Boost: Exponential distribution

The following functions return the pdf, cdf, qtf or boost class of the distribution with rate parameter \(\lambda_1 > 0\) and the support interval \((0, +\infty)\).

See also Wikipedia [1242], MathWorld [892], BoostMath [60], Ehrhardt [309] (3.9.7).

Ctx.exponential_pdf(x, lambda1)

where Ctx is Math53 or CtxBoost.

Returns \(\text{pdf}(x)\), the value of the probability density function (Pdf) of the exponential distribution:

\[\text{pdf}(x) = \lambda_1 \exp(-\lambda_1 x).\]

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("ExponentialPdf(x, a, b): ", ExponentialPdf(x, a, b))
>>> print ("dist_exponential(a, b).pdf(x): ", dist_exponential(a, b).pdf(x))
6.3563523462564525615615615614561356E+00

Ctx.exponential_cdf(x, lambda1)

where Ctx is Math53 or CtxBoost.

Returns \(\text{cdf}(x)\), the value of the cumulative distribution function (Cdf) of the exponential distribution:

\[\text{cdf}(x) = 1 - \exp(-\lambda_1 x) = -\text{expm1}(-\lambda_1 x).\]

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("ExponentialCdf(x, a, b): ", ExponentialCdf(x, a, b))
>>> print ("dist_exponential(a, b).cdf(x): ", dist_exponential(a, b).cdf(x))
6.3563523462564525615615615614561356E+00

Ctx.exponential_qtf(q, lambda1)

where Ctx is Math53 or CtxBoost.

Returns \(\text{qtf}(q)\), the value of the quantile function (Qtf) of the exponential distribution:

\[\text{qtf}(q) = - \text{log1p}(-q)/\lambda_1.\]

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; q = 0.6;
>>> print ("ExponentialQtf(q, a, b): ", ExponentialQtf(q, a, b))
>>> print ("dist_exponential(a, b).qtf(q): ", dist_exponential(a, b).qtf(q))
6.3563523462564525615615615614561356E+00

class ctx.dist_exponential(lambda1)

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

The exponential distribution is a continuous probability distribution with rate parameter \(\lambda_1 > 0\), and the support interval \([0, +\infty)\). See also Wikipedia [1242], MathWorld [892], BoostMath [60], Witkovský [1616], R (Statistical System) [548].

dist_exponential.pdf(x)

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

\[\text{pdf}_X(x) = \lambda_1 \exp(-\lambda_1 x).\]

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

dist_exponential.cdf(x)

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

\[\text{cdf}_X(x) = 1 - \exp(-\lambda_1 x) = -\text{expm1}(-\lambda_1 x).\]

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

dist_exponential.sf(x)

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

\[\text{sf}_X(x) = \exp(-\lambda_1 x).\]

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

dist_exponential.qtf(q)

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

\[\text{qtf}_X(q) = - \text{log1p}(-q)/\lambda_1.\]

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

dist_exponential.isf(q)

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

\[\text{isf}_X(q) = - \text{log}(q)/\lambda_1.\]

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

dist_exponential.c_x(t)

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

\[C_X(t) = \frac{\lambda}{\lambda - i t}.\]

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

dist_exponential.m_x(t)

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

\[M_X(t) = \frac{\lambda}{\lambda - t}, \quad \text{for } t < \lambda.\]

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

dist_exponential.k_x(t, k=0)

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

\[K_X(t) = \log \left( \frac{\lambda}{\lambda - t} \right) , \quad \text{for } t < \lambda.\]

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

dist_exponential.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following an exponential distribution. The rth moments only exists for \(n_2 > 2r\).

\[\mu'_{X}(n) = \frac{n!}{\lambda^n}.\]

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

dist_exponential.cumulants(k)

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

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