Chi-squared distribution
|
Normal distribution
|
Log-normal
|
Student's t
|
Laplace distribution
|
Weibull distribution
|
Pareto distribution
|
chi-squared
|
parameters |
— mean (location)
— variance (squared scale) |
, ![{\displaystyle \sigma >0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/762ecd0f0905dd0d4d7a07f80fa8bfb324b9b021) |
degrees of freedom (real) |
location (real)
scale (real) |
scale
shape |
xm > 0 scale (real) α > 0 shape (real) |
(known as "degrees of freedom")
|
pdf_image |
![Probability density function for the normal distribution](//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Normal_Distribution_PDF.svg/350px-Normal_Distribution_PDF.svg.png) The red curve is the standard normal distribution |
![Plot of the Lognormal PDF](//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/PDF-log_normal_distributions.svg/300px-PDF-log_normal_distributions.svg.png) Some log-normal density functions with identical parameter but differing parameters ![{\displaystyle \sigma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36) |
![](//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Student_t_pdf.svg/325px-Student_t_pdf.svg.png) |
![Probability density plots of Laplace distributions](//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Laplace_pdf_mod.svg/325px-Laplace_pdf_mod.svg.png) |
![Probability distribution function](//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Weibull_PDF.svg/325px-Weibull_PDF.svg.png) |
![Pareto Type I probability density functions for various α](//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Probability_density_function_of_Pareto_distribution.svg/325px-Probability_density_function_of_Pareto_distribution.svg.png) Pareto Type I probability density functions for various α with xm = 1. As α → ∞ the distribution approaches δ(x − xm) where δ is the Dirac delta function. |
|
pdf |
![{\displaystyle {\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\,e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a9287a082350af2fe84ea67da609e32f8591528) |
![{\displaystyle {\frac {1}{x\sigma {\sqrt {2\pi }}}}\ e^{-{\frac {\left(\ln x-\mu \right)^{2}}{2\sigma ^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c02672f3d3d174f3025fe93379bd92c6d6d5406f) |
|
|
| |
![{\displaystyle {\frac {\alpha \,x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}{\text{ for }}x\geq x_{m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aba33339c2e3833b21d12d4d6417311105030c00) |
|
cdf
|
![{\displaystyle {\tfrac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1839c06a3505568b7a4f141cb99ee5e0a0b39dd) |
![{\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\operatorname {erf} {\Big [}{\frac {\ln x-\mu }{{\sqrt {2}}\sigma }}{\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac1eb0032c5ba3af1ffbacf16a1a2ca275bdc657) |
where 2F1 is the hypergeometric function
|
|
|
![{\displaystyle 1-\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }{\text{ for }}x\geq x_{m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b1a644166c925cddc546fd309a6b3d8533a2c4) |
|
mean |
![{\displaystyle \quad \mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e0b88c13e96068427ea74158dc1ebd8ce63e8b5) |
![{\displaystyle \exp(\mu +\sigma ^{2}/2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37a3523df29ed51879b731cb975e063708172aac) |
0 for , otherwise undefined |
![{\displaystyle \mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161) |
![{\displaystyle \lambda \,\Gamma (1+1/k)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f5fcff3ff516836a57147e75a081078fae9309e) |
![{\displaystyle {\begin{cases}\infty &{\text{for }}\alpha \leq 1\\{\frac {\alpha \,x_{\mathrm {m} }}{\alpha -1}}&{\text{for }}\alpha >1\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/129c4b7aaace36bca691189802afdfd7575d58f4) |
|
variance |
![{\displaystyle \quad \sigma ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1aeca67847d56ce69da7d74761f189089819b8b) |
![{\displaystyle [\exp(\sigma ^{2})-1]\exp(2\mu +\sigma ^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b71d1959535c7b8ea00f302c3045c8dd941999b7) |
for , ∞ for , otherwise undefined |
![{\displaystyle 2b^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94998b0735ee7dc8e0249874c4a125b04f38ca68) |
![{\displaystyle \lambda ^{2}\left[\Gamma \left(1+{\frac {2}{k}}\right)-\left(\Gamma \left(1+{\frac {1}{k}}\right)\right)^{2}\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55fa6b5cdbe81bb9e6aa0452a2c619623cb23f14) |
![{\displaystyle {\begin{cases}\infty &{\text{for }}\alpha \in (0,2]\\{\frac {x_{\mathrm {m} }^{2}\alpha }{(\alpha -1)^{2}(\alpha -2)}}&{\text{for }}\alpha >2\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75ed52dee361942081ef1df0cc5ffef3b141f599) |
|
skewness |
![{\displaystyle \quad 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8b55a41700d68037e5603c1ee5c302a1f6a22c3) |
![{\displaystyle (e^{\sigma ^{2}}\!\!+2){\sqrt {e^{\sigma ^{2}}\!\!-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68a58ace97758726d2b43d70445f0d4e313de45d) |
0 for , otherwise undefined |
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) |
![{\displaystyle {\frac {\Gamma (1+3/k)\lambda ^{3}-3\mu \sigma ^{2}-\mu ^{3}}{\sigma ^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4687dd1de1cc08d3945ffb23108a6b84299e7e2) |
![{\displaystyle {\frac {2(1+\alpha )}{\alpha -3}}\,{\sqrt {\frac {\alpha -2}{\alpha }}}{\text{ for }}\alpha >3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55d575b8090d688eef7561cd5742cb2ebfc4e3a5) |
|
support |
![{\displaystyle x\in \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c6d458566aec47a7259762034790c8981aefab) |
![{\displaystyle x\in (0,+\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a84264a320783309b0360b749207851a58f148a) |
x ∈ (−∞; +∞) |
![{\displaystyle x\in (-\infty ;+\infty )\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3db4e72e0e12b8d2c961e72c720cd6c35636d77e) |
![{\displaystyle x\in [0,+\infty )\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7825a65e5be6c8a35a11eca156c1d69947afe3ca) |
![{\displaystyle x\in [x_{\mathrm {m} },+\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c00bced0717aafd229283255b17e938513c74bf1) |
|
median |
![{\displaystyle \quad \mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e0b88c13e96068427ea74158dc1ebd8ce63e8b5) |
![{\displaystyle \exp(\mu )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a12b59e62f4c73d184dab9960dbfeab8691d0c7) |
0 |
![{\displaystyle \mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161) |
![{\displaystyle \lambda (\ln 2)^{1/k}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ebc9bce539d77c38ca468b0a4f430b06fe73d77) |
![{\displaystyle x_{\mathrm {m} }{\sqrt[{\alpha }]{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef1a9e02a1d60cf9cd611b13188b078509904bc7) |
|
quantile |
![{\displaystyle \mu +\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(2F-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f09f40acec233b5b190f25968e19ac45d21d96b) |
- |
- |
- |
- |
- |
-
|