Birnbaum-Saunders Distribution

class ctx.dist_birnb_saunders(mu, sigma)

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

The Birnbaum-Saunders distribution is a continuous probability distribution with mean \(\mu \in \mathbb{R}\), standard deviation \(\sigma > 0\), and the support interval \((-\infty, +\infty)\). See also Wikipedia [1266], MathWorld [253], Balakrishnan and Kundi [30].

dist_birnb_saunders.pdf(x)

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

\[\text{pdf}_X(x) = \frac{1}{2 \sqrt{2\pi} \alpha \beta} \left[ \sqrt{\frac{t}{\beta}} + \sqrt{\frac{\beta^3}{t^3}} \right] \exp \left[ -\frac{1}{2 \alpha^2 } \left( \frac{t}{\beta} + \frac{\beta}{t}-2 \right)\right].\]

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

dist_birnb_saunders.cdf(x)

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

\[\text{cdf}_X(x) = \Phi \left[ \frac{1}{\alpha} \left( \sqrt{\frac{t}{\beta}} - \sqrt{\frac{\beta}{t}} \: \right) \right].\]

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

dist_birnb_saunders.sf(x)

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

\[\text{sf}_X(x) = 1-\Phi \left[ \frac{1}{\alpha} \left( \sqrt{\frac{t}{\beta}} - \sqrt{\frac{\beta}{t}} \: \right) \right].\]

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

dist_birnb_saunders.qtf(q)

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

\[\text{qtf}_X(q) = \frac{\beta}{4} \left[ \alpha Z + \sqrt{(\alpha Z)^2 + 4} \right]^2, \quad Z = \Phi^{-1}(q).\]

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

dist_birnb_saunders.isf(q)

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

\[\text{isf}_X(q) = \frac{\beta}{4} \left[ \alpha Z + \sqrt{(\alpha Z)^2 + 4} \right]^2, \quad Z = \Phi^{-1}(1-q).\]

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

dist_birnb_saunders.c_x(t)

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

\[C_X(t) = \frac{1}{2} \exp \left[ \frac{\beta}{\alpha} - \sqrt{ \frac{\beta^2}{\alpha^2} -2 i t \beta } \: \right] \left( 1 + \sqrt{\frac{1}{1 - 2 i t \alpha^2/\beta}} \: \right).\]

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

dist_birnb_saunders.m_x(t)

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

\[M_X(t) = \frac{1}{2} \exp \left[ \frac{\beta}{\alpha} - \sqrt{ \frac{\beta^2}{\alpha^2} -2 t \beta } \: \right] \left( 1 + \sqrt{\frac{1}{1 - 2 t \alpha^2/\beta}} \: \right).\]

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

dist_birnb_saunders.k_x(t, k=0)

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

\[K_X(t) = \frac{\beta}{\alpha} - \sqrt{ \frac{\beta^2}{\alpha^2} -2 t \beta } + \log \left( 1 + \sqrt{\frac{1}{1 - 2 t \alpha^2/\beta}} \: \right) - \log(2).\]

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

dist_birnb_saunders.moments(k)

Returns the first \(j\) central moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following an Birnbaum-Saunders distribution (Wikipedia). The raw moments are calculated from the central moments.

\[\mu_{X}(r) = \beta^r \sum_{j=0}^{r} \binom{2r}{2j} \sum_{i=0}^{j} \binom{i}{j} \frac{(2r-2j+2i)!}{2^{r-j+i}(r-j+i)!}.\]

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

dist_birnb_saunders.cumulants(k)

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

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