Question
In: Statistics and Probability
Use moment generating functions to decide whether or not the given random variables X and Y...
Use moment generating functions to decide whether or not the given random variables X and Y are equal in distribution.
a). The random variables Z1, Z2, Z3 are independent normal N(0,1), X = Z1 + Z2 + Z3 and Y = √3Z1
b). The random variables Z1, Z2, Z3 are independent Poisson with the same parameterλ, X = 3Z1 and Y=Z1 + Z2 + Z3
c). The random variables Z1, Z2 are independent normal N(0,1), X = Z1 + 2Z2 and Y= 2Z1 − Z2.
Solutions
orchestra answered 2 years agoRelated Solutions
1) Let X and Y be independent random variables with moment generator functions given by MX...
1) Let X and Y be independent random variables with moment
generator functions given by MX (t) = (0, 4e t + 0, 6) ^8 and MY
(t) = (0, 4e t + 0, 6) ^4 .
a) Determine a moment generating function of Z = 3Y + 2.
b) Determine a moment generating function of W = X + Y. What is
the distribution of W?
c) Find E [W] and Var [W].
2) The annual profits that companies...
Two random variable X; Y has the following joint moment generating function MX;Y (s; t) =...
Two random variable X; Y has the following joint moment
generating function
MX;Y (s; t) = 0:5 + 0:1et +
0:1es + 0:1es+t + 0:15e2s +
0:05e2s+t
Find the probability of Y = 1 given X < 2
Hint:
> Find the PGF
> Deduce the joint probability mass function PMF
fX,Y(x,y)
>Plot the PMF on a 3-D plane x, y and z =
fX,Y(x,y) or alternatively use Cartesian (x-y) plane
with fX,Y(x,y) plots
>Use conditional Probability rues to obtain...
Given below is a bivariate distribution for the random variables x and y. f(x, y) x...
Given below is a bivariate distribution for the random variables
x and y.
f(x,
y)
x
y
0.3
50
80
0.2
30
50
0.5
40
60
(a)
Compute the expected value and the variance for x and
y.
E(x)
=
E(y)
=
Var(x)
=
Var(y)
=
(b)
Develop a probability distribution for
x + y.
x + y
f(x +
y)
130
80
100
(c)
Using the result of part (b), compute
E(x +
y)
and
Var(x +
y).
E(x...
If the joint probability density function of the random variables X and Y is given by...
If the joint probability density function of the random
variables X and Y is given by f(x, y) = (1/4)(x + 2y) for 0 < x
< 2, 0 < y < 1, 0 elsewhere
(a) Find the conditional density of Y given X = x, and use it to
evaluate P (X + Y/2 ≥ 1 | X = 1/2)
(b) Find the conditional mean and the conditional variance of Y
given X = 1/2
(c) Find the variance...
(a) Suppose that Y is a random variable with moment generating function H(t). Suppose further that...
(a) Suppose that Y is a random variable with moment generating
function H(t). Suppose further that X is a random variable with
moment generating function M(t) given by M(t) = 1/3 (2e ^(3t) +
1)H(t). Given that the mean of Y is 10 and the variance of Y is 12,
then determine the mean and variance of X (Use H(0) = 1).
(b) Suppose that the Moment generating function for X is M(t) =
e^t/( 3 − 2e^t) . Find...
(a) Suppose that Y is a random variable with moment generating function H(t). Suppose further that...
(a) Suppose that Y is a random variable with moment generating
function H(t). Suppose further that X is a random variable with
moment generating function M(t) given by M(t) = 1 3 (2e 3t +
1)H(t). Given that the mean of Y is 10 and the variance of Y is 12,
then determine the mean and variance of X (Use H(0) = 1).
(b) Suppose that the Moment generating function for X is M(t) =
e t 3 − 2e...
2 .Define the probability density functions (pdfs) of the independent random variables X and Y as:...
2 .Define the probability density functions (pdfs) of the
independent random variables X and Y as:
fX(x)={ (1/5).(1-(x/10)), 0≤x≤10 fY(y)={
(1/5).(y/10) , 0≤y≤10
0 , otherwise for both fX(x) and fY(y)
(a) Suppose that you are playing a game where you want to maximize
your number of points , and you can have your score be X or Y .
Which would you choose?
(b) Suppose I am confused in part (a), so I flip an unfair coin,
with P...
X and Y are discrete random variables with joint distribution given below Y 1 Y 0...
X and Y are discrete random variables with joint distribution
given below Y 1 Y 0 Y 1 X 1 0 1/4 0
X 0 1/4 1/4 1/4
(a) Determine the conditional expectation E Y|X 1 . (b)
Determine the conditional expectation E X|Y y for each value of y.
(c) Determine the expected value of X using conditional expectation
results form part (b)
above. (d) Now obatin the marginal distribution of X and verify
your answers to part (c).
Given two random variables x and y State of Nature Probability variable x variable y I...
Given two random variables x and y
State of Nature Probability variable x
variable y
I 0.2 18 0
II 0. 2 5 -3
III 0.2 12 15
IV 0.2 4 12
V 0.2 6 1
(i) Calculate the mean and variance of each of these variables
and the covariance between them
(ii) Suppose x and y represent the returns from two assets.
Calculate the mean and variance for the following part folios.
% in x 125 100 75 50...
The joint probability density function for two random variables X and Y is given as, fx,y...
The joint probability density function for two random variables
X and Y is given as, fx,y (x, y) = (2/3)(1 + 2xy3 ), 0
< x < 1, 0 < y < 1
(a) Find the marginal probability density functions for X and Y
.
(b) Are X and Y independent? Justify your answer.
(c) Show that E[X] = 4/9 and E[Y ] = 7/15 .
(d) Calculate Cov(X, Y )
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