Following a subjective view of distribution, which is more amenable to reinterpretation I use an indicator function to place restrictions on the range of parameter of the PDF.
X \sim U[\alpha,\beta] \tag{C.1}
\mathbb{E}[X]=\frac{(\alpha+\beta)}{2} \tag{C.2}
\mathbb{V}ar[X]=\frac{(\beta-\alpha)^2}{12} \tag{C.3}
f(x)= \frac{1}{\alpha-\beta} \mathbb{I}_{\{\alpha \le x \le \beta\}}(x) \tag{C.4}
F(x\mid \alpha,\beta)=\begin{cases} 0, & \text{if }x < \alpha \\ \frac{x-\alpha}{\beta-\alpha}, & \text{if } x\in [\alpha,\beta]\\ 1, & \text{if } x > \beta \end{cases} \tag{C.5}
Since a number of families have the uniform as a special case we can use them as priors when we want a uniform prior:
Normal(0,1)= Beta(1,1)
The Beta distribution is used for random variables which take on values between 0 and 1. For this reason (and other reasons we will see later in the course), the Beta distribution is commonly used to model probabilities.
X \sim Beta(\alpha, \beta) \tag{C.6}
f(x \mid \alpha, \beta) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha−1}(1 − x)^{\beta−1}\mathbb{I}_{x\in(0,1)}\mathbb{I}_{\alpha\in\mathbb{R}^+}\mathbb{I}_{\beta\in\mathbb{R}^+} \qquad \text{(PDF)} \tag{C.7}
\begin{aligned} & F(x \mid \alpha,\beta) &= I_x(\alpha,\beta) && \text{(CDF)} \\ \text{where } & I_w(u,v) & &&\text{ is the regularized beta function: } \\ & I_w(u,v) &= \frac{B(w; u, v)}{B(u,v)} \\ \text{where }& B(w; u,v) &=\int_{0}^{w}t^{u-1}(1-t)^{v-1}\mathrm{d}t && \text{ is the incomplete beta function }\\ \text{and } & B(u,v)& && \text{ is the (complete) beta function} \end{aligned} \tag{C.8}
\mathbb{E}[X] = \frac{\alpha}{\alpha + \beta} \qquad (\text{expectation}) \tag{C.9}
\mathbb{V}ar[X] = \frac{\alpha\beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)} \qquad (\text{variance}) \tag{C.10}
\mathbb{M}_X(t) = 1+ \sum^\infty_{i=1} \left ( {\prod^\infty_{j=0} \frac{\alpha+j}{\alpha + \beta + j} } \right ) \frac{t^i}{i!} \tag{C.11}
where \Gamma(·) is the Gamma function introduced with the gamma distribution.
Note also that \alpha > 0 and \beta > 0.
The standard Uniform(0, 1) distribution is a special case of the beta distribution with \alpha = \beta = 1.
Uniform(0, 1) = Beta(1,1) \tag{C.12}
The Beta distribution is often used as a prior for parameters that are probabilities,since it takes values from 0 and 1.
During prior elicitation the parameters can be set using
\text{Cauchy}(y\mid\mu,\sigma) = \frac{1}{\pi \sigma} \ \frac{1}{1 + \left((y - \mu)/\sigma\right)^2} \mathbb{I}_{\mu \in \mathbb{R}}\mathbb{I}_{\sigma \in \mathbb{R}^+}\mathbb{I}_{y \in \mathbb{R}} \qquad \text{(PDF)} \tag{C.13}
F(x \mid \mu, \sigma) = \frac{1}{2} + \frac{1}{\pi}\text{arctan}\left(\frac{x-\mu}{\sigma}\right) \qquad \text{(CDF)} \tag{C.14}
\mathbb{E}(X) = \text{ undefined}
\mathbb{V}ar[X] = \text{ undefined}
The Cauchy despite having no mean or variance is recommended as a prior for regression coefficients in Logistic regression. see (Gelman et al. 2008) this is analyzed and discussed in (Ghosh, Li, and Mitra 2018)
\text{DoubleExponential}(y \mid \mu,\sigma) = \frac{1}{2\sigma} \exp \left( - \, \frac{\|y - \mu\|}{\sigma} \right) \qquad \text (PDF) \tag{C.15}
If X_1, X_2, ..., X_n are independent (and identically distributed \mathrm{Exp}(\lambda)) waiting times between successive events, then the total waiting time for all n events to occur Y = \sum X_i will follow a gamma distribution with shape parameter \alpha = n and rate parameter \beta = \lambda:
We denote this as:
Y =\sum^N_{i=0} \mathrm{Exp}_i(\lambda) \sim \mathrm{Gamma}(\alpha = N, \beta = \lambda) \tag{C.16}
f(y \mid \alpha , \beta) = \frac{\beta^\alpha} {\Gamma(\alpha)} y^{\alpha−1} e^{− \beta y} \mathbb{I}_{y \ge \theta }(y) \tag{C.17}
\mathbb{E}[Y] = \frac{\alpha}{ \beta} \tag{C.18}
\mathbb{V}ar[Y] = \frac{\alpha}{ \beta^2} \tag{C.19}
where \Gamma(·) is the gamma function, a generalization of the factorial function which can accept non-integer arguments. If n is a positive integer, then \Gamma(n) = (n − 1)!.
Note also that \alpha > 0 and $ > 0$.
The exponential distribution is a special case of the Gamma distribution with \alpha = 1. The gamma distribution commonly appears in statistical problems, as we will see in this course. It is used to model positive-valued, continuous quantities whose distribution is right-skewed. As \alpha increases, the gamma distribution more closely resembles the normal distribution.
\mathcal{IG}(y\mid\alpha,\beta) = \frac{1} {\Gamma(\alpha)}\frac{\beta^{\alpha}}{y^{\alpha + 1}} e^{- \frac{ \beta}{y}} \ \mathbb{I}_{\alpha \in \mathbb{R}^+}\mathbb{I}_{\beta \in \mathbb{R}^+}\mathbb{I}_{y \in \mathbb{R}^+} \qquad \text (PDF) \tag{C.20}
\mathbb{E}[X]=\frac{\beta}{\alpha - 1} \qquad \text{Expectation} \tag{C.21}
\mathbb{V}ar[X]=\frac{\beta^2}{(\alpha-1)^2(\alpha-2)}\qquad \text{Variance} \tag{C.22}
· The Standard normal distribution is given by:
Z \sim \mathcal{N}[1,0] \tag{C.23}
f(z) = \frac{1}{\sqrt{2 \pi}} e^{-\frac{z^2}{2}} \tag{C.24}
\mathcal{L}(\mu,\sigma)=\prod_{i=1}^{n}{1 \over 2 \pi \sigma}e^{−(x_i−\mu)^2 \over 2 \sigma^2} \tag{C.25}
\begin{aligned} \ell(\mu, \sigma) &= \log \mathcal{L}(\mu, \sigma) \\ &= -\frac{n}{2}\log(2\pi) - n\log\sigma - \sum_{i=1}^n \frac{(x_i-\mu)^2}{2\sigma^2} \end{aligned}\sigma^2 \tag{C.26}
\begin{aligned} \mathbb{E}(Z)&= 0 \quad \text{(Expectation)} \qquad \mathbb{V}ar(Z)&= 1 \quad \text{(Variance)} \end{aligned} \tag{C.27}
The normal, or Gaussian distribution is one of the most important distributions in statistics.
It arises as the limiting distribution of sums (and averages) of random variables. This is due to the sec-cl-theorem. Because of this property, the normal distribution is often used to model the “errors,” or unexplained variations of individual observations in regression models.
Now consider X = \sigma Z+\mu where \sigma > 0 and \mu is any real constant. Then \mathbb{E}[X] = \mathbb{E}[\sigma Z+\mu] = \sigma E[Z] + \mu = \sigma_0 + \mu = \mu and \mathbb{V}ar[X] = Var(\sigma^2 Z + \mu) = \sigma^2 Var(Z) + 0 = \sigma^2 \cdot 1 = \sigma^2
Then, X follows a normal distribution with mean \mu and variance \sigma^2 (standard deviation \sigma) denoted as
X \sim N[\mu,\sigma^2] \tag{C.28}
f(x\mid\mu,\sigma^2) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{1}{\sqrt{2 \pi \sigma^2}}(x-\mu)^2} \tag{C.29}
\mathbb{E}(x)= \mu \tag{C.30}
Var(x)= \sigma^2 \tag{C.31}
bell-shaped curve.The normal distribution has several desirable properties.
One is that if X_1 \sim N(\mu_1, \sigma^2_1) and X_2 ∼ N(\mu_2, \sigma^2_2) are independent, then X_1+X_2 \sim N(\mu_1+\mu_2, \sigma^2_1+\sigma^2_2).
Consequently, if we take the average of n Independent and Identically Distributed (IID) Normal random variables we have:
\bar X = \frac{1}{n}\sum_{i=1}^n X_i \sim N(\mu, \frac{\sigma^2}{n}) \tag{C.32}
import numpy as np
from scipy.stats import norm
import matplotlib.pyplot as plt
fig, ax = plt.subplots(1, 1)
n, p = 5, 0.4
mean, var, skew, kurt = norm.stats(moments='mvsk')
print(f'{mean=:1.2f}, {var=:1.2f}, {skew=:1.2f}, {kurt=:1.2f}')mean=0.00, var=1.00, skew=0.00, kurt=0.00x = np.linspace(norm.ppf(0.01),
norm.ppf(0.99), 100)
ax.plot(x, norm.pdf(x),
'r-', lw=5, alpha=0.6, label='norm pdf')
rv = norm()
ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
r = norm.rvs(size=1000)
ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2)(array([0.00737665, 0.00368832, 0.01106497, 0.05901319, 0.06270151,
0.10696141, 0.12909135, 0.23236443, 0.25080606, 0.35776746,
0.31719589, 0.35039081, 0.36145579, 0.37989741, 0.32088422,
0.2581827 , 0.20285784, 0.12909135, 0.05532487, 0.03688324,
0.02581827, 0.0147533 , 0.00368832, 0.00737665, 0.00368832]), array([-3.19698904, -2.92586321, -2.65473738, -2.38361155, -2.11248572,
-1.84135989, -1.57023406, -1.29910823, -1.0279824 , -0.75685656,
-0.48573073, -0.2146049 , 0.05652093, 0.32764676, 0.59877259,
0.86989842, 1.14102425, 1.41215008, 1.68327591, 1.95440174,
2.22552757, 2.4966534 , 2.76777923, 3.03890506, 3.31003089,
3.58115672]), [<matplotlib.patches.Polygon object at 0x73ba9cfc1e40>])ax.set_xlim([x[0], x[-1]])(-2.3263478740408408, 2.3263478740408408)ax.legend(loc='best', frameon=False)
plt.show()
If we have normal data, we can use (Equation eq-normal-sum) to help us estimate the mean \mu. Reversing the transformation from the previous section, we get:
\frac {\hat X - \mu}{\sigma / \sqrt(n)} \sim N(0, 1) \tag{C.33}
However, we may not know the value of \sigma. If we estimate it from data, we can replace it with S = \sqrt{\sum_i \frac{(X_i-\hat X)^2}{n-1}}, the sample standard deviation. This causes the expression (Equation eq-t-transform) to no longer be distributed as a Standard Normal; but as a standard t-distribution with ν = n − 1 degrees of freedom
X \sim t[\nu] \tag{C.34}
f(t\mid\nu) = \frac{\Gamma(\frac{\nu+1}{2})}{\Gamma(\frac{\nu}{2})\sqrt{\nu\pi}}\left (1 + \frac{t^2}{\nu}\right)^{-(\frac{\nu+1}{2})}\mathbb{I}_{t\in\mathbb{R}} \qquad \text{(PDF)} \tag{C.35}
\text{where }\Gamma(w)=\int_{0}^{\infty}t^{w-1}e^{-t}\mathrm{d}t \text{ is the gamma function}
f(t\mid\nu)={\frac {1}{{\sqrt {\nu }}\,\mathrm {B} ({\frac {1}{2}},{\frac {\nu }{2}})}}\left(1+{\frac {t^{2}}{\nu }}\right)^{-(\nu +1)/2}\mathbb{I}_{t\in\mathbb{R}} \qquad \text{(PDF)} \tag{C.36}
\text{where } B(u,v)=\int_{0}^{1}t^{u-1}(1-t)^{v-1}\mathrm{d}t \text{ is the beta function}
\begin{aligned} && F(t)&=\int _{-\infty }^{t}f(u)\,du=1-{\tfrac {1}{2}}I_{x(t)}\left({\tfrac {\nu }{2}},{\tfrac {1}{2}}\right) &&\text{(CDF)} \\ \text{where } && I_{x(t)}&= \frac{B(x; u, v)}{B(u,v)} &&\text{is the regularized Beta function} \\ \text{where } && B(w; u,v)&=\int_{0}^{w}t^{u-1}(1-t)^{v-1}\mathrm{d}t && \text{ is the incomplete Beta function } \\ \text {and } && B(u,v)&=\int_{0}^{1}t^{u-1}(1-t)^{v-1}\mathrm{d}t && \text{ is the (complete) beta function} \end{aligned} \tag{C.37}
\int _{-\infty }^{t}f(u)\,du={\tfrac {1}{2}}+t{\frac {\Gamma \left({\tfrac {1}{2}}(\nu +1)\right)}{{\sqrt {\pi \nu }}\,\Gamma \left({\tfrac {\nu }{2}}\right)}}\,{}_{2}F_{1}\left({\tfrac {1}{2}},{\tfrac {1}{2}}(\nu +1);{\tfrac {3}{2}};-{\tfrac {t^{2}}{\nu }}\right)
\mathcal{L}(\mu, \sigma, \nu) = \prod_{i=1}^n \frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\nu\pi}\Gamma(\frac{\nu}{2})} \left(1+\frac{(x_i-\mu)^2}{\sigma^2\nu}\right)^{-\frac{\nu+1}{2}} \tag{C.38}
\begin{aligned} \ell(\mu, \sigma, \nu) &= \log \mathcal{L}(\mu, \sigma, \nu) \\ &= \sum_{i=1}^n \left[\log\Gamma\left(\frac{\nu+1}{2}\right) - \log\sqrt{\nu\pi} - \log\Gamma\left(\frac{\nu}{2}\right) - \frac{\nu+1}{2}\log\left(1+\frac{(x_i-\mu)^2}{\sigma^2\nu}\right)\right] \\ &= n\log\Gamma\left(\frac{\nu+1}{2}\right) - n\log\sqrt{\nu\pi} - n\log\Gamma\left(\frac{\nu}{2}\right) - \frac{\nu+1}{2}\sum_{i=1}^n\log\left(1+\frac{(x_i-\mu)^2}{\sigma^2\nu}\right). \end{aligned} \tag{C.39}
\mathbb{E}[Y] = 0 \qquad \text{ if } \nu > 1 \tag{C.40}
\mathbb{V}ar[Y] = \frac{\nu}{\nu - 2} \qquad \text{ if } \nu > 2 \tag{C.41}
X=\mu+\sigma T
The resulting distribution is also called the non-standardized Student’s t-distribution.
this is another parameterization of the student-t with:
f(x \mid \mu, \sigma, \nu) = \frac{\left(\frac{\nu }{\nu +\frac{(x-\mu )^2}{\sigma ^2}}\right)^{\frac{\nu+1}{2}}}{\sqrt{\nu } \sigma B\left(\frac{\nu }{2},\frac{1}{2} \right)} \tag{C.42}
\text{where } B(u,v)=\int_{0}^{1}t^{u-1}(1-t)^{v-1}\mathrm{d}t \text{ is the beta function}
F(\mu, \sigma, \nu) = \begin{cases} \frac{1}{2} I_{\frac{\nu \sigma ^2}{(x-\mu )^2+\nu \sigma ^2}}\left(\frac{\nu }{2},\frac{1}{2}\right), & x\leq \mu \\ \frac{1}{2} \left(I_{\frac{(x-\mu )^2}{(x-\mu )^2+\nu \sigma ^2}}\left(\frac{1}{2},\frac{\nu }{2}\right)+1\right), & \text{Otherwise} \end{cases} \tag{C.43}
where I_w(u,v) is the regularized incomplete beta function:
I_w(u,v) = \frac{B(w; u, v)}{B(u,v)}
where B(w; u,v) is the incomplete beta function:
B(w; u,v) =\int_{0}^{w}t^{u-1}(1-t)^{v-1}\mathrm{d}t
And B(u,v) is the (complete) beta function
\mathbb{E}[X] = \begin{cases} \mu, & \text{if }\nu > 1 \\ \text{undefined} & \text{ otherwise} \end{cases} \tag{C.44}
\mathbb{V}ar[X] = \frac{\nu \sigma^2}{\nu-2} \tag{C.45}
The t distribution is symmetric and resembles the Normal Distribution but with thicker tails. As the degrees of freedom increase, the t distribution looks more and more like the standard normal distribution.
The student-t distribution is due to Gosset, William Sealy (1876-1937) who was an English statistician, chemist and brewer who served as Head Brewer of Guinness and Head Experimental Brewer of Guinness and was a pioneer of modern statistics. He is known for his pioneering work on small sample experimental designs. Gosset published under the pseudonym “Student” and developed most famously Student’s t-distribution – originally called Student’s “z” – and “Student’s test of statistical significance”.
He was told to use a Pseudonym and choose ‘Student’ after a predecessor at Guinness published a paper that leaked trade secrets. Gosset was a friend of both Karl Pearson and Ronald Fisher. Fisher suggested a correction to the student-t using the degrees of freedom rather than the sample size. Fisher is also credited with helping to publicize its use.
for a full biography see (Pearson et al. 1990)
The Exponential distribution models the waiting time between events for events with a rate lambda. Those events, typically, come from a Poisson process
The exponential distribution is often used to model the waiting time between random events. Indeed, if the waiting times between successive events are independent then they form an Exp(λ) distribution. Then for any fixed time window of length t, the number of events occurring in that window will follow a Poisson distribution with mean tλ.
X \sim Exp[\lambda] \tag{C.46}
f(x \mid \lambda) = \frac{1}{\lambda} e^{- \frac{x}{\lambda}}(x)\mathbb{I}_{\lambda\in\mathbb{R}^+ } \mathbb{I}_{x\in\mathbb{R}^+_0 } \quad \text{(PDF)} \tag{C.47}
F(x \mid \lambda) = 1 - e^{-\lambda x} \qquad \text{(CDF)}
\mathcal{L}(\lambda) = \prod_{i=1}^n \lambda e^{-\lambda x_i} \tag{C.48}
\begin{aligned} \ell(\lambda) &= \log \mathcal{L}(\lambda) \\ &= \sum_{i=1}^n \log(\lambda) - \lambda x_i \\ &= n\log(\lambda) - \lambda\sum_{i=1}^n x_i \end{aligned} \tag{C.49}
\mathbb{E}(x)= \lambda \tag{C.50}
\mathbb{V}ar[X]= \lambda^2 \tag{C.51}
\mathbb{M}_X(t)= \frac{1}{1-\lambda t} \qquad t < \frac{1}{\gamma} \tag{C.52}
import numpy as np
from scipy.stats import expon
import matplotlib.pyplot as plt
fig, ax = plt.subplots(1, 1)
n, p = 5, 0.4
mean, var, skew, kurt = expon.stats(moments='mvsk')
print(f'{mean=:1.2f}, {var=:1.2f}, {skew=:1.2f}, {kurt=:1.2f}')mean=1.00, var=1.00, skew=2.00, kurt=6.00x = np.linspace(expon.ppf(0.01), expon.ppf(0.99), 100)
ax.plot(x, expon.pdf(x), 'r-', lw=5, alpha=0.6, label='expon pdf')
rv = expon()
ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
r = expon.rvs(size=1000)
ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2)(array([0.79628554, 0.68381583, 0.60283764, 0.44088126, 0.37789822,
0.32391276, 0.29692003, 0.15745759, 0.15745759, 0.12596607,
0.10347213, 0.05848425, 0.06298304, 0.04498788, 0.08547698,
0.02249394, 0.02249394, 0.02699273, 0.0404891 , 0.00899758,
0. , 0.01349637, 0.00899758, 0.00449879, 0. ,
0.00449879, 0.01349637, 0. , 0. , 0. ,
0.00899758, 0. , 0. , 0.00449879]), array([2.39229972e-03, 2.24674371e-01, 4.46956443e-01, 6.69238515e-01,
8.91520587e-01, 1.11380266e+00, 1.33608473e+00, 1.55836680e+00,
1.78064887e+00, 2.00293095e+00, 2.22521302e+00, 2.44749509e+00,
2.66977716e+00, 2.89205923e+00, 3.11434130e+00, 3.33662338e+00,
3.55890545e+00, 3.78118752e+00, 4.00346959e+00, 4.22575166e+00,
4.44803373e+00, 4.67031581e+00, 4.89259788e+00, 5.11487995e+00,
5.33716202e+00, 5.55944409e+00, 5.78172617e+00, 6.00400824e+00,
6.22629031e+00, 6.44857238e+00, 6.67085445e+00, 6.89313652e+00,
7.11541860e+00, 7.33770067e+00, 7.55998274e+00]), [<matplotlib.patches.Polygon object at 0x73ba9cedc940>])ax.set_xlim([x[0], x[-1]])(0.010050335853501442, 4.605170185988091)ax.legend(loc='best', frameon=False)
plt.show()
The long normal arises when the a log transform is applied to the normal distribution.
\text{LogNormal}(y\mid\mu,\sigma) = \frac{1}{\sqrt{2 \pi} \ \sigma} \, \frac{1}{y} \ \exp \! \left( - \, \frac{1}{2} \, \left( \frac{\log y - \mu}{\sigma} \right)^2 \right) \ \mathbb{I}_{\mu \in \mathbb{R}}\mathbb{I}_{\sigma \in \mathbb{R}^+}\mathbb{I}_{y \in \mathbb{R}^+} \qquad \text (PDF) \tag{C.53}
\text{Pareto}(y|y_{\text{min}},\alpha) = \frac{\displaystyle \alpha\,y_{\text{min}}^\alpha}{\displaystyle y^{\alpha+1}} \mathbb{I}_{\alpha \in \mathbb{R}^+}\mathbb{I}_{y_{min} \in \mathbb{R}^+}\mathbb{I}_{y\ge y_{min} \in \mathbb{R}^+} \qquad \text (PDF) \tag{C.54}
\mathrm{Pareto\_Type\_2}(y|\mu,\lambda,\alpha) = \ \frac{\alpha}{\lambda} \, \left( 1+\frac{y-\mu}{\lambda} \right)^{-(\alpha+1)} \! \mathbb{I}_{\mu \in \mathbb{R}}\mathbb{I}_{\lambda \in \mathbb{R}^+}\mathbb{I}_{\alpha \in \mathbb{R}^+}\mathbb{I}_{y\ge \mu \in \mathbb{R}} \qquad \text (PDF) \tag{C.55}
\mathbb{E}[X]=\displaystyle{\frac{\alpha y_\mathrm{min}}{\alpha - 1}}\mathbb{I}_{\alpha>1} \qquad \text (expectation) \tag{C.56}
\mathbb{V}ar[X]=\displaystyle{\frac{\alpha y_\mathrm{min}^2}{(\alpha - 1)^2(\alpha - 2)}}\mathbb{I}_{\alpha>2} \qquad \text (variance) \tag{C.57}
\text{Weibull}(y|\alpha,\sigma) = \frac{\alpha}{\sigma} \, \left( \frac{y}{\sigma} \right)^{\alpha - 1} \, e^{ - \left( \frac{y}{\sigma} \right)^{\alpha}} \mathbb{I}_{\alpha \in \mathbb{R}^+}\mathbb{I}_{\sigma \in \mathbb{R}^+}\mathbb{I}_{y \in \mathbb{R}^+} \qquad \text (PDF)
The chi squared distribution is a special case of the gamma. It is widely used in hypothesis testing and the construction of confidence intervals. It is parameterized using parameter \nu for the degrees of predom
\text{ChiSquare}(y\mid\nu) = \frac{2^{-\nu/2}} {\Gamma(\nu / 2)} \, y^{\nu/2 - 1} \, \exp \! \left( -\, \frac{1}{2} \, y \right) \mathbb{I}_{\nu \in \mathbb{R}^+}\mathbb{I}_{y \in \mathbb{R}} \qquad \text (PDF) \tag{C.58}
{\frac {1}{\Gamma (\nu/2)}}\;\gamma \left({\frac {\nu}{2}},\,{\frac {x}{2}}\right) \qquad \text (CDF) \tag{C.59}
\mathbb{E}[X]=\nu \tag{C.60}
\mathbb{V}ar[X] = 2\nu \tag{C.61}
\text{Logistic}(y|\mu,\sigma) = \frac{1}{\sigma} \ \exp\!\left( - \, \frac{y - \mu}{\sigma} \right) \ \left(1 + \exp \!\left( - \, \frac{y - \mu}{\sigma} \right) \right)^{\!-2} \mathbb{I}_{\mu \in \mathbb{R}}\mathbb{I}_{\sigma \in \mathbb{R}^+}\mathbb{I}_{y \in \mathbb{R}} \qquad \text (PDF) \tag{C.62}
The F-distribution or F-ratio, arises frequently as the null distribution of a test statistic, in the analysis of variance (ANOVA) and other F-tests.
\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)} \tag{C.63}
\mathbb{I}_{\frac {d_{1}x}{d_{1}x+d_{2}}}\left({\tfrac {d_{1}}{2}},{\tfrac {d_{2}}{2}}\right) \tag{C.64}
\mathbb{E}[X]=\frac {d_{2}}{d_{2}-2} \tag{C.65}
\mathbb{V}ar[X] = {\frac {2\,d_{2}^{2}\,(d_{1}+d_{2}-2)}{d_{1}(d_{2}-2)^{2}(d_{2}-4)}} \tag{C.66}