Raw Moments#
See also Wikipedia [1225]
Calculating the raw moments from the pdf#
- ctx.rawmoments_from_pdf(x, cf)#
where
ctxisdec,mpm, orgmp.The n-th moment of a real-valued continuous function f(x) of a real variable about a value c is
\[\mu _{n}=\int _{-\infty }^{\infty }(x-c)^{n}\,f(x)\,\mathrm {d} x.\]It is possible to define moments for random variables in a more general fashion than moments for real values—see moments in metric spaces. The moment of a function, without further explanation, usually refers to the above expression with c = 0.
Calculating the raw moments from the pmf#
- ctx.rawmoments_from_pmf_vector(x, cf)#
where
ctxisdec,mpm, orgmp.If \(X\) is a purely discrete random variable, then it attains values \(x_{1},x_{2},\ldots\) with probability \(p_{i}=p(x_{i})\), and the CDF of \(X\) will be discontinuous at the points \(x_{i}\):
\[\mu _{n} = \sum _{x_{i}\leq x} x_{i}^n p(x_{i}).\]
Calculating the raw moments from the factorial moments#
- ctx.rawmoments_from_factorial_moments(x, cf)#
where
ctxisdec,mpm, orgmp.Calculates the raw moments \(\mu'_r\) from the factorial moments \(\mu'_{[r]}\)
\[\mu'_r = \sum_{j=0}^r S(r,j) \mu'_{[j]},\]where \(S(r,j)\) is the Stirling number of the second kind (see stirling2()).
Calculating the raw moments from the central moments#
- ctx.rawmoments_from_centralmoments(central, raw)#
where
ctxisipm,dec,mpm, orgmp.Calculates the raw moments \(\mu_r'\) from the central moments \(\mu_r\) (see Lee and Lin [426], Rinne [505], p. 36):
\[\mu_r' = \sum_{j=0}^r \binom{r}{j} \mu_{r-j} (\mu'_1)^{r}\]>>> from mpfunlab import dec, mpr, ivr, ivc >>> ivr.dps = 25; ivr.pretty = True >>> ivr.exp([-inf,0]) [0.0, 1.0] >>> ivr.exp([0,1]) [1.0, 2.71828182845904523536028749558]
Calculating the raw moments from the cumulants#
- ctx.rawmoments_from_cumulants()#
where
ctxisipm,dec,mpm, orgmp.Calculates the raw moments \(\mu_r'\) from the cumulants \(\kappa_r\) (see Lee and Lin [426], Rinne [505], p. 36):
\[\mu_r' = \kappa_r + \sum_{j=1}^{r-1} \binom{r-1}{j-1} \mu'_{r-j} \kappa_j\]>>> from mpfunlab import dec, mpr, ivr, ivc >>> ivr.dps = 25; ivr.pretty = True >>> ivr.exp([-inf,0]) [0.0, 1.0] >>> ivr.exp([0,1]) [1.0, 2.71828182845904523536028749558]
Calculating the raw moments from the moment-generating function#
- ctx.rawmoments_from_mgf()#
where
ctxisipm,dec,mpm, orgmp.The moment-generating function is so called because if it exists on an open interval around t = 0, then it is the exponential generating function of the moments of the probability distribution:
\[m_{n}=E\left(X^{n}\right)=M_{X}^{(n)}(0)=\left.{\frac {d^{n}M_{X}}{\mathrm{d} t^{n}}}\right|_{t=0}.\]That is, with n being a nonnegative integer, the nth moment about 0 is the nth derivative of the moment generating function, evaluated at t = 0.
Calculating the raw moments from the characteristic function#
- ctx.rawmoments_from_cf()#
where
ctxisipm,dec,mpm, orgmp.Characteristic functions can also be used to find moments of a random variable. Provided that the nth moment exists, the characteristic function can be differentiated n times and
\[\operatorname {E} \left[X^{n}\right]=i^{-n}\,\varphi _{X}^{(n)}(0)=i^{-n}\,\left[{\frac {d^{n}}{\mathrm{d} t^{n}}}\varphi _{X}(t)\right]{t=0}\,\!\]
Calculating the raw moments from the probability-generating function#
- ctx.rawmoments_from_pgf()#
where
ctxisipm,dec,mpm, orgmp.The kth raw moment of X is given by
\[\operatorname {E} (X^{k})=\left(z{\frac {\partial }{\partial z}}\right)^{k}G(z){\Big |}_{z=1^{-}}\]More generally, the kth factorial moment,
\(\operatorname {E} (X(X-1)\cdots (X-k+1))\) of \(X\) is given by
\[\operatorname {E} \left({\frac {X!}{(X-k)!}}\right)=G^{(k)}(1^{-}),\quad k\geq 0.\]