# 1. Let A, B and C are three events such that P(A) = 0.45, P(B) = 0.30, P(C) = 0.35,

Save this PDF as:

Size: px
Start display at page:

Download "1. Let A, B and C are three events such that P(A) = 0.45, P(B) = 0.30, P(C) = 0.35,"

## Transcription

1 1. Let A, B and C are three events such that PA =.4, PB =.3, PC =.3, P A B =.6, P A C =.6, P B C =., P A B C =.7. a Compute P A B, P A C, P B C. b Compute P A B C. c Compute the probability that exactly one of A, B and C happens. Solution. a P A B = P A + P B P A B =.1, P A C = P A + P C P A C =., P B C = P B + P C P B C =.1. b Since P A B C = P A + P B + P C P A B P A C P B C + P A B C we have P A B C = P A B C P A P B P C+P A B+P A C+P B C =.1. c We have P A B C = P A B P A B C =., P A B C = P A C P A B C =.1, P A B C = P B C P A B C =.. Now P A B C = P A P A B C P A B C P A B C =.. P A B C = P B P A B C P A B C P A B C =.1. P A B C = P C P A B C P A B C P A B C =.. So the answer is =.4.. You are dealt a -card hand from a well-shuffled deck of. a What is the probability of the union of the event that both cards are aces with the event that both cards are redhearts or diamons? b That is the probability that both cards have the same denomination e.g. both are s or both are jacks? c What is the conditional probability that the second card is a picture card J, Q, or K given that at least one of the two cards is a picture card? Solution. a Let A be the event that both cards are aces, and R be the event that both 4 6 cards are red. Then P A = since there are 4 aces. Likewise P R = 1. Next

2 P AR = 1 since there is only one good combination A, A. Hence the answer is b After the first card is drawn there are 1 cards left in the desk and 3 cards have the same denomination as the first card. Hence the answer is 3. 1 c Let B 1, B be the events that the first respectively, second card is a picture card. There are 1= 3 4 picture cards. Hence 1 P B 1 = P B = 1.3 and P B 1B =.. Thus if B is an event that at least one card is a picture card, then Hence P B 1 B = P B 1 B P B = P B 1 P B.6. P B = P B 1 + P B P B 1 B In a certain city there are two food stores. 7% of the the population use Cheap FoodStore and 3% use Green FoodStore. Among the shoopers of Cheap FoodStore 7% have household income of less than per year while among the shoopers of Green FoodStore % have household income of less than per year. a Find the probability that a randomly chosen citizen uses Green FoodStore and has household income or more per year. b What proportion of the citizens have household income or more per year. c Given that a person has a household income or more per year how likely he is to shop at Green FoodStore. Solution. Let C be event that a citizen uses Cheap FoodStore and G be event that a citizen uses Green FoodStore. Let R be the event that a citizen has a household income of or more per year. Note that P R C = 1 P R C =.3 and P R G = 1 P R G =.. a P RG = P GP R G =.3. =.1. b P R = P RG + P RC = =.36. c P G R = P RG/P R = /1. 4. Let X have cumulative distribution function F X x = x for x 1 and F X x = for x < and = 1 for x > 1,

3 3 a Find the density of f X x; b find the probability density function f V v of V = 3X + 1; c Compute the median, th and 7th percentiles of X. Solution. a F X x = x = x if x [, 1] and otherwise. b V takes value between = 1 and = 4. If v [1, 4] then P V < v = P 3X + 1 < v = P X < v 1 v 1 3 =. 3 Thus [ v f V v = dv ] 1 = v 1 1 dv = v. 9 c Equation for the k-th percentile is x = k so that x k 1 k =. Thus m =, x 1 7 = 3, x = 1.. The continuous random variable Y has density f Y y = yy + 1 for y 3 and 7 f Y y = otherwise. a Find the cumulative distribution function of Y. b Find E1/Y. c If Y 1, Y... Y 1 are independent random variables with distribution Y find the distribution of M = maxy 1, Y... Y 1. Solution. a f Y y = 7 y + y. Thus b EY = 7 F Y y = 7 3 y y + y dy = y 7 s + sds = 81 y y. 3 y + 1dy = y + 7 c P M < m = P Y 1 < m... Y 1 < m = P Y < m 1 = 3 = [ 81 m3 + 1 ] 1 7 m independent random variables W j, j = 1,..., 1 have been simulated, according to an exponential distribution with parameter λ = 1. a Let N be the number among the variables W j which are larger than ln. What is the exact probability distribution of N? b What is the approximate probability that N is 3 or less? c Answer questions a, b if the number ln is replaced by ln.

4 4 Solution. a P W > ln = e ln = 1. Accordingly N Bin1, 1 b Since 1 1 and 1 1 = is not too large we have N Pois. Thus P N 3 = P N = +P N = 1+P N = +P N = 3 e = c P W > ln = e ln = 1. Accordingly N Bin1, 1. 1.! + 1 1! +! + 3 3! P N 3 = P N = + P N = 1 + P N = + P N = ! ! 1 Since the last term is much larger than all other terms we get P N ! Let X, Y have joint density f X,Y x, y = xy + 8x 3 y 3 if x 1 and y 1 and f X,Y x, y = otherwise. a Compute the marginal density of X. b Compute EX and V X. c Find the CovX, Y. Solution. b EX = Likewise EY = EX = a p X x = x + x 3 xdx = x + x 3 x dx = Hence V X = = = c EXY = = = + 7 xydy + x dx + x 3 dx + xy + 8x 3 y 3 xydxdy = 8x 3 y 3 dy = x + x 3. = 1 1. Thus CovX, Y = x 4 dx = = x dx = = 7 1. x y dxdy + 11 = 1. 8x 4 y 4 dxdy

5 8. Toll rates at a certain bridge is \$ for axis vehicles, \$ 4 for 3 axis vehicles and \$ 1 for vehicles with 4 or more axis. Suppose that 8% of cars passing the bridge have axis, 1% have 3 axis and 1% have 4 or more axis. Let S be the toll collected from next 1 million cars passing the bridge. a Compute ES. b Compute V S. c Compute approximately P S > 33. Solution. Let X j be the toll paid by j-th car. Then S = X 1 + X +... X 1. Next, EX = = 3 so ES = 1EX = 3. Likewise V X = =.8, so V S = 1.8 = 8 and σ S = 8 = 48. By Central Limit Theorem S N3, 48, that is S Z where Z N, 1. Thus P S > 33 P Z > 33 = P 48Z > 3 = P Z > 1. = 1 P Z < = Suppose that X and Y are independent random variables with EX = 1, σx =, EY = 4, σy = 3. Let U = X 4Y. a Find the mean and variance of U. b Compute CovX, U. c If X and Y are each normally distributed find P U 1. Solution. a EU = EX 4EY = 16 = 11. V X = V X +4 V Y = = 98. b CovX, U = CovX, X CovX, 4Y = V X = 1. c By part a U N 11, 98. Hence U = 98Z 11 where Z N, 1. Accoridngly P U 1 = P 98Z 11 1 = P 98Z 1 = P Z 1 98 P Z 1.1 = 1 P Z < Let X 1, X,... X be a sample of some unknown distribution X. Let ˆµ = X 1 + X 4 + X X 4 + X be an estimator of the population mean a Is ˆµ unbiased or not? b Express V ˆµ in terms of V X. c Compare ˆµ with the sample mean. Solution. a Eˆµ = E X1 +E X 4 +E X3 8 +E X4 +E 16 X 1 = EX = EX 16

6 6 so ˆµ is unbiased. X1 X X3 X4 b V ˆµ = V + V + V + V + V = V X =.33937V X 16 c V X = V X estimator. X 16 =.V X. So the sampe mean has a smalller variance and hence is a better 11. Suppose that W is a discrete random variable such that P W = = 1 a/4, P W = 1 = 1 + a/, and P W = = 1 3a/4, where 1 < a < 1 is an unknown parameter. 3 a Compute EW. Let,, 1, 1, 1, be the sample from the distribution W. b Find the method of moment estimator for a. c Find the maximum likelyhood estimator of a. Solution. a EW = 1 a a + 1 3a = a. 4 4 b From part a we have X = â so â MOM = X = 7 = a 1 + a 1 3a c P,, 1, 1,, = 4 4 so the likelyhood function is Thus L a = if La = ln1 a + 3 ln1 + a + ln1 3a ln 4. L a = 1 1 a a 6 1 3a. 1 + a1 3a + 61 a1 3a 61 a1 + a =. That is 36a 9a 1 = Roots have form 9 ± 98. Only one of those roots belongs to the parameter interval, so 7 that â MLE = Let X be a random variable with density f X x = 1 θ + θx if x 1 and f X x = otherwise where < θ < 1 is an unknown parameter. a Compute EX and V X. Let X 1, X... X n be a sample of this distribution. b Find the method of moment estimator for θ. c Give an estimate for the variance of the estimator from part b.

7 Solution. a EX = EX = 1 θ + θxxdx = 1 θ + θ 3 = 1 + θ 6. 1 θ + θxx dx = 1 θ + θ 3 = θ 6. V X = θ θ = θ 36. b From part a we have X = 1 + ˆθ so that ˆθ = 6 X 3. 6 c V ˆθ = 6 V X = 36V X. So the estimate ˆV ˆθ = 36 V X. To estimate the variance of n n X we can use either S = 1 n n 1 j=1 X j X which is always a reasonable estimate for the population variance or a model specific estimate 1 ˆθ After examining bags of rice in a warehouse an inspector came with the following 9% confidence interval for the average weight of a bag 9.9 ±. lbs. a Compute 99% confidence interval for the average weight of a bag; b Compute 9% lower confidence bound for the average weight of a bag; c Estimate how many bags need to be examined so that the width of the 9% confidence interval is.1 lbs. Solution. 9% confidence interval has form X s ± z.. Thus X = 9.9 and 1.96s =.. Accordingly s =. =.7 Next, 99% confidence interval has form 1.96 s.8.7 x ± z.99 = 9.9 ± = 9.9 ±.6 = [9.64, 1.16]. n b Lower confidence bound is s X z.9 = 9.9 n = = s c The width of the 9% confidence interval is z N. N. So from equation 1.96s N N =.1 we get N = 1.96s N. Plugging our estimate for ŝ.1 N =.7 we get N = Researchers investigated the physiological changes that accompany laughter. Ninety 9 subjects years old watched film clips desighed to evoke laughter. During the laughing period, the researchers measured the heart rate beats per minute of subject, with the following summary results: x = 73.1, s = 3. It is well known that the mean resting heart rate of adults is 71 beats per minute. a Test whether the true mean heart rate during laughter exceeds 71 beats per minute using α =.. b Find the power of the test at µ = 7 7

8 8 c Estimate true mean heart rate during laughter in a way that conveys information about precision and reliability. Calculate 9% CI. Solution. a We have to test H = {µ = 71} vs H a = {µ > 71}. Test statistics is z = X 71 s/. 9 In our case z =.1 3/ = 6.6. Since z > z 9. = 1.6 we have sufficient evidence to reject H. That is we have a strong evidence that laughter causes the the average heart rate to exceed 71 beats per minute. b using the formula on page 314 of the book we get 71 7 β = Φ z. + 3/ = Φ Hence the power is 1 β =.93. c The confidence interval has the form s X ± z. = 73.1 ± = 73.1 ±.6 = [7.48, 73.7]. n 9 The upper confidence bound has the form s X ± z. = 73.1 ± = 73.1 ±.6 = [7.48, 73.7]. n 9 1. A business journal investigation of the performance and timing of corporate acquisitions discovered that in a random sample of, 863 firms, 848 announced one or more acquisitions during the year. Let p be the proportion of firms which made one or more acquisitions during the year. a Give a point estimate for p; b Construct 9% confidence interval for p; c Construct 9% upper confidence bound for p. Solution. a ˆp = X n = =.96. b The confidence interval takes form ˆp + z./ 863 ˆp1 ˆp/863 + z ±z. / z./ z./863 b The upper confidence bound takes form ˆp + z.1/ z.1/863 + z.1 ˆp1 ˆp/863 + z.1 / z.1/863 =.96±.14 = [.8,.31]. =.96 ±.11 = Strategic placement of lobster traps is one of keys for a successful lobster fisherman. A study was conducted of the average distance separating traps by lobster fishermen. The trap-spacing measurements in meters for a sample of seven teams from Blue Sea fishing

9 cooperative and came with the following data: x = 81. and s = 1. Suppose that the trap-spacing has normal distribution. a Test the hypotheis that the average distance between the traps is at least 9m at the significance level α =.. b What is the problem with using the normal z statistic to find a confidence interval for the average distance between the traps? c Construct 9% confidence interval for the average distance between the traps. d One team from Blue Sea cooperative was not working during the day of test. Given 9% prediction interval for the distance between the traps used by the missing team. Solution. a We have to test H = {µ = 9} vs H a = {µ < 9}. The test statistics is t = s x 9 which in our case equals to.13. The ctritiacal value t.,6 =.447. Since /7 t < t.,6 we do not reject H. That is we do not have sufficient evidence that the average distance between the traps is in fact less that 9 m. b We do not know the standard deviation and since the number of observations is small we can not estimate it accurately enough to use z statistic. c The confidence interval takes form 81 ±.447 s /7 = [7.69, 91.31]. Note that 9 is inside the confidence interval which is in agreement with the fact that we did not reject H. d The prediction interval takes form 81 ±.447 s = [1.84, 11.16]. 7 9

### Examination 110 Probability and Statistics Examination

Examination 0 Probability and Statistics Examination Sample Examination Questions The Probability and Statistics Examination consists of 5 multiple-choice test questions. The test is a three-hour examination

### MATH 201. Final ANSWERS August 12, 2016

MATH 01 Final ANSWERS August 1, 016 Part A 1. 17 points) A bag contains three different types of dice: four 6-sided dice, five 8-sided dice, and six 0-sided dice. A die is drawn from the bag and then rolled.

### EC 112 SAMPLING, ESTIMATION AND HYPOTHESIS TESTING. One and One Half Hours (1 1 2 Hours) TWO questions should be answered

ECONOMICS EC 112 SAMPLING, ESTIMATION AND HYPOTHESIS TESTING One and One Half Hours (1 1 2 Hours) TWO questions should be answered Statistical tables are provided. The approved calculator (that is, Casio

### CONFIDENCE INTERVALS FOR MEANS AND PROPORTIONS

LESSON SEVEN CONFIDENCE INTERVALS FOR MEANS AND PROPORTIONS An interval estimate for μ of the form a margin of error would provide the user with a measure of the uncertainty associated with the point estimate.

### Chapter 6: Point Estimation. Fall 2011. - Probability & Statistics

STAT355 Chapter 6: Point Estimation Fall 2011 Chapter Fall 2011 6: Point1 Estimat / 18 Chap 6 - Point Estimation 1 6.1 Some general Concepts of Point Estimation Point Estimate Unbiasedness Principle of

### 1. A survey of a group s viewing habits over the last year revealed the following

1. A survey of a group s viewing habits over the last year revealed the following information: (i) 8% watched gymnastics (ii) 9% watched baseball (iii) 19% watched soccer (iv) 14% watched gymnastics and

### Math 370, Spring 2008 Prof. A.J. Hildebrand. Practice Test 2 Solutions

Math 370, Spring 008 Prof. A.J. Hildebrand Practice Test Solutions About this test. This is a practice test made up of a random collection of 5 problems from past Course /P actuarial exams. Most of the

### Solutions for the exam for Matematisk statistik och diskret matematik (MVE050/MSG810). Statistik för fysiker (MSG820). December 15, 2012.

Solutions for the exam for Matematisk statistik och diskret matematik (MVE050/MSG810). Statistik för fysiker (MSG8). December 15, 12. 1. (3p) The joint distribution of the discrete random variables X and

### Math 370, Actuarial Problemsolving Spring 2008 A.J. Hildebrand. Practice Test, 1/28/2008 (with solutions)

Math 370, Actuarial Problemsolving Spring 008 A.J. Hildebrand Practice Test, 1/8/008 (with solutions) About this test. This is a practice test made up of a random collection of 0 problems from past Course

### Joint Exam 1/P Sample Exam 1

Joint Exam 1/P Sample Exam 1 Take this practice exam under strict exam conditions: Set a timer for 3 hours; Do not stop the timer for restroom breaks; Do not look at your notes. If you believe a question

### Confidence Intervals for Cp

Chapter 296 Confidence Intervals for Cp Introduction This routine calculates the sample size needed to obtain a specified width of a Cp confidence interval at a stated confidence level. Cp is a process

### Math 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 3 Solutions

Math 37/48, Spring 28 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 3 Solutions About this problem set: These are problems from Course /P actuarial exams that I have collected over the years,

### MINITAB ASSISTANT WHITE PAPER

MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. One-Way

### Covariance and Correlation. Consider the joint probability distribution f XY (x, y).

Chapter 5: JOINT PROBABILITY DISTRIBUTIONS Part 2: Section 5-2 Covariance and Correlation Consider the joint probability distribution f XY (x, y). Is there a relationship between X and Y? If so, what kind?

### Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)

Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume

### Joint Distributions. Tieming Ji. Fall 2012

Joint Distributions Tieming Ji Fall 2012 1 / 33 X : univariate random variable. (X, Y ): bivariate random variable. In this chapter, we are going to study the distributions of bivariate random variables

### Chapter 4 Expected Values

Chapter 4 Expected Values 4. The Expected Value of a Random Variables Definition. Let X be a random variable having a pdf f(x). Also, suppose the the following conditions are satisfied: x f(x) converges

### ST 371 (VIII): Theory of Joint Distributions

ST 371 (VIII): Theory of Joint Distributions So far we have focused on probability distributions for single random variables. However, we are often interested in probability statements concerning two or

### 3. Continuous Random Variables

3. Continuous Random Variables A continuous random variable is one which can take any value in an interval (or union of intervals) The values that can be taken by such a variable cannot be listed. Such

### Confidence Intervals for One Standard Deviation Using Standard Deviation

Chapter 640 Confidence Intervals for One Standard Deviation Using Standard Deviation Introduction This routine calculates the sample size necessary to achieve a specified interval width or distance from

### Definition The covariance of X and Y, denoted by cov(x, Y ) is defined by. cov(x, Y ) = E(X µ 1 )(Y µ 2 ).

Correlation Regression Bivariate Normal Suppose that X and Y are r.v. s with joint density f(x y) and suppose that the means of X and Y are respectively µ 1 µ 2 and the variances are 1 2. Definition The

### MAS108 Probability I

1 QUEEN MARY UNIVERSITY OF LONDON 2:30 pm, Thursday 3 May, 2007 Duration: 2 hours MAS108 Probability I Do not start reading the question paper until you are instructed to by the invigilators. The paper

### Review Solutions, Exam 3, Math 338

Review Solutions, Eam 3, Math 338. Define a random sample: A random sample is a collection of random variables, typically indeed as X,..., X n.. Why is the t distribution used in association with sampling?

### Lecture 6: Discrete & Continuous Probability and Random Variables

Lecture 6: Discrete & Continuous Probability and Random Variables D. Alex Hughes Math Camp September 17, 2015 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September

### Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density

HW MATH 461/561 Lecture Notes 15 1 Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density and marginal densities f(x, y), (x, y) Λ X,Y f X (x), x Λ X,

### Data Modeling & Analysis Techniques. Probability & Statistics. Manfred Huber 2011 1

Data Modeling & Analysis Techniques Probability & Statistics Manfred Huber 2011 1 Probability and Statistics Probability and statistics are often used interchangeably but are different, related fields

### 4. Joint Distributions of Two Random Variables

4. Joint Distributions of Two Random Variables 4.1 Joint Distributions of Two Discrete Random Variables Suppose the discrete random variables X and Y have supports S X and S Y, respectively. The joint

### 7 CONTINUOUS PROBABILITY DISTRIBUTIONS

7 CONTINUOUS PROBABILITY DISTRIBUTIONS Chapter 7 Continuous Probability Distributions Objectives After studying this chapter you should understand the use of continuous probability distributions and the

### Confidence Intervals for Cpk

Chapter 297 Confidence Intervals for Cpk Introduction This routine calculates the sample size needed to obtain a specified width of a Cpk confidence interval at a stated confidence level. Cpk is a process

### Joint Probability Distributions and Random Samples (Devore Chapter Five)

Joint Probability Distributions and Random Samples (Devore Chapter Five) 1016-345-01 Probability and Statistics for Engineers Winter 2010-2011 Contents 1 Joint Probability Distributions 1 1.1 Two Discrete

### Probability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0

Probability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0 This primer provides an overview of basic concepts and definitions in probability and statistics. We shall

### Sampling & Confidence Intervals

Sampling & Confidence Intervals Mark Lunt Arthritis Research UK Centre for Excellence in Epidemiology University of Manchester 18/10/2016 Principles of Sampling Often, it is not practical to measure every

### Topic 4: Multivariate random variables. Multiple random variables

Topic 4: Multivariate random variables Joint, marginal, and conditional pmf Joint, marginal, and conditional pdf and cdf Independence Expectation, covariance, correlation Conditional expectation Two jointly

### Definition: Let S be a solid with cross-sectional area A(x) perpendicular to the x-axis at each point x [a, b]. The volume of S is V = A(x)dx.

Section 7.: Volume Let S be a solid and suppose that the area of the cross-section of S in the plane P x perpendicular to the x-axis passing through x is A(x) for a x b. Consider slicing the solid into

### Name: Date: Use the following to answer questions 2-4:

Name: Date: 1. A phenomenon is observed many, many times under identical conditions. The proportion of times a particular event A occurs is recorded. What does this proportion represent? A) The probability

### MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters

MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters Inferences about a population parameter can be made using sample statistics for

### The Big 50 Revision Guidelines for S1

The Big 50 Revision Guidelines for S1 If you can understand all of these you ll do very well 1. Know what is meant by a statistical model and the Modelling cycle of continuous refinement 2. Understand

### Continuous Probability Distributions (120 Points)

Economics : Statistics for Economics Problem Set 5 - Suggested Solutions University of Notre Dame Instructor: Julio Garín Spring 22 Continuous Probability Distributions 2 Points. A management consulting

### Random Variables, Expectation, Distributions

Random Variables, Expectation, Distributions CS 5960/6960: Nonparametric Methods Tom Fletcher January 21, 2009 Review Random Variables Definition A random variable is a function defined on a probability

### Regression Analysis: A Complete Example

Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty

### ECE302 Spring 2006 HW7 Solutions March 11, 2006 1

ECE32 Spring 26 HW7 Solutions March, 26 Solutions to HW7 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in italics where

### Continuous Random Variables

Chapter 5 Continuous Random Variables 5.1 Continuous Random Variables 1 5.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Recognize and understand continuous

### The t Test. April 3-8, exp (2πσ 2 ) 2σ 2. µ ln L(µ, σ2 x) = 1 σ 2. i=1. 2σ (σ 2 ) 2. ˆσ 2 0 = 1 n. (x i µ 0 ) 2 = i=1

The t Test April 3-8, 008 Again, we begin with independent normal observations X, X n with unknown mean µ and unknown variance σ The likelihood function Lµ, σ x) exp πσ ) n/ σ x i µ) Thus, ˆµ x ln Lµ,

### Statistical Inference

Statistical Inference Idea: Estimate parameters of the population distribution using data. How: Use the sampling distribution of sample statistics and methods based on what would happen if we used this

### Recitation 4. 24xy for 0 < x < 1, 0 < y < 1, x + y < 1 0 elsewhere

Recitation. Exercise 3.5: If the joint probability density of X and Y is given by xy for < x

### Math 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 5 Solutions

Math 370/408, Spring 2008 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 5 Solutions About this problem set: These are problems from Course 1/P actuarial exams that I have collected over the

### Lecture Notes 1. Brief Review of Basic Probability

Probability Review Lecture Notes Brief Review of Basic Probability I assume you know basic probability. Chapters -3 are a review. I will assume you have read and understood Chapters -3. Here is a very

### Statistics I for QBIC. Contents and Objectives. Chapters 1 7. Revised: August 2013

Statistics I for QBIC Text Book: Biostatistics, 10 th edition, by Daniel & Cross Contents and Objectives Chapters 1 7 Revised: August 2013 Chapter 1: Nature of Statistics (sections 1.1-1.6) Objectives

### P(a X b) = f X (x)dx. A p.d.f. must integrate to one: f X (x)dx = 1. Z b

Continuous Random Variables The probability that a continuous random variable, X, has a value between a and b is computed by integrating its probability density function (p.d.f.) over the interval [a,b]:

### The problems of first TA class of Math144

The problems of first TA class of Math144 T eaching Assistant : Liu Zhi Date : F riday, F eb 2, 27 Q1. According to the journal Chemical Engineering, an important property of a fiber is its water absorbency.

### Confidence Intervals for Spearman s Rank Correlation

Chapter 808 Confidence Intervals for Spearman s Rank Correlation Introduction This routine calculates the sample size needed to obtain a specified width of Spearman s rank correlation coefficient confidence

### Confidence Intervals for Exponential Reliability

Chapter 408 Confidence Intervals for Exponential Reliability Introduction This routine calculates the number of events needed to obtain a specified width of a confidence interval for the reliability (proportion

### INTRODUCTION TO PROBABILITY AND STATISTICS

INTRODUCTION TO PROBABILITY AND STATISTICS Conditional probability and independent events.. A fair die is tossed twice. Find the probability of getting a 4, 5, or 6 on the first toss and a,,, or 4 on the

### An Introduction to Basic Statistics and Probability

An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random

### Hypothesis Testing COMP 245 STATISTICS. Dr N A Heard. 1 Hypothesis Testing 2 1.1 Introduction... 2 1.2 Error Rates and Power of a Test...

Hypothesis Testing COMP 45 STATISTICS Dr N A Heard Contents 1 Hypothesis Testing 1.1 Introduction........................................ 1. Error Rates and Power of a Test.............................

### 4. Introduction to Statistics

Statistics for Engineers 4-1 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation

### Jointly Distributed Random Variables

Jointly Distributed Random Variables COMP 245 STATISTICS Dr N A Heard Contents 1 Jointly Distributed Random Variables 1 1.1 Definition......................................... 1 1.2 Joint cdfs..........................................

### Random Variables. Chapter 2. Random Variables 1

Random Variables Chapter 2 Random Variables 1 Roulette and Random Variables A Roulette wheel has 38 pockets. 18 of them are red and 18 are black; these are numbered from 1 to 36. The two remaining pockets

### INTRODUCTORY STATISTICS

INTRODUCTORY STATISTICS Questions 290 Field Statistics Target Audience Science Students Outline Target Level First or Second-year Undergraduate Topics Introduction to Statistics Descriptive Statistics

### Point and Interval Estimates

Point and Interval Estimates Suppose we want to estimate a parameter, such as p or µ, based on a finite sample of data. There are two main methods: 1. Point estimate: Summarize the sample by a single number

### Expectation Discrete RV - weighted average Continuous RV - use integral to take the weighted average

PHP 2510 Expectation, variance, covariance, correlation Expectation Discrete RV - weighted average Continuous RV - use integral to take the weighted average Variance Variance is the average of (X µ) 2

### Chapter 4 - Lecture 1 Probability Density Functions and Cumul. Distribution Functions

Chapter 4 - Lecture 1 Probability Density Functions and Cumulative Distribution Functions October 21st, 2009 Review Probability distribution function Useful results Relationship between the pdf and the

### Math 431 An Introduction to Probability. Final Exam Solutions

Math 43 An Introduction to Probability Final Eam Solutions. A continuous random variable X has cdf a for 0, F () = for 0 <

### Generating Random Numbers Variance Reduction Quasi-Monte Carlo. Simulation Methods. Leonid Kogan. MIT, Sloan. 15.450, Fall 2010

Simulation Methods Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Simulation Methods 15.450, Fall 2010 1 / 35 Outline 1 Generating Random Numbers 2 Variance Reduction 3 Quasi-Monte

### Joint Probability Distributions and Random Samples. Week 5, 2011 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage

5 Joint Probability Distributions and Random Samples Week 5, 2011 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Two Discrete Random Variables The probability mass function (pmf) of a single

### Answer keys for Assignment 10: Measurement of study variables

Answer keys for Assignment 10: Measurement of study variables (The correct answer is underlined in bold text) 1. In a study, participants are asked to indicate the type of pet they have at home (ex: dog,

### 4. Continuous Random Variables, the Pareto and Normal Distributions

4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random

### Chapter 3 RANDOM VARIATE GENERATION

Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.

### MULTIVARIATE PROBABILITY DISTRIBUTIONS

MULTIVARIATE PROBABILITY DISTRIBUTIONS. PRELIMINARIES.. Example. Consider an experiment that consists of tossing a die and a coin at the same time. We can consider a number of random variables defined

### Covariance and Correlation

Covariance and Correlation ( c Robert J. Serfling Not for reproduction or distribution) We have seen how to summarize a data-based relative frequency distribution by measures of location and spread, such

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 18. A Brief Introduction to Continuous Probability

CS 7 Discrete Mathematics and Probability Theory Fall 29 Satish Rao, David Tse Note 8 A Brief Introduction to Continuous Probability Up to now we have focused exclusively on discrete probability spaces

### Confidence Intervals for the Difference Between Two Means

Chapter 47 Confidence Intervals for the Difference Between Two Means Introduction This procedure calculates the sample size necessary to achieve a specified distance from the difference in sample means

### Statistics - Written Examination MEC Students - BOVISA

Statistics - Written Examination MEC Students - BOVISA Prof.ssa A. Guglielmi 26.0.2 All rights reserved. Legal action will be taken against infringement. Reproduction is prohibited without prior consent.

### STA 256: Statistics and Probability I

Al Nosedal. University of Toronto. Fall 2014 1 2 3 4 5 My momma always said: Life was like a box of chocolates. You never know what you re gonna get. Forrest Gump. Experiment, outcome, sample space, and

### Statistiek (WISB361)

Statistiek (WISB361) Final exam June 29, 2015 Schrijf uw naam op elk in te leveren vel. Schrijf ook uw studentnummer op blad 1. The maximum number of points is 100. Points distribution: 23 20 20 20 17

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Final Exam Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A researcher for an airline interviews all of the passengers on five randomly

### Lecture 14. Chapter 7: Probability. Rule 1: Rule 2: Rule 3: Nancy Pfenning Stats 1000

Lecture 4 Nancy Pfenning Stats 000 Chapter 7: Probability Last time we established some basic definitions and rules of probability: Rule : P (A C ) = P (A). Rule 2: In general, the probability of one event

### Contents. 3 Survival Distributions: Force of Mortality 37 Exercises Solutions... 51

Contents 1 Probability Review 1 1.1 Functions and moments...................................... 1 1.2 Probability distributions...................................... 2 1.2.1 Bernoulli distribution...................................

### 10 BIVARIATE DISTRIBUTIONS

BIVARIATE DISTRIBUTIONS After some discussion of the Normal distribution, consideration is given to handling two continuous random variables. The Normal Distribution The probability density function f(x)

### PRACTICE PAPER MATHEMATICS Extended Part Module 1 (Calculus and Statistics) Question-Answer Book

PP-DSE MATH EP M1 Please stick the barcode label here. HONG KONG EXAMINATIONS AND ASSESSMENT AUTHORITY HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION Candidate Number PRACTICE PAPER MATHEMATICS Extended

### MATH4427 Notebook 2 Spring 2016. 2 MATH4427 Notebook 2 3. 2.1 Definitions and Examples... 3. 2.2 Performance Measures for Estimators...

MATH4427 Notebook 2 Spring 2016 prepared by Professor Jenny Baglivo c Copyright 2009-2016 by Jenny A. Baglivo. All Rights Reserved. Contents 2 MATH4427 Notebook 2 3 2.1 Definitions and Examples...................................

### Exact Confidence Intervals

Math 541: Statistical Theory II Instructor: Songfeng Zheng Exact Confidence Intervals Confidence intervals provide an alternative to using an estimator ˆθ when we wish to estimate an unknown parameter

### Section 6.1 Joint Distribution Functions

Section 6.1 Joint Distribution Functions We often care about more than one random variable at a time. DEFINITION: For any two random variables X and Y the joint cumulative probability distribution function

### Let m denote the margin of error. Then

S:105 Statistical Methods and Computing Sample size for confidence intervals with σ known t Intervals Lecture 13 Mar. 6, 009 Kate Cowles 374 SH, 335-077 kcowles@stat.uiowa.edu 1 The margin of error The

### Solutions to Homework 3 Statistics 302 Professor Larget

s to Homework 3 Statistics 302 Professor Larget Textbook Exercises 3.20 Customized Home Pages A random sample of n = 1675 Internet users in the US in January 2010 found that 469 of them have customized

### Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment 2-3, Probability and Statistics, March 2015. Due:-March 25, 2015.

Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment -3, Probability and Statistics, March 05. Due:-March 5, 05.. Show that the function 0 for x < x+ F (x) = 4 for x < for x

### UNIVERSITY of TORONTO. Faculty of Arts and Science

UNIVERSITY of TORONTO Faculty of Arts and Science AUGUST 2005 EXAMINATION AT245HS uration - 3 hours Examination Aids: Non-programmable or SOA-approved calculator. Instruction:. There are 27 equally weighted

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) (a) 2. (b) 1.5. (c) 0.5-2.

Stats: Test 1 Review Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the given frequency distribution to find the (a) class width. (b) class

### Math 370, Spring 2008 Prof. A.J. Hildebrand. Practice Test 1 Solutions

Math 70, Spring 008 Prof. A.J. Hildebrand Practice Test Solutions About this test. This is a practice test made up of a random collection of 5 problems from past Course /P actuarial exams. Most of the

### Continuous Random Variables

Probability 2 - Notes 7 Continuous Random Variables Definition. A random variable X is said to be a continuous random variable if there is a function f X (x) (the probability density function or p.d.f.)

### Notes for STA 437/1005 Methods for Multivariate Data

Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.

### Section 2: Estimation, Confidence Intervals and Testing Hypothesis

Section 2: Estimation, Confidence Intervals and Testing Hypothesis Carlos M. Carvalho The University of Texas at Austin McCombs School of Business http://faculty.mccombs.utexas.edu/carlos.carvalho/teaching/

### WISE Power Tutorial All Exercises

ame Date Class WISE Power Tutorial All Exercises Power: The B.E.A.. Mnemonic Four interrelated features of power can be summarized using BEA B Beta Error (Power = 1 Beta Error): Beta error (or Type II

### Numerical Summarization of Data OPRE 6301

Numerical Summarization of Data OPRE 6301 Motivation... In the previous session, we used graphical techniques to describe data. For example: While this histogram provides useful insight, other interesting

### Example: If we roll a dice and flip a coin, how many outcomes are possible?

12.5 Tree Diagrams Sample space- Sample point- Counting principle- Example: If we roll a dice and flip a coin, how many outcomes are possible? TREE DIAGRAM EXAMPLE: Use a tree diagram to show all the possible

### Continuous Random Variables and Probability Distributions. Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage

4 Continuous Random Variables and Probability Distributions Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage Continuous r.v. A random variable X is continuous if possible values

### Math 115 HW #8 Solutions

Math 115 HW #8 Solutions 1 The function with the given graph is a solution of one of the following differential equations Decide which is the correct equation and justify your answer a) y = 1 + xy b) y

### Joint distributions Math 217 Probability and Statistics Prof. D. Joyce, Fall 2014

Joint distributions Math 17 Probability and Statistics Prof. D. Joyce, Fall 14 Today we ll look at joint random variables and joint distributions in detail. Product distributions. If Ω 1 and Ω are sample