5.3. Characteristic Function

from sympy import symbols, exp, diff, simplify, I
w = symbols('w')

def raw_moment(cf, k):
    return simplify((-I)**k * diff(cf, w, k).subs(dict(w=0)))

def central_moment(cf, k):
    m = raw_moment(cf, 1)
    return simplify(raw_moment(cf/exp(I*w*m),k))

n1, t1 = symbols('n1, t1')
cf1 = exp(n1*(1/(1 - I*w*t1) - 1))
cf1

\[\displaystyle e^{n_{1} \left(-1 + \frac{1}{- i t_{1} w + 1}\right)}\]

raw_moment(cf1, 1)

\[\displaystyle n_{1} t_{1}\]

central_moment(cf1, 2)

\[\displaystyle 2 n_{1} t_{1}^{2}\]

central_moment(cf1, 3)

\[\displaystyle 6 n_{1} t_{1}^{3}\]

n1, t1, n2, t2, t0, a, b = symbols('n1, t1, n2, t2, t0, a, b')
cf2 = exp(a*n1*(1/(1 - I*w*t1) - 1) + b*n2*(1/(1 - I*w*t2) - 1) + I*w*t0)
cf2

\[\displaystyle e^{a n_{1} \left(-1 + \frac{1}{- i t_{1} w + 1}\right) + b n_{2} \left(-1 + \frac{1}{- i t_{2} w + 1}\right) + i t_{0} w}\]

raw_moment(cf2, 1)

\[\displaystyle a n_{1} t_{1} + b n_{2} t_{2} + t_{0}\]

central_moment(cf2, 2)

\[\displaystyle 2 a n_{1} t_{1}^{2} + 2 b n_{2} t_{2}^{2}\]