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HW1_Solutions
School: Wisconsin
Course: Introduction To Mathematical Statistics
HW1 Solutions Section 1.2 #4,5,6,7,12,14 4. (a) In three tosses of a coin the rst outcome is a head. (b) In three tosses of a coin the same side turns up on each toss. (c) In three tosses of a coin exactly one tail turns up. (d) In three tosses of a coin

311.midterm.sample.01
School: Wisconsin
Course: Statistics
STAT 311 Old midterm problems and answers 1. If A and B are independent events, with P (A) = 1 and P (B ) = 1 , nd the following: 3 4 (a) P (Ac B c ) Solution 1. Since A and B are independent, Ac and B c are independent. So P (Ac B c ) = P (Ac )P (B c ) =

311.midterm.actual.02.2005.solution
School: Wisconsin
Course: Statistics
STAT 311 Midterm 2 solutions, 2005 Nov 17. 2005 1. Two players A and B ip a fair coin alternately and the rst player to obtain a head wins. Suppose A ips rst. (a) Let X be the number of tosses player A is making till he wins the game. Determine the probab

311.midterm.actual.01.2005.solution
School: Wisconsin
Course: Statistics
STAT 311 2005 fall semester midterm problems and answers 1. 10% of the glass bottles coming o a production line haven serious aws in the glass. If two bottles are randomly selected, nd the mean and variance of the number of bottles that have serious aws b

311.final.sample.01
School: Wisconsin
Course: Statistics
STAT 311 Old nal exam problems Dec 13. 2005 1. Suppose that X1 , , Xn are a random sample of size n from Bernoulli distribution with P (Xi = 1) = p, 0 < p < 1. (a) Compute the correlation coecient of Y1 = X1 + X2 and Y2 = X1 + X3 . 2 2 Are Y1 and Y2 indep

311.16.2005
School: Wisconsin
Course: Statistics
Stat 311Lecture 14 Moment generating functions Dec 5, 2005 1. For random variable X , the moment generating function of X is a function of t given by MX (t) = EetX for any t c. This function can be used in determining any nth moment automatically. The

311.14.2005
School: Wisconsin
Course: Statistics
Stat 311Lecture 14 Conditional distributions Nov 10, 2005 1. The conditional probability mass function of Y given X = x is given by p(y x) = p(x, y ) . pX (x) The conditional expectation of Y given X = x is then E(Y X = x) = yp(y x). y If X and Y are

311.13.2005
School: Wisconsin
Course: Statistics
Stat 311Lecture 13 Sum of random variables Nov 3, 2005 1. For random variables X and Y and their joint distribution f (x, y ), the probability of P (X Y ) = f (x, y ) dxdy. xy If X and Y are independent U nif (0, 1), the joint distribution is a uniform r

311.12.2005
School: Wisconsin
Course: Statistics
Stat 311Lecture 12 Other Continuous Random Variables Oct 27, 2005 1. Exponential random variable. X exp(). Density function: f (x) = ex , x 0. c.d.f. F (x) = 1 exp(x), x 0. EX n = n EX n1 . 1 EX = , VX = 1 . 2 2. Problem. The density function of an expon

311.11.2005
School: Wisconsin
Course: Statistics
Stat 311Lecture 11 Central Limit Theorem Oct 24, 2005 1. If the cdfs of X and Y are identical, two random variables are identically distributed. This does not imply X = Y which is nonsense. To denote the equality of distribution, we will use notation X Y

311.midterm.sample.02
School: Wisconsin
Course: Statistics
STAT 311 Old midterm problems and answers Nov 15. 2005 1. A man with n keys wants to open his door and tries the keys at random. Exactly one key will open the door. Find the mean and the variance of the number of trials if unsuccessful keys are eliminated

HW11_Solutions
School: Wisconsin
Course: Introduction To Mathematical Statistics
HW11 Solutions Section 9.2 1. EX = 7/2, EX 2 = 91/2, V ar(X ) = 91/2 (7/2)2 = 35/12 ES24 = 7/2(24) = 84, V ar(S24 ) = 35/12(24) = 70 (a) P (S24 > 84) = P (S24 85) 85 .5 84 P (Z ) 70 P (Z .0598) .4762 (b) 1 84 84 P (S24 = 84) ( ) 70 70 1 1 = 70 2 .0

HW2_Solutions
School: Wisconsin
Course: Introduction To Mathematical Statistics
HW2 Solutions Section 2.2 #2,6,8,12 2. (a) 10 Cxdx = 1 2 10 C x2 =1 22 C (102 22 ) =1 2 48C = 1 C = 1/48 (b) b P (E ) = x/48dx a b x2 96 a 2 b a2 = 96 = (c) 102 52 = 75/96 = 25/32 96 72 22 P (X < 7) = = 45/96 = 15/32 96 P (X > 5) = P (X 2 12X + 35 > 0) =

HW3_Solutions
School: Wisconsin
Course: Introduction To Mathematical Statistics
HW3 Solutions Section 3.1 4. P (at least 2 US Presidents died on the same day) = 1 P (all died on dierent day) # nonrepeating day assignments # possible day assignments (365)(364).(365 38 + 1) = 36538 (365)38 = (365)38 .1359 P (all dierent) = P (at leas

HW4_Solutions
School: Wisconsin
Course: Introduction To Mathematical Statistics
HW4 Solutions Section 4.1 2. (a) P (2H 1st H) = P (1H in last 2 tosses) = 1/2 (b) P (2H 1st T) = P (2H in last 2 tosses) = 1/4 (c) P (2H 1st 2H) = P (T in last toss) = 1/2 (d) P (2H 1st 2T) = 0 (e) P (2H 1st H 3rd H) = P (2nd T) = 1/2 4. (a) P (heart

HW5_Solutions
School: Wisconsin
Course: Introduction To Mathematical Statistics
HW5 Solutions Section 4.2 2. P (T t) = F (t) t .1e.1x dx = 0 = e.1x = 1 e.1t (a) (2 pts) P (T 10T > 1) = 1 P (T > 10T > 1) P (T > 10 T > 1) =1 P (T > 1) P (T > 10) =1 P (T > 1) e1 = 1 .1 e = 1 e.9 .5934 Or, using the memoryless property P (T 10T > 1)

HW6_Solutions
School: Wisconsin
Course: Introduction To Mathematical Statistics
HW6 Solutions Section 5.1 7. (a) T has a geometric distribution. P (T = t) = 1 6 5 6 t1 (b) P (T > 3) = 1 P (T 3) = 1 (1/6) (5/6)(1/6) (5/6)2 (1/6) 125 .5787 = 216 (c) P (T > 6, T > 3) P (T > 3) 1 P (T 6) = P (T > 3) 1 P (T 3) P (T = 4) P (T = 5) P (T =

HW7_Solutions
School: Wisconsin
Course: Introduction To Mathematical Statistics
HW7 Solutions Section 5.2 2. (a) Y= 1 U +1 1 u+1 1 y= u+1 1y u= y 1y 1 (y ) = y (u) = Since is strictly decreasing, FY (y ) = 1 FU (1 (y ) 1y = 1 FU y 1y =1 y 0 y < 1/2 2y 1 = 1/2 y 1 y 1 otherwise fY (y ) = = 1 dF dy 1 y2 0 1/2 y 1 otherwise (b) Y = log(

HW8_Solutions
School: Wisconsin
Course: Introduction To Mathematical Statistics
HW8 Solutions Section 6.1 8. Children B GB GGB GGG Probability 1/2 1/4 1/8 1/8 E (B ) = 1/2 + 1/4 + 1/8 = 7/8 E (G) = 1/4 + 2(1/8) + 3(1/8) = 7/8 18. 1 1 1 1 1 1 7 E (K ) = 1 + 2 + 3 + 4 + 5 + 6 = 6 6 6 6 6 6 2 Section 6.2 5. (a) E (F ) = 60(1/10) + 61(2/

HW9_Solutions
School: Wisconsin
Course: Introduction To Mathematical Statistics
HW9 Solutions Section 7.1 2. pX +X = = 2 1 11 21 1 44 42 0 1 11 11 1 + 24 8 22 8 + 24 1 8 11 22 11 88 2 3 11 1 + 22 8 28 1 8 2 1 0 1 2 3 4 1/16 1/4 5/16 3/16 9/64 1/32 1/64 5. (a) pY + X = = 3 11 34 13 34 4 + 11 34 13 34 5 + 6 11 34 13 34 3 4 5 6 1/12 1/3

HW10_Solutions
School: Wisconsin
Course: Introduction To Mathematical Statistics
HW10 Solutions Section 8.1 1. P (X 50 15) 15 1 = 2 15 9 5. P (X  k ) .01 V (X ) = .01 k2 k = 10 6. P (Sn np ) np(1 p) 2 np(1 p) n2 2 p(1 p) Sn P (  p ) n n2 P (Sn np n ) 7. d p(1 p) = 1 2p = 0 dp p = 1/2 max p(1 p) = (1/2)(1 1/2) = 1/4 0<p<1

Sample Quiz Solutions
School: Wisconsin
Stat 311 Sample Quiz Solutions 1. A coin is ipped three times. Let A = cfw_rst ip is heads B = cfw_second ip is heads C = cfw_third ip is heads Express each of the following events in terms of A, B , and C . D = cfw_all three ips are tails = Ac B c C c E

Formula Cheat Sheet
School: Wisconsin
STAT 311 October 29, 2007 Discrete Distributions Distribution Probability Mass Function p(x) Binomial binomial(n, p) n x px q nx , x = 0, 1, , n Mean Variance Moment Generating Function np npq (pet + q )n Geometric (i) pq x , x = 0, 1, (i) q/p (i) q/p2 (

Example Quiz Problems
School: Wisconsin
Stat 311 Sample Quiz Problems 1. A coin is ipped three times. Let A = cfw_rst ip is heads B = cfw_second ip is heads C = cfw_third ip is heads Express each of the following events in terms of A, B , and C . D = cfw_all three ips are tails E = cfw_exactly

Week10Notes
School: Wisconsin
Course: Stat
Stat 311 Introduction to Mathematical Statistics Zhengjun Zhang Department of Statistics, University of Wisconsin, Madison, WI 53706, USA November 35, 2009 Functions of a Random Variable Theorem 5.1 Let X be a continuous random variable, and suppose that

Week08Notes
School: Wisconsin
Course: Stat
Stat 311 Introduction to Mathematical Statistics Zhengjun Zhang Department of Statistics, University of Wisconsin, Madison, WI 53706, USA October 2022, 2009 Example continue Consider a circle of radius R and chosen a point at random and denote X and Y to

Week05Notes
School: Wisconsin
Course: Stat
Stat 311 Introduction to Mathematical Statistics Zhengjun Zhang Department of Statistics, University of Wisconsin, Madison, WI 53706, USA September 29October 1, 2009 Extra Examples for Chapter 3 Example 3.6 Charles claims that he can distinguish between

Week03Notes
School: Wisconsin
Course: Stat
Stat 311 Introduction to Mathematical Statistics Zhengjun Zhang Department of Statistics, University of Wisconsin, Madison, WI 53706, USA September 1517, 2009 Mutually exclusive sets: No common element between any pair of sets. Exhaustive sets: The unio

Week09Notes
School: Wisconsin
Course: Stat
Stat 311 Introduction to Mathematical Statistics Zhengjun Zhang Department of Statistics, University of Wisconsin, Madison, WI 53706, USA October 2729, 2009 Example Let X1 , X2 , . . . , Xn be n mutually independent random variables, each of which is uni

Week07Notes
School: Wisconsin
Course: Stat
Stat 311 Introduction to Mathematical Statistics Zhengjun Zhang Department of Statistics, University of Wisconsin, Madison, WI 53706, USA October 1315, 2009 Example Suppose we roll two fair sixsided dice, one red and one blue. Let A be the event that tw

Ch12
School: Wisconsin
MILLER AND FREUND'S PROBABILITY AND STATISTICS FOR ENGINEERS Richard Johnson Department of Statistics University of WisconsinMadison 2 Contents 1 Probability Distributions 1.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . 1.2 The

311.midterm.actual.01.2005.solution
School: Wisconsin
STAT 311 2005 fall semester midterm problems and answers 1. 10% of the glass bottles coming off a production line haven serious flaws in the glass. If two bottles are randomly selected, find the mean and variance of the number of bottles that have se

311.final.sample.01
School: Wisconsin
STAT 311 Old nal exam problems Dec 13. 2005 1. Suppose that X1 , , Xn are a random sample of size n from Bernoulli distribution with P (Xi = 1) = p, 0 < p < 1. (a) Compute the correlation coecient of Y1 = X1 + X2 and Y2 = X1 + X3 . 2 2 Are Y1 and

Solution05
School: Wisconsin
Course: Stat
Solution 05 Section 4.1: 2. (a)1/2; (b)1/4; (c)1/2; (d)0; (e)1/2. 4. (a)1/2; (b)4/13; (c)1/13. 7. (a) We have 1 P (A B ) = P (A C ) = P (B C ) = , 4 1 P (A)P (B ) = P (A)P (C ) = P (B )P (C ) = , 4 1 1 P (A B C ) = = P (A)P (B )P (C ) = . 4 8 (b) We have

Example Midterm
School: Wisconsin
Name: Stat 311 Sample Midterm Examination Part I. Do all problems in the space provided. Clearly dene all random variables and other notation that you use, and clearly specify what you are calculating at each step in a calculation. Justify all answers. Cl

Confidence Interval Review
School: Wisconsin
Stat 311 Approximate condence intervals for the expectation Let X1 , . . . , Xn be independent, identically distributed random variables with expectation and variance 2 . The method of moments estimator for is just the sample mean, that is X1 + + Xn X n

Calc Pop Quiz
School: Wisconsin
Stat 311 Kurtz Calculus Review 1. Calculate the following: (a) (b) (c) 2 0 2 1 axb dx 3e3x dx 0 3xe3x dx 2. Let f (x) = 2 x(3 9 x), 0, 0x3 otherwise (a) Graph f . (b) Calculate 0 .5 3 x 2 y 3. Let f (x, y ) = 6e . Calculate R in which (x, y ) satises 0

Week04Notes
School: Wisconsin
Course: Stat
Stat 311 Introduction to Mathematical Statistics Zhengjun Zhang Department of Statistics, University of Wisconsin, Madison, WI 53706, USA September 2224, 2009 Example Suppose we have 6 cards cfw_A,K,Q,J,10,9 and we select a pair of dierent cards from the

Solution02
School: Wisconsin
Course: Stat
Solution 02 1 Section 1.2 16. [2 points] 10 per cent. An example: 10 lost eye, ear, hand, and leg; 15 eye, ear, and hand; 20 eye, ear, and leg; 25 eye, hand, and leg; 30 ear, hand, and leg. 18. [2 points] (a) The righthand side is the sum of the probabil

Solution 01
School: Wisconsin
Course: Stat
Solution 01 4. (4 points) (a) In three tosses of a coin the first outcome is a head. (b) In three tosses of a coin the same side turns up on each toss. (c) In three tosses of a coin exactly one tail turns up. (d) In three tosses of a coin at least one tai

Practice_Midterm(2)_Sol_STAT311
School: Wisconsin
Course: Stat
STAT 311 Practice Questions Q1. We are drawing balls from the urn with 2 white balls and 3 red balls with replacement. X be the number of red balls drawn in the first two draw, then X=0, 1, 2. Y can go on forever until you get the first red ball in your d

Prac311mid02
School: Wisconsin
Course: Stat
STAT311 Practice Questions II Question 1 An urn contains 2 white ball and 3 red balls. Two balls are randomly drawn from the urn with replacement. Let X be the number of red balls drawn, and let Y be the draw on which the rst red ball is drawn. (For examp

311.16.2005
School: Wisconsin
Stat 311Lecture 14 Moment generating functions Dec 5, 2005 1. For random variable X, the moment generating function of X is a function of t given by MX (t) = EetX for any t c. This function can be used in determining any nth moment automaticall