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STAT 311 - Wisconsin Study Resources
  • 3 Pages HW1_Solutions
    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

  • 1 Page 311.midterm.sample.01
    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 ) =

  • 1 Page 311.midterm.actual.02.2005.solution
    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

  • 1 Page 311.midterm.actual.01.2005.solution
    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

  • 1 Page 311.final.sample.01
    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

  • 1 Page 311.16.2005
    311.16.2005

    School: Wisconsin

    Course: Statistics

    Stat 311-Lecture 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 n-th moment automatically. The

  • 1 Page 311.14.2005
    311.14.2005

    School: Wisconsin

    Course: Statistics

    Stat 311-Lecture 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

  • 1 Page 311.13.2005
    311.13.2005

    School: Wisconsin

    Course: Statistics

    Stat 311-Lecture 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

  • 1 Page 311.12.2005
    311.12.2005

    School: Wisconsin

    Course: Statistics

    Stat 311-Lecture 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

  • 1 Page 311.11.2005
    311.11.2005

    School: Wisconsin

    Course: Statistics

    Stat 311-Lecture 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

  • 1 Page 311.midterm.sample.02
    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

  • 5 Pages HW11_Solutions
    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

  • 7 Pages HW2_Solutions
    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) =

  • 5 Pages HW3_Solutions
    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) # non-repeating day assignments # possible day assignments (365)(364).(365 38 + 1) = 36538 (365)38 = (365)38 .1359 P (all dierent) = P (at leas

  • 4 Pages HW4_Solutions
    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

  • 8 Pages HW5_Solutions
    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 10|T > 1) = 1 P (T > 10|T > 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 10|T > 1)

  • 5 Pages HW6_Solutions
    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 =

  • 8 Pages HW7_Solutions
    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(

  • 7 Pages HW8_Solutions
    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/

  • 7 Pages HW9_Solutions
    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

  • 7 Pages HW10_Solutions
    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

  • 3 Pages Sample Quiz Solutions
    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

  • 2 Pages Formula Cheat Sheet
    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 (

  • 2 Pages Example Quiz Problems
    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

  • 6 Pages Week10Notes
    Week10Notes

    School: Wisconsin

    Course: Stat

    Stat 311 Introduction to Mathematical Statistics Zhengjun Zhang Department of Statistics, University of Wisconsin, Madison, WI 53706, USA November 3-5, 2009 Functions of a Random Variable Theorem 5.1 Let X be a continuous random variable, and suppose that

  • 8 Pages Week08Notes
    Week08Notes

    School: Wisconsin

    Course: Stat

    Stat 311 Introduction to Mathematical Statistics Zhengjun Zhang Department of Statistics, University of Wisconsin, Madison, WI 53706, USA October 20-22, 2009 Example continue Consider a circle of radius R and chosen a point at random and denote X and Y to

  • 4 Pages Week05Notes
    Week05Notes

    School: Wisconsin

    Course: Stat

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

  • 9 Pages Week03Notes
    Week03Notes

    School: Wisconsin

    Course: Stat

    Stat 311 Introduction to Mathematical Statistics Zhengjun Zhang Department of Statistics, University of Wisconsin, Madison, WI 53706, USA September 15-17, 2009 Mutually exclusive sets: No common element between any pair of sets. Exhaustive sets: The unio

  • 9 Pages Week09Notes
    Week09Notes

    School: Wisconsin

    Course: Stat

    Stat 311 Introduction to Mathematical Statistics Zhengjun Zhang Department of Statistics, University of Wisconsin, Madison, WI 53706, USA October 27-29, 2009 Example Let X1 , X2 , . . . , Xn be n mutually independent random variables, each of which is uni

  • 9 Pages Week07Notes
    Week07Notes

    School: Wisconsin

    Course: Stat

    Stat 311 Introduction to Mathematical Statistics Zhengjun Zhang Department of Statistics, University of Wisconsin, Madison, WI 53706, USA October 13-15, 2009 Example Suppose we roll two fair six-sided dice, one red and one blue. Let A be the event that tw

  • 70 Pages Ch1-2
    Ch1-2

    School: Wisconsin

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

  • 1 Page 311.midterm.actual.01.2005.solution
    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

  • 1 Page 311.final.sample.01
    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

  • 12 Pages old-quiz1
    Old-quiz1

    School: Wisconsin

    Course: Stat

  • 1 Page solution05
    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

  • 7 Pages Example Midterm
    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

  • 2 Pages Confidence Interval Review
    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

  • 1 Page Calc Pop Quiz
    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

  • 6 Pages Week04Notes
    Week04Notes

    School: Wisconsin

    Course: Stat

    Stat 311 Introduction to Mathematical Statistics Zhengjun Zhang Department of Statistics, University of Wisconsin, Madison, WI 53706, USA September 22-24, 2009 Example Suppose we have 6 cards cfw_A,K,Q,J,10,9 and we select a pair of dierent cards from the

  • 1 Page solution03
    Solution03

    School: Wisconsin

    Course: Stat

  • 1 Page solution02
    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 right-hand side is the sum of the probabil

  • 3 Pages solution 01
    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

  • 6 Pages Practice_Midterm(2)_Sol_STAT311
    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

  • 2 Pages prac311mid02
    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

  • 1 Page 311.16.2005
    311.16.2005

    School: Wisconsin

    Stat 311-Lecture 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 n-th moment automaticall

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