Lindley distribution (generalized)

class ctx.dist_lindley(b)

where ctx is fpm, mpm, ipm, dec, gmp or apm.

The Lindley distribution is a continuous probability distribution with scale \(b > 0\), and the support interval \((0,+\infty)\).

See also: Al-Babtain et al. [7], Zakerzadeh and Dolati [1654], Lindley [431].

dist_lindley.pdf(x)

Returns \(\text{pdf}_X(x)\), the probability density function (pdf) of a random variable \(X\), following an Lindley distribution:

\[\text{pdf}_X(x) = \frac{\theta^2}{\eta + \theta k} \left(\frac{k(\theta x)^{\alpha-1}}{\Gamma(\alpha)} + \frac{\eta(\theta x)^{\beta-1}}{\theta \Gamma(\beta)} \right) e^{-\theta x}, \quad x>0.\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("pdf: ", lindley(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_lindley.cdf(x)

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following an Lindley distribution:

\[\text{cdf}_X(x) = \frac{1}{\eta + \theta k} \left(\theta k P(\alpha, \theta x) + \eta P(\beta, \theta x) \right), \quad x>0.\]

\[\text{cdf}_X(x) = w \cdot P(\alpha, \theta x) + (1-w) \cdot P(\beta, \theta x), \quad w = \frac{\theta k}{\eta + \theta k}.\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("cdf: ", lindley(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_lindley.sf(x)

Returns \(\text{sf}_X(x)\), the survival function (sf) of a random variable \(X\), following an Lindley distribution:

\[\text{sf}_X(x) = 1 - \frac{1}{\eta + \theta k} \left(\theta k P(\alpha, \theta x) + \eta P(\beta, \theta x) \right), \quad x>0.\]

\[\text{sf}_X(x) = w \cdot Q(\alpha, \theta x) + (1-w) \cdot Q(\beta, \theta x), \quad w = \frac{\theta k}{\eta + \theta k}.\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print (" sf: ", lindley(mu, sigma).pdf(x))
sf: 6.3563523462564525615615615614561356E-20

dist_lindley.qtf(q)

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following an Lindley distribution:

\[\text{qtf}_X(q) = \frac{1}{\theta} \left( w \cdot P^{-1}(\alpha, q) + (1-w) \cdot P^{-1}(\beta, q) \right), \quad w = \frac{\theta k}{\eta + \theta k}.\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("qtf: ", lindley(mu, sigma).qtf(q))
qtf: 6.3563523462564525615615615614561356E+00

dist_lindley.isf(q)

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following an Lindley distribution:

\[\text{isf}_X(q) = \frac{1}{\theta} \left( w \cdot Q^{-1}(\alpha, q) + (1-w) \cdot Q^{-1}(\beta, q) \right), \quad w = \frac{\theta k}{\eta + \theta k}.\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("isf: ", lindley(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_lindley.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following an Lindley distribution:

\[C_X(t) = \frac{1}{\eta + \theta k} \left[\theta k \left(1-\frac{it}{\theta}\right)^{-\alpha} + \eta \left(1-\frac{it}{\theta}\right)^{-\beta} \right].\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("c_x: ", lindley(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_lindley.m_x(t)

Returns \(M_X(t)\), the moment generating function of a random variable \(X\), following an Lindley distribution:

\[M_X(t) = \frac{1}{\eta + \theta k} \left[\theta k \left(1-\frac{t}{\theta}\right)^{-\alpha} + \eta \left(1-\frac{t}{\theta}\right)^{-\beta} \right].\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("m_x: ", lindley(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_lindley.k_x(t, k=0)

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following an Lindley distribution:

\[K_X(t) = \log \left( M(t) \right).\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3; k = 6;
>>> print ("c_x: ", lindley(mu, sigma).k_x(t, k))
6.3563523462564525615615615614561356E+00

dist_lindley.moments(k)

Returns the first \(j\) raw moments, \(\mu'_j, j = 1 \ldots k\), of a random variable \(X\), following an Lindley distribution (Wikipedia). The raw moments are given by

\[\mu'_X(k) = \frac{\alpha(\alpha+1) \cdots (\alpha+r-1)\theta k + \beta(\beta+1) \cdots (\beta+r-1)\eta}{\theta^r (\eta+\theta k)}.\]

\[\mu'_X(k) = \frac{w}{\theta^r} \frac{\Gamma(\alpha+r)}{\Gamma(\alpha)} + \frac{1-w}{\theta^r} \frac{\Gamma(\beta+r)}{\Gamma(\beta)}\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", lindley(mu, sigma).moments(k))
6.3563523462564525615615615614561356E+00

dist_lindley.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following an Lindley distribution. The cumulants are calculated from the moments.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", lindley(mu, sigma).cumulants(k))
6.3563523462564525615615615614561356E+00