Question

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 expected value of X, to find the average time required to upgrade a computer system.

d) Find the moment generating function mx(t) for X, where t < 2.

e) Use your answer to part d) to verify your answer to part c)

Answer #1

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:

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 )

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).

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.

Let random variable X denote the time (in years) it takes to
develop a software. Suppose that X has the following probability
density function: f(x)= 5??4 if 0 ≤ x ≤ 1, and 0 otherwise. b.
Write the CDF of the time it takes to develop a software. d.
Compute the variance of the number of years it takes to develop a
software.

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

let X be a random variable that denotes the life (or time to
failure) in hours of a certain electronic device. Its probability
density function is given by
f(x){ 0.1 e−0.1x, x > 0 , 0 , elsewhere
(a) What is the mean lifetime of this type of device?
(b) Find the variance of the lifetime of this device.
(c) Find the expected value of X2 − 20X + 100.

(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)

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 1 minute ago

asked 10 minutes ago

asked 20 minutes ago

asked 21 minutes ago

asked 24 minutes ago

asked 37 minutes ago

asked 47 minutes ago

asked 52 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago