Question

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).

Answer #1

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 )

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

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).

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)

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 two discrete random variables. The range of X
is {0, 1, 2}, while the range of Y is {1, 2, 3}. Their joint
probability mass function P(X,Y) is given in the table below:
X\Y 1
2
3
0
0
.25 0
1
.25
0
.25
2
0
.25 0
Compute E[X], V[X], E[Y], V[Y], and Cov(X, Y).

Let X be a discrete random variable that takes on the values −1,
0, and 1. If E (X) = 1/2 and Var(X) = 7/16, what is the probability
mass function of X?

Given
f(x) = (
c(x + 1) if 1 < x < 3
0 else
as a probability function for a continuous random variable;
find
a. c.
b. The moment generating function MX(t).
c. Use MX(t) to find the variance and the standard deviation of
X.

1. Let the random variable X denote the time (in hours) required
to upgrade a computer system. Assume that the probability density
function for X is given by: p(x) = Ce^-2x for 0 < x <
infinity (and p(x) = 0 otherwise).
a) Find the numerical value of C that makes this a valid
probability density function.
b) Find the probability that it will take at most 45 minutes to
upgrade a given system.
c) Use the definition of the...

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