1. Let X and Y be random variables such that E[X]= 2, E[Y] = 3, E[X2] 5. E[Y?] = 10 and E[XY] = 7. %3D %3D (a) Are X and Y independent? (b) Calculate E[(X+ Y)(X-Y)] (c) Calculate El YI 3V 11)(Y 1 2Y
1. Let X and Y be random variables such that E[X]= 2, E[Y] = 3, E[X2] 5. E[Y?] = 10 and E[XY] = 7. %3D %3D (a) Are X and Y independent? (b) Calculate E[(X+ Y)(X-Y)] (c) Calculate El YI 3V 11)(Y 1 2Y
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Question
![1. Let X and Y be random variables such that E[X]= 2, E[Y] = 3, EX2]=
5. E[Y²] = 10 and E[XY] = 7.
(a) Are X and Y independent?
(b) Calculate E[(X+Y)(X-Y)]
(c) Calculate E[(X+3Y+1)(X+2Y+4)]
2. Trinomial distribution is a special multinomial distribution whcre an ex-
perimental outcome can be any one of three categories. Let n be the
number of independent expcriments, X1 be the number of category 1
outcomes. and X2 be the number of category 2 outcomes (the num-
ber of category 3 outcomes, X3, becomes X3
(X1, X2)
of category 1 and 2 in cach experiment, respectively. Show
= n- X-X2), then
- trinomial(n, p1, p2), where pi and p2 are the probabilities
(a) X1~ binomial (n, p1)
(b) X3 ~ binomial (n, 1- Pi- P2).
binomial (n, 1 – pi - P2).
|
(c) X1X2~ binomial (n - X2, P)
1-P2
(d) X3X1~ binomial (n- X1, 2).
1-p1-P2).
1-p1
3. Random variables, X and Y, are independently distributed as N(0, o?).
Let R= VX² + Y² and C =
.
%3D
R'
(a) Derive the joint distribution function of R2 and C or R and C.
(b) Show R and C are independent.
(c) Generate two independent Uniform (0, 1) variables Uj and U2. Let
R = V-2 log U2 and 0 =
Rsin 0 are two independent N(0, 1) random variables.
2TU1. Show that X = Rcos e and Y =
4. Let the bivariate random vector(X, Y) be normally distributed as N2(u. E),
where the subscript indicate the dimemsion of this distribution, i is the
ux, lyT and E is the covariance matrix as E =
mcan vector as i =
%3D
po xoy
. Show that
po xoY](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6da60a5-ae94-432a-910a-4d6290f9da61%2Fc498d258-3ac5-414f-abe3-3472d1c6ef56%2F6r3fw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Let X and Y be random variables such that E[X]= 2, E[Y] = 3, EX2]=
5. E[Y²] = 10 and E[XY] = 7.
(a) Are X and Y independent?
(b) Calculate E[(X+Y)(X-Y)]
(c) Calculate E[(X+3Y+1)(X+2Y+4)]
2. Trinomial distribution is a special multinomial distribution whcre an ex-
perimental outcome can be any one of three categories. Let n be the
number of independent expcriments, X1 be the number of category 1
outcomes. and X2 be the number of category 2 outcomes (the num-
ber of category 3 outcomes, X3, becomes X3
(X1, X2)
of category 1 and 2 in cach experiment, respectively. Show
= n- X-X2), then
- trinomial(n, p1, p2), where pi and p2 are the probabilities
(a) X1~ binomial (n, p1)
(b) X3 ~ binomial (n, 1- Pi- P2).
binomial (n, 1 – pi - P2).
|
(c) X1X2~ binomial (n - X2, P)
1-P2
(d) X3X1~ binomial (n- X1, 2).
1-p1-P2).
1-p1
3. Random variables, X and Y, are independently distributed as N(0, o?).
Let R= VX² + Y² and C =
.
%3D
R'
(a) Derive the joint distribution function of R2 and C or R and C.
(b) Show R and C are independent.
(c) Generate two independent Uniform (0, 1) variables Uj and U2. Let
R = V-2 log U2 and 0 =
Rsin 0 are two independent N(0, 1) random variables.
2TU1. Show that X = Rcos e and Y =
4. Let the bivariate random vector(X, Y) be normally distributed as N2(u. E),
where the subscript indicate the dimemsion of this distribution, i is the
ux, lyT and E is the covariance matrix as E =
mcan vector as i =
%3D
po xoy
. Show that
po xoY
Transcribed Image Text:(a) The marginal distribtion of X or Y is a normal distribution.
(b) Find the conditional distribution of Y given X = r.
(c) Find the distribution of aX + bY, whcre a and b are constants.
5. X and Y are two independent random variables with .Y exponential()
and Y erponential (u). It is impossible to obtain direct obseravations
of X and Y. Instcad. wc obscrve the random variables Z and W. where
Z = min{X,Y}, and W = 1 if Z = X and W = 0 if Z =Y.
%D
(a) Find the joint distribution of Z and W.
(b) Prove that Z and W are independent.
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