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 )

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.

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!

Use the moment generating function Mx(t) to find the mean u and
variance o^2. Do not find the infinite series.
Mx(t) = e^[5*((e^t)-1)]

Poisson Distribution: p(x,
λ) = λx exp(-λ)
/x! , x = 0, 1, 2, …..
Find the moment generating function Mx(t)
Find E(X) using the moment generating function
2. If X1 , X2 ,
X3 are independent and have means 4, 9, and
3, and variencesn3, 7, and 5. Given that Y =
2X1 - 3X2 +
4X3. find the
mean of Y
variance of Y.
3. A safety engineer claims that 2 in 12 automobile accidents
are due to driver fatigue. Using the formula for Binomial
Distribution find the...

Consider a discrete random variable X with probability mass
function P(X = x) = p(x) = C/3^x, x = 2, 3, 4, . . . a. Find the
value of C. b. Find the moment generating function MX(t). c. Use
your answer from a. to find the mean E[X]. d. If Y = 3X + 5, find
the moment generating function MY (t).

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