Chapter 11
Analysis of Variance and Regression
11.1 a. The rst order Taylors series approximation is Var[g (Y )] [g ()]2 VarY = [g ()]2 v (). b. If we choose g (y ) = g (y ) =
y a
1
v (x)
dx, then
a
dg () d = d d
1 v (x)
dx =
1 v ()
,
by the Fundamental
Chapter 3
Common Families of Distributions
3.1 The pmf of X is f (x) =
1 N1 N0 +1 , N1
x = N0 , N0 + 1, . . . , N1 . Then
N1 N0 1
EX
=
x=N 0
x
1 1 = N1 N 0 +1 N1 N 0 +1
x
x=1 x=1
x
= =
1 N1 N 0 +1 N1 + N0 . 2
N 1
N1 (N 1 +1) (N 0 1)(N 0 1 + 1) 2 2
Similar
Chapter 4
Multiple Random Variables
4.1 Since the distribution is uniform, the easiest way to calculate these probabilities is as the ratio of areas, the total area being 4. a. The circle x2 + y 2 1 has area , so P (X 2 + Y 2 1) = . 4 2 b. The area below
Chapter 5
Properties of a Random Sample
5.1 Let X = # color blind people in a sample of size n. Then X binomial(n, p), where p = .01. The probability that a sample contains a color blind person is P (X > 0) = 1 P (X = 0), where P (X = 0) = n (.01)0 (.99)n
Chapter 6
Principles of Data Reduction
6.1 By the Factorization Theorem, |X | is sucient because the pdf of X is f (x| 2 ) =
2 2 2 2 1 1 ex /2 = e|x| /2 = g ( |x| 2 ) 1 . 2 2
h(x)
6.2 By the Factorization Theorem, T (X ) = mini (Xi /i) is sucient because
Chapter 7
Point Estimation
7.1 For each value of x, the MLE is the value of that maximizes f (x|). These values are in the following table. x01 2 34 1 1 2 or 3 3 3 At x = 2, f (x|2) = f (x|3) = 1/4 are both maxima, so both = 2 or = 3 are MLEs. 7.2 a. L(
Chapter 8
Hypothesis Testing
8.1 Let X = # of heads out of 1000. If the coin is fair, then X binomial(1000, 1/2). So
1000
P (X 560) =
x=560
1000 x
1 2
x
1 2
nx
.0000825,
where a computer was used to do the calculation. For this binomial, E X = 1000p = 50
Chapter 9
Interval Estimation
9.1 Denote A = cfw_x : L(x) and B = cfw_x : U (x) . Then A B = cfw_x : L(x) U (x) and 1 P cfw_A B = P cfw_L(X ) or U (X ) P cfw_L(X ) or L(X ) = 1, since L(x) U (x). Therefore, P (A B ) = P (A)+ P (B ) P (A B ) = 1 1 +1 2 1
Chapter 10
Asymptotic Evaluations
10.1 First calculate some moments for this distribution. EX = /3, E X 2 = 1/3, VarX = 1 2 . 3 9
So 3Xn is an unbiased estimator of with variance Var(3Xn ) = 9(VarX )/n = (3 2 )/n 0 as n . So by Theorem 10.1.3, 3Xn is a c
Chapter 12
Regression Models
12.1 The point ( , y ) is the closest if it lies on the vertex of the right triangle with vertices (x , y ) x and (x , a + bx ). By the Pythagorean theorem, we must have (x x ) + y (a + bx )
2 2
+ (x x ) +( y ) y
2
2
= (x x )