Question

# Define the nth moment of the random variable X. Dene the nth central moment of a...

Define the nth moment of the random variable X. Dene the nth central moment

of a random variable X. Finally, dene 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?

solution; from the given data,

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. please give a like, thank you

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