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.

1) Let the random variables ? be the sum of independent Poisson
distributed random variables, i.e., ? = ∑ ? (top) ?=1(bottom) ?? ,
where ?? is Poisson distributed with mean ?? .
(a) Find the moment generating function of ?? . (b) Derive the
moment generating function of ?. (c) Hence, find the probability
mass function of ?.
2)The moment generating function of the random variable X is
given by ??(?) = exp{7(?^(?)) − 7} and that of ?...

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 ]

Let X, Y, and Z be independent and identically distributed
discrete random variables, with each having a probability
distribution that puts a mass of 1/4 on the number 0, a mass of 1/4
at 1, and a mass of 1/2 at 2.
a. Compute the moment generating function for S= X+Y+Z
b. Use the MGF from part a to compute the second moment of S,
E(S^2)
c. Compute the second moment of S in a completely different way,
by expanding...

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!

Show that if two binomial random variables X ∼ Bin(a,p) and Y ∼
Bin(b,p) are independent, then X + Y ∼ Bin(a + b, p), using the
technique of moment generating function.

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.

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