In the following `sd`

represents a generic one-dimensional statistical distribution. We loosely refer to `sd`

for random variables distributed
according to `sd`

. Each `sd`

has associated parameters which may be required to satisfy numerical constraints. If such constraints are violated an
error is thrown.

Returns the minimum and the maximum possible theoretical values (infinite values allowed) assumed by `sd`

.

The argument to the function must be a prng or a qrng object. In the first case the function generates
independent and identically distributed random numbers according to the law of `sd`

. In the second case the function returns samples which uniformly cover
the range of sd according to the law of `sd`

. Please notice that some statistical distributions do **not** support sampling via a qrng, in which
case this fact is noted in the relevant documentation.

Returns the value of the probability density function (for continuous distributions) or of the mass probability function (for discrete distributions) computed at
`x`

for `sd`

. The domain of this function is the real line.

Returns the value of the natural logarithm of `sd:pdf()`

. It is preferable to use this function instead of `math.log(sd:pdf(x))`

. The domain of
this function is the real line.

Returns the mean of `sd`

. Returns an infinite number where appropriate or a `nan`

number if the mean does not exist.

Returns the variance of `sd`

. Returns an infinite number where appropriate or a `nan`

number if the variance does not exist.

Returns an independent copy of `sd`

which is initialized with the same parameters of `sd`

.

Constraint: \( a < b \). Returns a continuous uniform distribution which is characterized by the pdf: $$ f(x) = \frac{1}{b - a}, \quad x\in(a,b) $$

Constraint: \( \lambda > 0 \). Returns a continuous exponential distribution which is characterized by the pdf: $$ f(x) = \lambda e^{-\lambda x}, \quad x\in(0,+\infty) $$

Constraint: \( \sigma > 0 \). Returns a continuous normal (Gaussian) distribution which is characterized by the pdf: $$ f(x) = \frac{1}{\sqrt{2\pi}\sigma}\exp{\{-\frac{(x - \mu)^2}{2\sigma^2}\}}, \quad x\in(-\infty,+\infty) $$

Constraint: \( \sigma > 0 \). Returns a continuous log-normal distribution which is characterized by the pdf: $$ f(x) = \frac{1}{\sqrt{2\pi}\sigma x}\exp{\{-\frac{(\ln{x} - \mu)^2}{2\sigma^2}\}}, \quad x\in(0,+\infty) $$

Constraint: \( \nu > 0 \). Returns a continuous student's t-distribution which is characterized by the pdf: $$ f(x) = \frac{1}{\sqrt{\nu}\text{B}\left(\frac{1}{2},\frac{\nu}{2}\right)}\left(1 + \frac{x^2}{2}\right)^{-\frac{\nu + 1}{2}}, \quad x\in(-\infty,+\infty) $$ where \( \text{B} \) is the beta function.

Constraint: \( \alpha > 0 \) and \( \beta > 0 \). Returns a continuous gamma distribution with shape parameter \( \alpha \) and **rate** parameter \(
\beta \) which is characterized by the pdf: $$ f(x) = \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}, \quad x\in(0,+\infty) $$ where \( \Gamma \) is the
gamma function. This distribution does **not** allow sampling via a qrng.

Constraint: \( \alpha > 0 \) and \( \beta > 0 \). Returns a continuous beta distribution which is characterized by the pdf: $$ f(x) =
\frac{1}{\text{B}(\alpha,\beta)}x^{\alpha-1}(1-x)^{\beta-1}, \quad x\in(0,1) $$ where \( \text{B} \) is the beta function. This
distribution does **not** allow sampling via a qrng