Exponentially Modified Gaussian (EMG) distribution

class ctx.dist_emg(n1, n2, lambda, **kwargs)

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

These functions return PDF, CDF, and ICDF of the exponentially modified Gaussian distribution with location \(a\), scale \(b > 0\), and the support interval \((-\infty,+\infty)\) :

See also: Wikipedia [1293].

dist_emg.pdf(x)

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

\[\text{pdf}_X(x) = f(x;\mu ,\sigma ,\lambda )={\frac {\lambda }{2}}e^{{\frac {\lambda }{2}}(2\mu +\lambda \sigma ^{2}-2x)}\operatorname {erfc} \left({\frac {\mu +\lambda \sigma ^{2}-x}{{\sqrt {2}}\sigma }}\right)\]

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

dist_emg.cdf(x)

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

\[\text{cdf}_X(x) = \Phi(x, \mu, \sigma) - {\tfrac{1}{2}}e^{{\frac {\lambda }{2}}(2\mu +\lambda \sigma ^{2}-2x)}\operatorname {erfc} \left({\frac {\mu +\lambda \sigma ^{2}-x}{{\sqrt {2}}\sigma }}\right)\]

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

dist_emg.sf(x)

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

\[\text{sf}_X(x) = \Phi(-x, \mu, \sigma) + {\tfrac{1}{2}}e^{{\frac {\lambda }{2}}(2\mu +\lambda \sigma ^{2}-2x)}\operatorname {erfc} \left({\frac {\mu +\lambda \sigma ^{2}-x}{{\sqrt {2}}\sigma }}\right)\]

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

dist_emg.qtf(q)

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

\[\text{qtf}_X(q) = ??\]

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

dist_emg.isf(q)

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

\[\text{isf}_X(q) = ??\]

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

dist_emg.c_x(t)

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

\[C_X(t) = \left(1 - \frac{it}{\lambda} \right)^{-1} \exp \left(i \mu t - \tfrac{1}{2} \sigma^2 t^2 \right)\]

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

dist_emg.m_x(t)

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

\[M_X(t) = \left(1 - \frac{t}{\lambda} \right)^{-1} \exp \left(\mu t + \tfrac{1}{2} \sigma^2 t^2 \right)\]

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

dist_emg.k_x(t, k=0)

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

\[K_X(t) = \log (M_X(t))\]

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

dist_emg.moments(k)

Returns the first \(j\) central moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following an exponentially modified Gaussian distribution. The moments are calculated from their definition:

\[\mu'_X(r) = E(X^r) = \int_{0}^{1} x^r \text{pdf}_X(x) \mathrm{d} x\]

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

dist_emg.cumulants(k)

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

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