Question

**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)

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.

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)

Suppose that a random variable X has the following
moment generating function,
M X (t) = (1 −
3t)−8, t < 1/3. (a)
Find the mean of X (b) Find the Varience of X. Please explain
steps. :) Thanks!

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

(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 random variable X has moment generating function
ϕX(t)=(0.44e^t+1−0.44)^8
Provide answers to the following to two decimal places
(a) Evaluate the natural logarithm of the moment generating
function of 3X at the point t=0.4.
(b) Hence (or otherwise) find the expectation of 3X.
(c) Evaluate the natural logarithm of the moment generating
function of 3X+6 at the point t=0.4.

The random variable X has moment generating function
ϕX(t)=exp((9t)^2)/2)+15t)
Provide answers to the following to two decimal places
(a) Evaluate the natural logarithm of the moment generating
function of 2X at the point t=0.62.
(b) Hence (or otherwise) find the expectation of 2X.
c) Evaluate the natural logarithm of the moment generating
function of 2X+7 at the point t=0.62.

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:

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.

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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