16:960:583, Section H6
Methods of Statistical Inference
Review for Midterm Examination
These questions are taken from previous semesters midterms.
1. For i = 1, 2, 3, Xi and Yi are bivariate normal, Xi ~ N(-1, 5), Yi is a standard normal random variable,
16:960:583, Section H6
Methods of Statistical Inference
Homework from 21 July 2011
Due 26 July
Chapter 4, # 5 (Var(X) only), 44, 48, 62(b), 75, 77(b, c), 79.
A) Z has standard normal distribution; Pcfw_I = 1 = Pcfw_I = -1 = ; Z and I are
independent, and
16:960:583, Section H6
Methods of Statistical Inference
Homework from 19 July 2011
Solutions
2.63. To simplify typing here, I use X instead of . Let Y = tan X. FY(y) = Pcfw_Y y
= Pcfw_tan X y = Pcfw_X tan-1 y = 1/ (tan-1 y + /2) = + (tan-1 y) / (for
y R).
16:960:583, Section H6
Methods of Statistical Inference
Homework from 14 July 2011
Solutions
1
2.40 a) c = x 2 dx
0
0 if x < 0
3
= 3 . b) F ( x) = x if 0 x < 1 . c) 0.5 - 0.1 = 0.124.
1 if x 1
1
3.1 See the solutions in Rice. Ask in the chat room if you
16:960:583, Section H6
Methods of Statistical Inference
Homework from 14 July 2011
Due 19 July
Chapter 2, #40.
Chapter 3, # 1, 12, 17.
A) By definition, S 2 =
n
1
n 1 i =1
(
2
1 n
X n X i .
X i X . Verify that 2 i =1
i =1
S=
n 1
)
2
n
2
i
B) A biased t
16:960:583, Section H6
Methods of Statistical Inference
Homework from 12 July 2011
Solutions
1.49. a) Pcfw_HHH, HHT, HTH, THH / (1 - Pcfw_TTT) = 4/7. b) 1/7 by a similar argument.
1.57. There are six equally likely outcomes: ai, bi and bi for i = 1, 2, wh
16:960:583, Section H6
Methods of Statistical Inference
Homework from 12 July 2011
Due 19 July
Chapter 1, # 49, 57, 61, 63.
Chapter 2, # 2(b, d), 13, 14, 31.
A) The problem from class: Xi | ~ Ber(), and the Xi are conditionally independent
given . Pcfw_ =
16:960:583, Section H6
Methods of Statistical Inference
Homework from 26 July 2011
Due 2 August
Chapter 6, # 3, 4, 8.
A) X1, X2, X3,. ~ Exp() are independent. We dont know . Our model is that X1,
X2, X3,. | ~ Exp() are c.i.i.d., with ~ (, ). Show that | X
16:960:583, Section H6
Methods of Statistical Inference
Homework from 26 July 2011
Solutions
6.3. X ~ N(0, 1/16). 0.75 = Pcfw_ X < c = Pcfw_ X /4< c/4 = (c/4). From the table,
c/4 .67 (or 17 .0.67 + 14 .0.68 0.675 by linear interpolation). c 2.68 (or
31
3
Theorem. If m is a median for X, then E|X m| E|X a| for all a.
Proof. First consider the case where a > m.
Let Y = |X a| - |X m|, so that we are to show that EY is nonnegative. Let I1, I2, and
I3 respectively indicate cfw_X m, cfw_m < X a, and cfw_X > a.
16:960:583, Section H6
Methods of Statistical Inference
Homework from 9 August 2011
Solutions
A) X is an unbiased estimator, and Var( X ) = Var (X) / n = /n. I() = 1/ (from 4
August, Problem I). Thus, Var( X ) = 1 / [n I()].
B)
2
i) (n 1)S | ~ n 1 = (n 1)
16:960:583, Section H6
Methods of Statistical Inference
Homework from 9 August 2011
Due 16 August
A) Show that the sample mean is an efficient estimator of the parameter of a Poisson
distribution.
B) Consider a random sample from a normal distribution who
16:960:583, Section H6
Methods of Statistical Inference
Homework from 4 August 2011
Solutions
i ( x)
4. (c)
i ( x)
i ( x)
i ( x)
0
1
2
3
2 2(1 ) 1
, where ij(x) indicates x = j.
p ( x) =
3 3 3 3
( ) s ( x ) (1 ) t ( x ) , where s(x) = k [i0(xk) + i1
16:960:583, Section H6
Methods of Statistical Inference
Homework from 4 August 2011
Due 9 August
Chapter 8, # 4(c, e), 16(b), 21(b), 31.
(Note that the answer to 4(e) is the Bayes estimator under the 0-1 loss function.)
For all these problems, determine w
16:960:583, Section H6
Methods of Statistical Inference
Homework from 28 July 2011
Solutions
n
n
A) E[ X ] = 0 x
x 1e x
+ n 1 x
( + n) [ n ]
dx =
x
e dx =
= n.
( )
( ) 0
( ) + n
B)
2 1
i) T = c 2
(n 1) S 2
2
X ) 2 , and
~ n 1 = ( n2 1 , 1 ) . Thus,
2
16:960:583, Section H6
Methods of Statistical Inference
Homework from 28 July 2011
Due 2 August
A) Using the bracket exponential notation introduced in lecture, determine a formula for
the nth moment (where n is a nonnegative integer) of the Gamma distrib
16:960:583, Section H6, Summer 2011
Methods of Statistical Inference
Stephen J. Herschkorn, Instructor
Midterm Examination
Solutions
1. We wish to explore a Poisson distribution with unknown parameter . Initially,
Pcfw_ = 2 = 0.2 and Pcfw_ = 5 = Pcfw_ = 6
16:960:583:H6
Methods of Statistical Inference
Stephen J. Herschkorn
Homework from 2 August 2012
Reading for the next lecture: Sections 8.8.2 and 7.3.3.
Chapter 8, # 4(c, e), 16(a, b, d), 21, 31.
(Note that the answer to 4(e) is the Bayes estimator under
16:960:583:H6
Methods of Statistical Inference
Stephen J. Herschkorn
Homework from 31 July 2012
Reading for the next lecture: Sections 8.5 (before 8.5.1) and 8.8.2 (review 8.8), pp. 217220.
Chapter 4, # 41.
Chapter 5, # 3, 4, 10, 16.
A) X1, X2, and X3 are
16:960:583:H6
Methods of Statistical Inference
Stephen J. Herschkorn
Homework from 18 July 2013
Solutions
Chapter 6
4. (both a and b) Pcfw_T < t0 = 0.95. From the table, t0 1.895.
2
Z
Z2
2
6. T =
, where Z ~ N(0, 1) and X ~ n are independent. By
=
X /n
16:960:583:H6
Methods of Statistical Inference
Stephen J. Herschkorn
Homework from 18 July 2013
Chapter 6, # 4, 6, 8.
Chapter 8, # 65.
A) Given unknown , X1, X2, X3,. are conditionally independent Poisson() random
variables.
i) What is a sufficient statis
16:960:583
Methods of Statistical Inference
Stephen J. Herschkorn, Ph.D.
Homework from 16 July 2013
Solutions
Chapter 4
25. 2 [Var(X) + (EX)] = 2 (1 + ) / .
54. Cov(U, V) = ; U ,V
2
Z
2
2
= 1 + X 2 1 + Y 2
Z
Z
1 / 2
.
75. 1/(2), 5/(12), where is the
16:960:583
Methods of Statistical Inference
Stephen J. Herschkorn, Ph.D.
Homework from 16 July 2013
Chapter 4, # 25, 54, 75, 79, 88.
Chapter 8, # 71, 73.
A) Given , X is uniformly distributed on [0, ]. f(t) t-4 for t 4. Determine the
unconditional density
16:960:583
Methods of Statistical Inference
Stephen J. Herschkorn, Ph.D.
Solutions for Homework from 11 July 2013
Chapter 2, #58. f(x) = 2x for 0 < x < 1. (U ~ Beta(2, 1).)
Chapter 3
14.
a) fX(x) = e-x for x 0. fY(y) = 1/ (y + 1) for y 0. X and Y are not
16:960:583
Methods of Statistical Inference
Stephen J. Herschkorn, Ph.D.
Homework from 11 July 2013
Chapter 2, # 45, 58, 61.
Chapter 3, # 1, 14, 18*, 64.
*Add (e) to 3.18: Determine Pcfw_X + Y < 1
A) X ~ Exp(). Determine the density of X / (1 + X).
B) Det
16:960:583:H6
Methods of Statistical Inference
Stephen J. Herschkorn, Ph.D.
Solutions to Homework from 9 July 2013
Note: If a solution is omitted in these documents, consult the back of the textbook.
Chapter 1
57. 1/3 / (1/3 + 1/6) = 2/3.
A
B
C
Gold
1/3
0