Question

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 ?.

Answer #1

(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

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)

Independent random variables X and Y follow binomial
distributions with parameters(n1,θ) and (n2,θ). Let Z =X+Y. What
will be the distribution of Z?
Hint: Use moment generating function.

Let X and Y be independent random variables with density functions given by fX (x) = 1/2, −1 ≤ x ≤ 1 and fY (y) = 1/2, 3 ≤ y ≤ 5. Find the density function of X-Y.

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

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!

Given a random permutation of the elements of the set
{a,b,c,d,e}, let X equal the number of elements that are in their
original position (as listed). The moment generating function is X
is: M(t) = 44/120 + 45/120e^t + 20/120e^2t + 10/120e^3t+1/120e^5t
Explain Why there is not (e^4t) term in the moment generating
function of X ?

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 Mx(t) be a moment generating function. Let
Sx (t) = [Mx (t)]2− Mx
(t). Prove that S ′x(0) = µX.

Let X and Y be independent random variables following Poisson
distributions, each with parameter λ = 1. Show that the
distribution of Z = X + Y is Poisson with parameter λ = 2. using
convolution formula

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