Question

An exponential density function of random variable ? is given
by: ??(?)={???−??(?−?),?>??, ?????????

Determine Moment Generating Function ?(?) (MGF) which is given by
?(?)=?[???]. Use this MGF to examine and find the Variance of ?
(Hint: Find 1st and 2nd order moments first).

Answer #1

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 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)

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

Find the moment generating function of each of the following
random variables. Then, use it to find the mean and variance of the
random variable
1. Y, a discrete random variable with P(X = n) = (1-p)p^n, n
>= 0, 0 < p < 1.
2. Z, a discrete random variable with P(Z = -1) = 1/5, P(Z = 0)
= 2/5 and P(Z = 2) = 2/5.

10pts) Let Y be a continuous random variable with density
function f(y) = 1 2 e −|y| , −∞ < y < ∞ 0, elsewhere (a) Find
the moment-generating function of Y . (b) Use the moment-generating
function you find in (a) to find the V (Y ).

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...

Find the mean and the variance of the binomial distribution.
(Hint: Use the Moment-Generating Function
to make life easy for you!).

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?

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. 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

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