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.

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?

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?

Consider a random variable X following the exponential
distribution X ~ f(x), where f(x) = ae^( -ax) for x > 0 and 0
otherwise, a > 0. Derive its moment-generating function MX(t)
and specify its domain (where it is defined or for what t does the
integral exist). Use it to compute the first four non-central
moments of X and then derive the general formula for the nth
non-central moment for any positive integer n. Also, write down the
expression...

Suppose that the moment generating function of a random variable
X is of the form MX (t) = (0.4e^t + 0.6)8 . What is the moment
generating function, MZ(t), of the random variable Z = 2X + 1?
(Hint: think of 2X as the sum two independent random variables).
Find E[X]. Find E[Z ]. Compute E[X] another way - try to recognize
the origin of MX (t) (it is from a well-known distribution)

X is a random variable with Moment Generating Function M(t) =
exp(3t + t2).
Calculate P[ X > 3 ]

Let X denote a random variable with probability density
function
a. FInd the moment generating function of X
b If Y = 2^x, find the mean E(Y)
c Show that moments E(X ^n) where n=1,4 is given by:

Suppose that a random variable X has the following
moment generating function,
M X (t) = (1 −
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Find the mean of X (b) Find the Varience of X. Please explain
steps. :) Thanks!

The moment generating function for the random variable X is
MX(t) = (e^t/ (1−t )) if |t| < 1. Find the variance of X.

Question 1: Compute the moment generating
function M(t) for a Poisson random variable.
a) Use M’(t) to compute E(X)
b) Use M’’(t) to compute Var(X)

An exponential density function of random variable ? is given
by: ??(?)={???−??(?−?),?>??, ?????????
Determine Moment Generating Function ?(?) (MGF) which is given by
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