ R⁺ | x ∈ [1,0] | \frac{1}{B(\alpha,\beta)}x^{\alpha-1}(1-x)^{\beta-1} | \frac{\alpha}{\alpha+\beta} | \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)} | |
| **Gamma** | α ∈ R⁺ (_shape_), β ∈ R⁺ (_rate_) | x ∈ R⁺ | \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x} | \frac{\alpha}{\beta} | \frac{\alpha}{\beta^2} | |

**Beta function** : B(\alpha,\beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} = \frac{\alpha + \beta}{\alpha \beta}   
**Gamma function** : \Gamma(x)=(x-1)! ~ \forall x \in N

| Distribution | Julia | Python (stats.[distributionName]) | R |
| --- | --- | --- | --- |
| Uniform | `Uniform(lRange,uRange)` | `uniform(lRange,uRange)` | `unif(lRange,uRange)` |
| Exponential | `Exponential(rate)` | `expon(rate)` | `exp(rate)` |
| Normal | `Normal(μ,sqrt(σsq))` | `norm(μ,math.sqrt(σsq))` | `norm(μ,sqrt(σsq))` |
| Erlang | `Erlang(n,rate)` | `erlang(n,rate)` | Use gamma |
| Cauchy | `Cauchy(μ, σ)` | `cauchy(μ, σ)` | `cauchy(μ,σ)` |
| Chisq | `Chisq(df)` | `chi2(df)` | `chisq(df)` |
| T Dist | `TDist(df)` | `t(df)` | `t(df)` |
| F Dist | `FDist(df1, df2)` | `f(df1, df2)` | `f(df1,df2)` |
| Beta Dist | `Beta(shapeα,shapeβ)` | `beta(shapeα,shapeβ)` | `beta(shapeα,shapeβ)` |
| Gamma Dist | `Gamma(shapeα,1/rateβ)` | `gamma(shapeα,1/rateβ)` | `gamma(shapeα,1/rateβ)` |

Note: The Negative Binomial returns the number of failures before n successes instead of the total trials to n successes as the Pascal distribution

## Usage

y = CDF(x), i.e. y ∈ [0,1]

| | Julia | Python | R |
| --- | --- | --- | --- |
| **Mean** | `mean(d)` | `d.mean()` | |
| **Variance** | `var(d)` | `d.var()` | |
| **Median** | `median(d)` | `d.median()` | |
| **Sample** | `rand(d)` | `d.rvs()` | `r[distributionName](1,distributionParameters)`, e.g. `runif(1,10,20)` |
| **Quantiles** (F^{-1}(y)) | `quantile(d,y)` | `d.ppf(y)` | `q[distributionName](y, distributionParameters)`, e.g. `qunif(0.2,10,20)` |
| **PDF/PMF** | `pdf(d,x)` | `d.pmf(x)` for discrete r.v. and `d.pdf(x)` for continuous ones | `d[distributionName](x, distributionParameters)`, e.g. `dunif(15,10,20)` |
| **CDF** | `cdf(d,x)` | `d.cdf(x)` | `p[distributionName](x, distributionParameters)`, e.g. `punif(15,10,20)` |

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**Author:** ![jzr](https://avatars.discourse-cdn.com/v4/letter/j/eb9ed0/32.png) [@jzr](https://discourse.julialang.org/u/jzr)\
**Post date:** [May 21, 2021, 9:06pm UTC](https://discourse.julialang.org/t/a-comparison-of-common-distributions-in-julia-python-and-r/61604/2 "2021-05-21T21:06:41Z")

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@cscherrer 's MeasureTheory.jl ecosystem might be another Julia option?