Plot Distributions

Use the tool above to plot statistical distributions online that you can download as PDFs. The charts show the probability density (or mass) function and the cumulative distribution function. You can also generate and plot random samples from the distributions.

To get started, choose a distribution from the drop-down list and enter parameter values. For additional help click on the icon at the top right.

Supported distributions include:

Beta (Shape α, Shape β)
Binomial (Trials n, Probability p)
Cauchy (Location a, Scale γ)
Chi-squared (Degrees of freedom k)
Erlang (Shape k, Rate λ)
Exponential (Rate λ)
F (Degrees of freedom n, Degrees of freedom m)
Gamma (Shape k, Scale θ)
Laplace (Location μ, Scale s)
Levy (Location μ, Scale c)
Logarithmic (Probability p)
Logistic (Mean μ, Scale s)
Log-logistic (Shape α, Scale β)
Log-normal (Mean μ, Scale σ)
Negative binomial (Failures r, Probability of success p)
Normal (Mean μ, Scale σ)
Pareto (Scale k, Shape α)
Poisson (Rate λ)
Rayleigh (Scale σ)
Student's T (Degrees of freedom v)
Triangular (Maximum a, Minimum b, Mode c)
Uniform (Minimum a, Maximum b)
Weibull (Shape k, Scale λ)

Click on the "Details" button below the distribution selection to see full parameterization details for each distribution.

About

The charts are constructed using the d3 Javascript library. The underlying distributions are part of the Essy Distribution Javascript library.

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Failuresr ∈ ℕ _{> 0}

Probability of successp ∈ (0, 1)

k ∈ {0, 1, 2, ...}

x + r − 1

r − 1

p^r (1 − p)^x

1 − I_p(k + 1, r)

where I is the regularized incomplete beta function.

r(1 − p)

1 +

r − 1

r(1 − p)

p²

(2 − p)

√r(1 − p)

p² − 6p + 6

n(1 − p)

Locationμ ∈ ℝ

Scalec > 0

x ∈ [μ, ∞)

√c / 2π

e ^{−

c
2(x − μ)}

(x − μ)^3/2

erfc( √c / 2(x − μ) )

where erfc is the complementary error function.

∞

c / 3for μ = 0

∞

undefined

Population sizeN ∈ ℕ _{> 0}

Number of successesM ∈ ℕ _{≤ N}

Number of trialsn ∈ ℕ _{≤ N}

k ∈ {max(0, n + M − N), ..., min(n, M)}

N − M

n − k

if 0 ≤ k ≤ M ∧ M + n − N ≤ k ≤ M ∧ 0 ≤ k ≤ n ∧ M + n − N ≤ k ≤ n

1 −

k + 1

N − n

M − k − 1

₃F₂

1, k + 1 - M, k + 1 − n

k + 2, N + k + 2 − M − n

; 1

where ₃F₂ is the regularized hypergeometric function.

nM(1 − M / n)(N − n)

N(N − 1)

√N − 1 (N − 2M)(N − 2n)

(N − 2) √nM(N − M)(N − n)

Shapek ∈ (0, ∞)

Scaleλ ∈ (0, ∞)

x ∈ [0, ∞)

(k / λ)(x / λ)^k−1e^{−(x / λ)^k}

1 − e^{−(x / λ)^k}

Minimum a−∞ < a < b

Maximum ba < b < ∞

x ∈ [a, b]

b − a

x − a

b − a

(a + b)

Minimum aa ∈ (−∞, ∞)

Maximum ba < b

Mode ca ≤ c ≤ b

a ≤ x ≤ b

0for x < a

2(x − a)

(b − a)(c − a)

for a ≤ x ≤ c

b − a

for x = c

2(b − x)

(b − a)(b − c)

for c < x ≤ b

0for b < x

0for x ≤ a

(x − a)²

(b − a)(c − a)

for a < x ≤ c

1 −

(b − x)²

(b − a)(b − c)

for c < x < b

1for b ≤ x

a + b + c

Degrees of freedomv > 0

x ∈ (−∞, ∞)

Γ((v + 1) / 2)

√vπ Γ(v / 2)

1 +

x²

^{− v + 1
2}

+ xΓ

v + 1

₂F₁

v + 1

;

; −

x²

√vπ Γ(v / 2)

where ₂F₁ is the hypergeometric function.

0if v > 1, else undefined.

Scaleσ > 0

x ∈ [0, ∞)

σ²

e^{−x² / 2σ²}

1 − e^{−x² / 2σ²}

σ √π / 2

Rateλ > 0

k ∈ ℤ

λ^ke^−λ

Γ(⌊k + 1⌋, λ)

⌊k⌋!

for k ≥ 0

where Γ is the lower incomplete gamma function.

Scalek > 0

Shapeα > 0

x ∈ [k, ∞)

αk²

x^{α + 1}

for x ≥ k.

1 − (k / x)^α for x ≥ k.

∞for α ≤ 1

αk

α − 1

for α > 1

Meanμ ∈ ℝ

Scaleσ > 0

x ∈ ℝ

√2σ²π

e^{−

(x − μ)²
2σ²}

1 + erf

x − μ

σ √2

where erf is the error function.

σ²

Meanμ ∈ ℝ

Scaleσ > 0

x ∈ (0, ∞)

xσ √2π

e^{− (ln x − μ)²
2σ²}

ln x − μ

where Φ is the CDF of the standard normal distribution.

e^{μ + σ² / 2}

(e^σ² − 1) e^{2μ + σ²}

√e^σ² − 1 (e^σ² + 2)

2e^2σ² + 2e^3σ² + e^4σ² − 3

Shapeα > 0

Scaleβ > 0

x ∈ [0, ∞)

(β / α)(x / α)^β−1

(1 + (x / α)^β)²

1 + (x / α)^−β

(απ / β)

sin(x / β)

if β > 1, else undefined.

α² (2b / sin(2b) − b² / sin²(b)) for β > 2

where b = π / β

Meanμ ∈ ℝ

Scales > 0

x ∈ (−∞, ∞)

e^{− x−μ
s}

s(1 + e^{− x−μ
s})²

1 + e^{− x−μ
s}

π²s²

4.2

Probabilityp ∈ [0, 1]

k ∈ {1, 2, 3, ...}

− 1

ln(1 − p)

p^k

1 +

Β(p; k + 1, 0)

ln(1 − p)

where Β is the incomplete beta function.

− 1

ln(1 − p)

1 − p

Locationμ ∈ ℝ

Scales > 0

x ∈ (−∞, ∞)

e^{−

|x − μ|
s}

e^{x − μ
s}

2s²

Shapek > 0

Scaleθ > 0

x ∈ (0, ∞)

Γ(k)θ^k

x^k−1 e^−x/θ

where Γ is the gamma function.

Γ(k)

γ k,

where Γ is the gamma function and γ is the lower incomplete gamma function.

kθ

kθ²

√k

Degrees of freedom 1n > 0

Degrees of freedom 2m > 0

x ∈ [0, ∞)

√((nx)ⁿm^m) / (nx + m)^{n + m}

xΒ

I_{nx
nx + m}

m − 2

for m > 2, else indeterminate

2m²(n + m − 2)

(m − 4)(m − 2)²n

if m > 4, else indeterminate

2 √2 √m − 4(m + 2n − 2)

(m − 6) √n √m + n − 2

if m > 6, else indeterminate

12(5m − 22)n(m + n − 2) + (m − 4)(m − 2)²

(m − 8)(m − 6)n(m + n − 2)

if m > 8, else indeterminate

Rateλ > 0

x ∈ [0, ∞)

λe^−λx

1 − e^−λx

λ²

Shapek ∈ ℕ

Rateλ > 0

x ∈ [0, ∞)

λ^k x^k−1 e^−λx

(k − 1)!

γ(k, λx)

(k − 1)!

where γ is the lower incomplete gamma function.

λ²

√k

Degrees of freedomk ∈ ℕ_{> 0}

x ∈ (0, ∞)if k = 1

x ∈ [0, ∞)if k ≠ 1

2^k/2 Γ

x^{k/2 − 1} e^{− x/2}

2 √2 √1 / k

Locationa ∈ ℝ

Scaleγ > 0

x ∈ (−∞, ∞)

πγ1 +

x − a

arctan

x − a

indeterminate

Trialsn ∈ ℕ _{> 0}

Probabilityp ∈ [0, 1]

k ∈ {0, 1, 2, ...}

p^k(1 − p)^{n − k}

I_{1 − p}(n − k, k + 1)

np(1 − p)

1 − 2p

√np(1 − p)

1 − 6(1 − p)p

n(1 − p)p

+ 3

Shapeα > 0

Shapeβ > 0

x ∈ [0, 1]

x^α−1(1 − x)^β−1

Β(α, β)

where Β(α, β) =

Γ(α)Γ(β)

Γ(α + β)

I_x(α, β)

where I is the regularized incomplete beta function.

α + β

αβ

(α + β)² (α + β + 1)

2(β − α)√α + β + 1

√α√β(α + β + 2)

6[(α − β)²(α + β + 1) − αβ(α + β + 2)]

αβ(α + β + 2)(α + β + 3)

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