Moment generating function

Moment generating function#

See also Wikipedia [1226].

Calculating the moment-generating function from the pdf#

ctx.mgf_from_pdf(x, cf)#

where ctx is dec, mpm, or gmp.

The moment-generating function’s definition expands to

\[M_{X}(t)=\operatorname {E} \left[e^{tX}\right]=\int _{-\infty }^{\infty }e^{tx}f_{X}(x)\,\mathrm{d} x\]

Calculating the moment-generating function from the characteristic function#

ctx.mgf_from_cf(x, cf)#

where ctx is dec, mpm, or gmp.

The characteristic function \(\varphi _{X}(t)\) is related to the moment-generating function via \(\varphi _{X}(t)=M_{iX}(t)=M_{X}(it)\): the characteristic function is the moment-generating function of \(iX\) or the moment generating function of \(X\) evaluated on the imaginary axis. This function can also be viewed as the Fourier transform of the probability density function, which can therefore be deduced from it by inverse Fourier transform.

Calculating the moment-generating function from the cumulant-generating function#

ctx.mgf_from_cgf(x, cf)#

where ctx is dec, mpm, or gmp.

The cumulant-generating function is defined as the logarithm of the moment-generating function; some instead define the cumulant-generating function as the logarithm of the characteristic function, while others call this latter the second cumulant-generating function.

Calculating the moment-generating function from the probability-generating function#

ctx.mgf_from_pgf(x, cf)#

where ctx is dec, mpm, or gmp.

The probability-generating function is defined as \(G(z)=E\left[z^{X}\right]\). This immediately implies that \(G(e^{t})=E\left[e^{tX}\right]=M_{X}(t).\)

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

Calculating the moment-generating function from the raw moments#

ctx.mgf_from_rawmoments(x, cf)#

where ctx is dec, mpm, or gmp.

\[M_{X}(t) = 1+tm_{1}+{\frac {t^{2}m_{2}}{2!}}+{\frac {t^{3}m_{3}}{3!}}+\cdots +{\frac {t^{n}m_{n}}{n!}}+\cdots\]

where \(m_{n}\) is the nth moment. Differentiating \(M_{X}(t) i\) times with respect to \(t\) and setting \(t = 0\), we obtain the ith moment about the origin.

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