Question

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:

Answer #1

An exponential density function of random variable ? is given
by: ??(?)={???−??(?−?),?>??, ?????????
Determine Moment Generating Function ?(?) (MGF) which is given by
?(?)=?[???]. Use this MGF to examine and find the Variance of ?
(Hint: Find 1st and 2nd order moments first).

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...

Let X be a continuous random variable with a probability density function
fX (x) = 2xI (0,1) (x) and let it be the function´
Y (x) = e^−x
a. Find the expression for the probability density function fY (y).
b. Find the domain of the probability density function fY (y).

10pts) Let Y be a continuous random variable with density
function f(y) = 1 2 e −|y| , −∞ < y < ∞ 0, elsewhere (a) Find
the moment-generating function of Y . (b) Use the moment-generating
function you find in (a) to find the V (Y ).

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).

Let X be a random variable with probability density function
f(x) = {3/10x(3-x) if 0<=x<=2
.........{0 otherwise
a) Find the standard deviation of X to four decimal
places.
b) Find the mean of X to four decimal places.
c) Let y=x2 find the probability density function
fy of Y.

(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)

Y is a continuous random variable with a probability
density function f(y)=a+by and 0<y<1. Given E(Y^2)=1/6,
Find:
i) a and b.
ii) the moment generating function of Y. M(t)=?

Find the moment generating function of each of the following
random variables. Then, use it to find the mean and variance of the
random variable
1. Y, a discrete random variable with P(X = n) = (1-p)p^n, n
>= 0, 0 < p < 1.
2. Z, a discrete random variable with P(Z = -1) = 1/5, P(Z = 0)
= 2/5 and P(Z = 2) = 2/5.

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