Question

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)

Answer #1

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

(i) If a discrete random variable X has a moment generating
function
MX(t) = (1/2+(e^-t+e^t)/4)^2, all t
Find the probability mass function of X. (ii) Let X and Y be two
independent continuous random variables with moment generating
functions
MX(t)=1/sqrt(1-t) and MY(t)=1/(1-t)^3/2, t<1
Calculate E(X+Y)^2

The range of a discrete random variable X is {−1, 0, 1}. Let MX
(t) be the moment generating function of X, and let MX(1) = MX(2) =
0.5. Find the third moment of X, E(X^3).

The range of a discrete random variable X is {−1, 0, 1}. Let
MX(t) be the moment generating function of X, and let MX(1) = MX(2)
= 0.5. Find the third moment of X, E(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:

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

Let ? and ? be two independent random variables with moment
generating functions ?x(?) = ?t^2+2t and
?Y(?)=?3t^2+t . Determine the
moment generating function of ? = ? + 2?. If possible, state the
distribution name (and include parameter values) of the
distribution of ?.

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)

What is the standard deviation of the random variable X
associated with the following mean generating function?
MX(t)=3/3−t

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?

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