Question

Define the nth moment of the random variable X. Define the nth central moment of a random variable X. Finally, define the moment generating function, M(t). Write down a few terms of the series expansion of a general M(t). Why is the series expansion relevant in terms of calculating moments

Answer 1

nth moment: The nth moment of a random variable X is defined as

nth Central moment: The nth central moment of a random variable X is defined as

Moment generating function: The moment generating function(MGF) M(t) of a random variable X is defined as:

M(t) exist if there exist a positive constant 'a' such that

M(t) is finite for all t € [-a , a].

Expansion of M(t):

we can see in the expansion of mgf coefficients of is the rth moment of random variable X that's why series expansion of MGF is relevant in terms of calculating moments.