Chapter 2: Probability
2.1 A = cfw_FF, B = cfw_MM, C = cfw_MF, FM, MM. Then, AB = 0 , BC = cfw_MM, C B = / cfw_MF, FM, A B =cfw_FF,MM, A C = S, B C = C. a . A B b. A B c. A B d. ( A B ) ( A B )
2.2 2.3
2.4
a.
b.
8
Chapter 2: Probability
9
Instructors Solu
Chapter 3: Discrete Random Variables and Their Probability Distributions
35
Instructors Solutions Manual
3.33
a. E ( aY + b ) = E ( aY ) + E (b ) = aE (Y ) + b = a + b. b. V ( aY + b ) = E[( aY + b a b ) 2 ] = E[( aY a ) 2 ] = a 2 E[(Y ) 2 ] = a 2 2 .
3.3
Chapter 4: Continuous Variables and Their Probability Distributions
0.0
0.2
0.4
F(y)
0.6
0.8
1.0
4.1
y <1
0
.4 1 y < 2
a. F ( y ) = P(Y y ) = .7 2 y < 3
.9 3 y < 4
1
y4
0
1
2
b. The graph is above.
4.2
3
4
5
y
a. p(1) = .2, p(2) = (1/4)4/5 = .2, p(3) = (1
Solutions to HW1: MTH 540 2.5 a. ( A B ) ( A B ) = A ( B B ) = A S = A . b. B ( A B ) = ( B A) ( B B ) = ( B A) = A . c. ( A B ) ( A B ) = A ( B B ) = 0 . The result follows from part a. / / d. B ( A B ) = A ( B B ) = 0 . The result follows from part b. 2
Chapter 4: Continuous Variables and Their Probability Distributions
y <1 0 .4 1 y < 2 a. F ( y ) = P(Y y ) = .7 2 y < 3 .9 3 y < 4 1 y4
1.0 F(y) 0.0 0 0.2 0.4 0.6 0.8
4.1
1
2 y
3
4
5
b. The graph is above. 4.2
a. p(1) = .2, p(2) = (1/4)4/5 = .2, p(3) = (
Chapter 5: Multivariate Probability Distributions
97
Instructors Solutions Manual
1
5.23
3
32
a. f 2 ( y 2 ) = 3 y1 dy1 = 2 2 y 2 , 0 y 2 1 .
y2
b. Defined over y2 y1 1, with the constant y2 0.
y1
2
c. First, we have f 1 ( y1 ) = 3 y1 dy 2 = 3 y 2 , 0 y1
Chapter 5: Multivariate Probability Distributions
5.1
a. The sample space S gives the possible values for Y1 and Y2:
S
AA
AB
AC
BA
BB
BC
CA
CB
CC
(y1, y2) (2, 0) (1, 1) (1, 0) (1, 1) (0, 2) (1, 0) (1, 0) (0, 1) (0, 0)
Since each sample point is equally li
Chapter 6: Functions of Random Variables
6.1 The distribution function of Y is FY ( y ) = 2(1 t )dt = 2 y y 2 , 0 y 1.
0 y
a. FU1 (u ) = P(U 1 u ) = P( 2Y 1 u ) = P(Y
u +1 2
+ + + ) = FY ( u2 1 ) = 2( u21 ) ( u2 1 ) 2 . Thus,
fU1 (u ) = FU1 (u ) = 1u , 1
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Chapter 5: Multivariate Probability Distributions
Instructors Solutions Manual
5.67
5.68
5.69
P( a < Y1 b, c < Y2 d ) = F (b, d ) F (b, c ) F ( a , d ) + F ( a , c )
= F1 (b ) F2 ( d ) F1 (b ) F2 (c ) F1 ( a ) F2 ( d ) + F1 ( a ) F2 ( c )
= F1 (b )[F2
MSPE PROGRAM
ECON 506
FALL 2006
Solution to part C of HW 7
Ali Toossi
443348339
Theoretical (teta)
Theoretical STD of teta_hat
E(teta_hat)=
E(teta_hat)-teta=
VAR(teta_hat)=
MSE=
MEAN
MEAN
3.9
3.9
0.6245
3.9100
3.7700
0.0100
-0.1300
0.3942
0.6789
0.3943
0.
Economics 506-Sections M1 & M2
Homework 6
Due on November 27 2012, at the beginning of the class
A. Make an excel file. The purpose of this exercise is to generate 100 samples each with
10 items. Then find a number of different sample statistics and find
Chapter 2: Probability
2.1 A = cfw_FF, B = cfw_MM, C = cfw_MF, FM, MM. Then, AB = 0 , BC = cfw_MM, C B = / cfw_MF, FM, A B =cfw_FF,MM, A C = S, B C = C. a . A B b. A B c. A B d. ( A B ) ( A B )
2.2 2.3
2.4
a.
b.
8
Chapter 2: Probability
9
Instructors Solu