# Examination 110 Probability and Statistics Examination

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 based on material usually covered in undergraduate mathematics courses in mathematical probability and statistics. This booklet provides examples of the types of test questions found in the Probability and Statistics Examination. Although the particular selection of topics covered in the examination may vary from year to year, the types and the general forms of questions will remain the same. After covering the study material thoroughly, candidates may wish to test themselves on each set of questions by trying to answer correctly as many questions as possible in the allotted time of three hours. In so doing, candidates can identify those areas requiring further work and study as well as gain experience in answering multiple-choice test questions. The Examination Committees do not intend to publish new sample examination questions each year. The committees intend, however, to update from time to time the sample questions included in this booklet. The Probability and Statistics Examination is jointly administered by the Society of Actuaries and the Casualty Actuarial Society. It is jointly sponsored by : the American Academy of Actuaries, the Canadian Institute of Actuaries, the Casualty Actuarial Society, the Conference of Consulting Actuaries and the Society of Actuaries. GENERAL INFORMATION. n is the number of combinations of n objects taken r at a time. r 2. lnx is the natural logarithm of x. 3. If A and B are sets, then the union A B of sets A and B is the the set of all elements belonging either to set A or to set B or to both; the intersection A B of sets A and B is the the set of all elements belonging both to set A and to set B; and the complement A of set A is the set of all elements that do not belong to set A.. is the null or empty set. 5. All problems involving cards refer to an ordinary deck of 52 playing cards, consisting of suits of 3 cards each. A bridge hand consists of 3 cards. 6. All problems involving dice refer to the ordinary six-sided die. 7. All problems involving a fair coin refer to a coin for which the probability of heads is /2 and the probability of tails is /2. 8. The mean of a random variable X is denoted by µ X = EX; the variance of a random variable X is denoted by σ 2 X = VarX; the covariance of two random variables X and Y is denoted by σ XY = CovX, Y ; and the correlation coefficient of two random variables X and Y is denoted by n ρ XY = CorrX, Y. The mean of a sample X,..., X n is denoted by X = n X i.

2 9. The table below gives the value of Φx = 2π x e w2 /2 dw for certain values of x. The integer of x is given in the left column, and the first decimal place of x is given in the top row. Since the density function of x is symmetric, the value of the cumulative distribution function for negative x can be obtained by substracting from unity the value of the cumulative distribution for x. x x Selected Points of the Normal Distribution Φx x

3 0. The table below gives the value c for which P [ χ 2 < c ] = p for a given number of degrees of freedom and a given value p. c Value of p df

4 . The table below gives the value t for which P [ < T < t] = p for a given number of degrees of freedom and a given value of p. t Value of p df

5 2. The table below gives the value f for which P F < f = p for a given numbers of degrees of freedom and a given value of p. f Value Degrees of freedom Degrees of freedom for numerator of p for denominator

6 . A box contains 0 balls, of which 3 are red, 2 are yellow, and 5 are blue. Five balls are randomly selected with replacement. Calculate the probability that fewer than 2 of the selected balls are red. A B C D E Let X be a normal random variable with mean 0 and variance a > 0. Calculate PX 2 < a. A 0.3 B 0.2 C 0.68 D 0.8 E Workplace accidents are categorized in three groups : minor, moderate and severe. The probability that a given accident is minor is 0.5, that it is moderate is 0., and that it is severe is 0.. Two accidents occur independently in one month. Calculate the probability that neither accident is severe and at most one is moderate. A 0.25 B 0.0 C 0.5 D 0.56 E Let X, X 2, X 3 be uniform random variables on the interval 0, with CovX i, X j = /2 for i, j =, 2, 3, i j. Calculate the variance of X + 2X 2 X 3. A /6 B / C 5/2 D /2 E /2 5. Let A, B, C and D be events such that B = A, C D =, and PA = /, PB = 3/, PC A = /2, PC B = /8, PD A = / and PD B = /8. Calculate PC D. A 5/32 B / C 3/8 D 3/ E 6. Let X, Y,..., X 8, Y 8 be a random sample from a bivariate normal distribution with means µ X and µ Y and nonzero variances. The null hypothesis H 0 : µ X = µ Y is rejected in favor of the alternative hypothesis H : µ X µ Y if 8 X Y > k. 8 [ Xi Y i X Y ] 2 7 Determine the value of k for which the significance level size of the test is A.6 B.90 C.96 D 2.3 E A class contains 8 boys and 7 girls. The teacher selects 3 of the children at random and without replacement. Calculate the probability that the number of boys selected exceeds the number of girls selected. A B C 8 5 D E Let X and Y be discrete random variables with joint probability function px, y given by the following table : x = 2 x = 3 x = x = 5 y = y = y = For this joint distribution EX = 2.85 and EY =. Calculate CovX, Y. A 0.20 B 0.5 C 0.95 D 2.70 E

7 9. You are given the model EY i = θx i + x 2 i and the following data : Calculate the least squares estimate of θ. i x i y i A 3 92 B C 3 0 D 9 2 E Let X, X 2 and X 3 be independent discrete random variables with probability functions PX i = k = ni p k k p ni k k = 0,,..., n i for i =, 2, 3 where 0 < p <. Determine the probability function of S = X + X 2 + X 3, where positive. A B C D E n + n 2 + n 3 p s s p n+n2+n3 s n i n + n 2 + n 3 ni p s p ni s s ni p s p ni s s n n 2 n 3 s ni p s p ni s s p s p nn2n3 s. Let X be a random variable with distribution function 0 for x < 0 Fx = x/8 for 0 x < 2 + x/8 for x < x/2 for 2 x < 3 for x 3 Calculate P X 2. A /8 B 3/8 C 7/6 D 3/2 E 9/2 7

8 2. Let X,..., X n be a random sample from a continuous distribution with density function fx = α2 α x α+ for x 2 where α > 0. Determine the maximum likelihood estimator of α. A min X,...,X n B C n n X i n n lnx i D maxx,...,x n E n n lnx i n ln2 3. Let X,..., X n be a random sample from a discrete distribution with probability function { θ θ x for x = 0,, 2,... px = where 0 < θ <. Determine the form of the likelihood ratio test for rejecting the null hypothesis H 0 : θ = 0.25 in favor of the alternative hypothesis H : θ n ΣXi A 3+X < k B C D E +X +X X n ΣXi 3+X < k n ΣXi +X +X 2 2X < k X n 3 X n+σxi +X 2 < k ΣXi < k. Let X,...,X 7 and Y,...,Y be independent random samples from normal distributions with common mean µ = 30 and common variance σ 2 Y 302 > 0. The statistic W = 2 has an F distribution X 30 2 with c and d degrees of freedom. Determine c and d. A c =, d = B c = 6, d = 3 C c = 7, d = D c = 3, d = 6 E c =, d = 7 8

9 5. Let X,...,X n be independent Poisson random variables with expectations λ,..., λ n, respectively. Let Z = n ax i, where a is a constant. Determine the moment generating function of Z. A B C D E exp t n aλ i + n 2 t2 a 2 λ i n exp aλ i e t exp t n aλ i + n 2 t2 a 2 λ 2 i n exp λ i e at [ n λ i ]e at n 6. Let X be a single observation from a continuous distribution with density function fx = 2 e x θ for < x <. The null hypothesis H 0 : θ = 0 is tested against the alternative hypothesis H : θ =. The null hypothesis is rejected if X > k. The probability of a Type I error is Calculate the probability of a Type II error. A 0.08 B C 0.86 D E Let X and Y be continuous random variables with joint density function fx, y, where f is nonzero for x and y. Let Z = XY and W = X/Y. Determine the joint density of Z and W, where positive. A f zw, z w B 2w f zw, z w C f zw, z w D 2wf zw, z w E f zw, z w 8. Let X be a random variable with mean 0 and variance. Calculate the largest possible value of P X 8, according to Chebyshev s inequality. A /6 B /8 C / D 3/ E 5/6 9. Let X,...,X n and Y,..., Y n be independent random samples from normal distributions with means µ X and µ Y and variances 2 and respectively. The null hypothesis H 0 : µ X = µ Y is rejected in favor of the alternative hypothesis H : µ X > µ Y if X Y > k. Determine the smallest value of n for which a test of significance level size has power of at least 0.5 when µ X = µ Y + 2. A 3 B C 5 D 6 E Five hypotheses are to be tested using five independent test statistics. A common significance level size for each test is desired which ensured that the probability of rejecting at least one hypothesis is 0., when all five hypothese are true. Determine the desired common significance level size. A 0.00 B C D 0.67 E Let X be a Poisson random variable with EX = ln2. Calculate E[cosπX]. A 0 B / C /2 D E 2 ln2 9

10 22. Let X and Y be continuous random variables with joint density function for 0 < x < 2 and 0 < y < x fx, y = Calculate VarX. A /8 B /6 C /3 D E 7/6 23. Let A and B be events such that PA = 0.7 and PB = 0.9. Calculate the largest possible value of PA B PA B. A 0.20 B 0.3 C 0.0 D 0.60 E Let X and Y be continuous random variables having a bivariate normal distribution with means µ X and µ Y, common variance σ 2, and correlation coefficient ρ XY. Let F X and F Y be the cumulative distribution functions of X and Y, respectively. Determine which of the following is a necessary and sufficient condition for F X t F Y t. A µ X µ Y B µ X µ Y C µ X ρ XY µ Y D µ X ρ XY µ Y E ρ XY The sum of the sample mean and median for ten distinct data points is equal to 20. The largest data is equal to 5. Calculate the sum of the sample mean and median if the largest data point were replaced by 25. A 20 B 2 C 22 D 30 E Let X,..., X n be a random sample from a discrete distribution with probability function θ for x = px = θ for x = 2 2θ for x = 3 where 0 < θ < /2. Determine the method of moments estimator of θ. A 3 X 3 B X C 2X 3 6 D X E 2 X 27. Let X be a continuous random variable with density function 2/x 2 for x 2 fx = Determine the density function of Y = X for 0 < y. A y 2 B 2 y+ 2 C 2 y 2 D 2y 2 y+ 2. E 2y+2 y 28. Let X be the number of independent Bernouilli trials performed until a success occurs. Let Y be the number of independent Bernouilli trials performed until 5 successes occur. A success occurs with probability p and VarX = 3/. Calculate VarY. A 3 20 B 3 5 C 3 D 5 E Let X and Y be discrete random variables with joint probability function 2 x y+ 9 for x =, 2 and y =, 2 px, y =. Calculate EX/Y. A 8/9 B 5/ C /3 D 25/8 E 5/3 0

11 30. Let X and Y be two independent random variables with moment generating functions M X t = e t2 +2t M Y t = e 3t2 +t Determine the moment generating function of X + 2Y. A e t2 +2t + 2e 3t2 +t B e t2 +2t + e 2t2 +2t C e 7t2 +t D 2e t2 +3t E e 3t2 +t 3. Let X and Y be independent random variables with µ X =, µ Y =, σ 2 = /2 and X σ2 Y E [ X + 2 Y 2]. A B 9/2 C 6 D 7 E Let X be a single observation from a continuous distribution with density function λe λx for x > 0 fx = = 2. Calculate where λ > 0. The null hypothesis H 0 : λ = is tested against the alternative H : λ >. Determine the critical region corresponding to the uniformly most powerful test of significance level size A X < 0.05 B X > 0.05 C X < D X > E X > Let X,..., X n be a random sample from a continuous distribution with density function e x for 0 < x < θ e fx = θ where 0 < θ <. Determine the maximum likelihood estimator of θ. A n /n X i B X C ln D min X,...,X n E max X,..., X n e x 3. Let X and Y be continuous random variables with joint density function 2 for 0 < x < y < fx, y =. Determine the conditional density function of Y given X = x, where 0 < x <. A x for x < y < B 2 x for x < y < C 2 for x < y < D y for x < y < E y for x < y < 35. Let X be a continuous random variable with density function 6x x for 0 < x < fx =. Calculate P [ X 2 > ]. A B C D E

12 36. Let X and Y be continuous random variables with joint cumulative distribution function Fx, y = 20xy x 2 y xy for 0 x 5 and 0 y 5. Determine PX > 2. A 3/25 B /50 C 2/25 D y 3y 2 E y 2y Let X,..., X n be a random sample from a normal distribution with mean µ = 0 and variance σ 2 = 00. Let c n X i 0 2 have a χ 2 distribution with d degrees of freedom. Determine c and d. A c = 0.0, d = n B c = 0.0, d = n C c = 0., d = n D c = 0., d = n E c = n /2, d = n 38. A coin is tossed repeatedly and independently until the fourth head occurs. The null hypothesis H 0 : π = 0.2, where π is the probability of heads, is rejected if the number of tosses required is less than or equal to 6. Determine the value of the power function of this test at π = 0.3. A B C D E 6 i= 6 i= 6 i= i=7 6 i= i i i i i i i i i i A bowl contains 0 balls, of which are red and 6 are white. Balls are randomly selected with replacement from the bowl until red balls have been selected. Let X be the number of white balls drawn before the fourth red ball is selected. Calculate PX = 6. A B C 0.06 D E Let X,..., X 00 be a random sample from a exponential distribution with mean /2. Determine the 00 approximative value of P X i > 57 using the Central Limit Theorem. A 0.08 B 0.6 C 0.3 D 0.38 E 0.6 2

13 . Let X be a continuous random variable with density function 2 e x/2 for x > 0 fx =. Determine the 25 th percentile of the distribution of X. A ln 9 B ln 6 9 C ln D 2 E ln 6 2. Let X be a continuous random variable with density function fx = 2π e x2 /2 for < x <. Calculate EX X > 0. A 0 B / 2π C /2 D 2/π E 3. Let A, B and C be independent events such that PA = 0.5, PB = 0.6 and PC = 0.. Calculate PA B C. A 0.69 B 0.7 C 0.73 D 0.98 E. The probability that a particular machine breaks down on any day is 0.2 and is independent of the breakdowns on any other day. The machine can break down only once per day. Calculate the probability that the machine breaks down two or more times in ten days. A B C D E The weights of the animals in a population are normally distributed with variance. A random sample of 6 of the animals is taken. The mean weight of the sample is 200 pounds. Calculate the lower bound of the symmetric 90% confidence interval for the mean weight of the population. A 0.96 B 9.2 C 9.75 D E Keys 0 à 05 C C E C C 06 à 0 E E B B A à 5 E E A A D 6 à 20 C B A D C 2 à 25 B B C B B 26 à 30 A B D D E 3 à 35 E A E A C 36 à 0 C B A D A à 5 B D C D D Document original obtenu du site Society of Actuaries et reproduit en L A TEX. Jacques Carel Département de mathématiques Cégep de Lévis-Lauzon mai

### 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.

### 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

### 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

### 3.6: General Hypothesis Tests

3.6: General Hypothesis Tests The χ 2 goodness of fit tests which we introduced in the previous section were an example of a hypothesis test. In this section we now consider hypothesis tests more generally.

### 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

### 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,

### Summary of Probability

Summary of Probability Mathematical Physics I Rules of Probability The probability of an event is called P(A), which is a positive number less than or equal to 1. The total probability for all possible

### Exercises in Probability Theory Nikolai Chernov

Exercises in Probability Theory Nikolai Chernov All exercises (except Chapters 16 and 17) are taken from two books: R. Durrett, The Essentials of Probability, Duxbury Press, 1994 S. Ghahramani, Fundamentals

### FALL 2005 EXAM C SOLUTIONS

FALL 005 EXAM C SOLUTIONS Question #1 Key: D S ˆ(300) = 3/10 (there are three observations greater than 300) H ˆ (300) = ln[ S ˆ (300)] = ln(0.3) = 1.0. Question # EX ( λ) = VarX ( λ) = λ µ = v = E( λ)

### 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

### 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

### 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

### MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions...

MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo c Copyright 2004-2012 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook 3 3 3.1 Definitions............................................

### 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

### 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

### 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

### Expected Value. Let X be a discrete random variable which takes values in S X = {x 1, x 2,..., x n }

Expected Value Let X be a discrete random variable which takes values in S X = {x 1, x 2,..., x n } Expected Value or Mean of X: E(X) = n x i p(x i ) i=1 Example: Roll one die Let X be outcome of rolling

### Topic 2: Scalar random variables. Definition of random variables

Topic 2: Scalar random variables Discrete and continuous random variables Probability distribution and densities (cdf, pmf, pdf) Important random variables Expectation, mean, variance, moments Markov and

### ACTM State Exam-Statistics

ACTM State Exam-Statistics For the 25 multiple-choice questions, make your answer choice and record it on the answer sheet provided. Once you have completed that section of the test, proceed to the tie-breaker

### 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.

### 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

### Statistics for Management II-STAT 362-Final Review

Statistics for Management II-STAT 362-Final Review Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1. The ability of an interval estimate to

### For a partition B 1,..., B n, where B i B j = for i. A = (A B 1 ) (A B 2 ),..., (A B n ) and thus. P (A) = P (A B i ) = P (A B i )P (B i )

Probability Review 15.075 Cynthia Rudin A probability space, defined by Kolmogorov (1903-1987) consists of: A set of outcomes S, e.g., for the roll of a die, S = {1, 2, 3, 4, 5, 6}, 1 1 2 1 6 for the roll

### Homework Zero. Max H. Farrell Chicago Booth BUS41100 Applied Regression Analysis. Complete before the first class, do not turn in

Homework Zero Max H. Farrell Chicago Booth BUS41100 Applied Regression Analysis Complete before the first class, do not turn in This homework is intended as a self test of your knowledge of the basic statistical

### Data Analysis and Uncertainty Part 3: Hypothesis Testing/Sampling

Data Analysis and Uncertainty Part 3: Hypothesis Testing/Sampling Instructor: Sargur N. University at Buffalo The State University of New York srihari@cedar.buffalo.edu Topics 1. Hypothesis Testing 1.

### MATH4427 Notebook 3 Spring MATH4427 Notebook Hypotheses: Definitions and Examples... 3

MATH4427 Notebook 3 Spring 2016 prepared by Professor Jenny Baglivo c Copyright 2009-2016 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MATH4427 Notebook 3 3 3.1 Hypotheses: Definitions and Examples............................

### Contents. TTM4155: Teletraffic Theory (Teletrafikkteori) Probability Theory Basics. Yuming Jiang. Basic Concepts Random Variables

TTM4155: Teletraffic Theory (Teletrafikkteori) Probability Theory Basics Yuming Jiang 1 Some figures taken from the web. Contents Basic Concepts Random Variables Discrete Random Variables Continuous Random

### 0.1 Solutions to CAS Exam ST, Fall 2015

0.1 Solutions to CAS Exam ST, Fall 2015 The questions can be found at http://www.casact.org/admissions/studytools/examst/f15-st.pdf. 1. The variance equals the parameter λ of the Poisson distribution for

### Bivariate Distributions

Chapter 4 Bivariate Distributions 4.1 Distributions of Two Random Variables In many practical cases it is desirable to take more than one measurement of a random observation: (brief examples) 1. What is

### Multiple Choice Questions

Multiple Choice Questions 1. A fast- food restaurant chain with 700 outlets in the United States describes the geographic location of its restaurants with the accompanying table of percentages. A restaurant

### 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

### 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.

### Lecture 18 Chapter 6: Empirical Statistics

Lecture 18 Chapter 6: Empirical Statistics M. George Akritas Definition (Sample Covariance and Pearson s Correlation) Let (X 1, Y 1 ),..., (X n, Y n ) be a sample from a bivariate population. The sample

### Chi-Square & F Distributions

Chi-Square & F Distributions Carolyn J. Anderson EdPsych 580 Fall 2005 Chi-Square & F Distributions p. 1/55 Chi-Square & F Distributions... and Inferences about Variances The Chi-square Distribution Definition,

### P (x) 0. Discrete random variables Expected value. The expected value, mean or average of a random variable x is: xp (x) = v i P (v i )

Discrete random variables Probability mass function Given a discrete random variable X taking values in X = {v 1,..., v m }, its probability mass function P : X [0, 1] is defined as: P (v i ) = Pr[X =

### Statistics 100A Homework 6 Solutions

Chapter 5 Statistics A Homework Solutions Ryan Rosario 3. The time in hours) required to repair a machine is an exponential distributed random variable with paramter λ. What is Let X denote the time in

### E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

### 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..........................................

### Common probability distributionsi Math 217/218 Probability and Statistics Prof. D. Joyce, 2016

Introduction. ommon probability distributionsi Math 7/8 Probability and Statistics Prof. D. Joyce, 06 I summarize here some of the more common distributions used in probability and statistics. Some are

### Introduction to Probability

Introduction to Probability EE 179, Lecture 15, Handout #24 Probability theory gives a mathematical characterization for experiments with random outcomes. coin toss life of lightbulb binary data sequence

### POL 571: Expectation and Functions of Random Variables

POL 571: Expectation and Functions of Random Variables Kosuke Imai Department of Politics, Princeton University March 10, 2006 1 Expectation and Independence To gain further insights about the behavior

### Math 576: Quantitative Risk Management

Math 576: Quantitative Risk Management Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 4 Haijun Li Math 576: Quantitative Risk Management Week 4 1 / 22 Outline 1 Basics

### L10: Probability, statistics, and estimation theory

L10: Probability, statistics, and estimation theory Review of probability theory Bayes theorem Statistics and the Normal distribution Least Squares Error estimation Maximum Likelihood estimation Bayesian

### 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.)

### MA 485-1E, Probability (Dr Chernov) Final Exam Wed, Dec 10, 2003 Student s name Be sure to show all your work. Full credit is given for 100 points.

MA 485-1E, Probability (Dr Chernov) Final Exam Wed, Dec 10, 2003 Student s name Be sure to show all your work. Full credit is given for 100 points. 1. (10 pts) Assume that accidents on a 600 miles long

### 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

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

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

### PROBABILITY. Chapter Overview Conditional Probability

PROBABILITY Chapter. Overview.. Conditional Probability If E and F are two events associated with the same sample space of a random experiment, then the conditional probability of the event E under the

### Random Variables of The Discrete Type

Random Variables of The Discrete Type Definition: Given a random experiment with an outcome space S, a function X that assigns to each element s in S one and only one real number x X(s) is called a random

### Probability is concerned with quantifying the likelihoods of various events in situations involving elements of randomness or uncertainty.

Chapter 1 Probability Spaces 11 What is Probability? Probability is concerned with quantifying the likelihoods of various events in situations involving elements of randomness or uncertainty Example 111

### MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS

MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS CONTENTS Sample Space Accumulative Probability Probability Distributions Binomial Distribution Normal Distribution Poisson Distribution

### 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

### ( ) = 1 x. ! 2x = 2. The region where that joint density is positive is indicated with dotted lines in the graph below. y = x

Errata for the ASM Study Manual for Exam P, Eleventh Edition By Dr. Krzysztof M. Ostaszewski, FSA, CERA, FSAS, CFA, MAAA Web site: http://www.krzysio.net E-mail: krzysio@krzysio.net Posted September 21,

### 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

### Continuous Random Variables. and Probability Distributions. Continuous Random Variables and Probability Distributions ( ) ( ) Chapter 4 4.

UCLA STAT 11 A Applied Probability & Statistics for Engineers Instructor: Ivo Dinov, Asst. Prof. In Statistics and Neurology Teaching Assistant: Neda Farzinnia, UCLA Statistics University of California,

### Probability and Statistics

CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be

### Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics

Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in

### 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

### Math/Stat 370: Engineering Statistics, Washington State University

Math/Stat 370: Engineering Statistics, Washington State University Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 2 Haijun Li Math/Stat 370: Engineering Statistics,

### Probability and Statistics Vocabulary List (Definitions for Middle School Teachers)

Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) B Bar graph a diagram representing the frequency distribution for nominal or discrete data. It consists of a sequence

### Statistical tables for use in examinations

Statistical tables for use in examinations Royal Statistical Society 2004 Contents Table Page 1 Binomial Cumulative Distribution Function 3 2 Poisson Cumulative Distribution Function 4 3 Normal Cumulative

### Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the

### Test Code MS (Short answer type) 2004 Syllabus for Mathematics. Syllabus for Statistics

Test Code MS (Short answer type) 24 Syllabus for Mathematics Permutations and Combinations. Binomials and multinomial theorem. Theory of equations. Inequalities. Determinants, matrices, solution of linear

### 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

### Math 421: Probability and Statistics I Note Set 2

Math 421: Probability and Statistics I Note Set 2 Marcus Pendergrass September 13, 2013 4 Discrete Probability Discrete probability is concerned with situations in which you can essentially list all the

### Probabilities and Random Variables

Probabilities and Random Variables This is an elementary overview of the basic concepts of probability theory. 1 The Probability Space The purpose of probability theory is to model random experiments so

### The Chi-Square Distributions

MATH 183 The Chi-Square Distributions Dr. Neal, WKU The chi-square distributions can be used in statistics to analyze the standard deviation " of a normally distributed measurement and to test the goodness

### 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...................................

### Math 135A Homework Problems

Math 35A Homework Problems Homework Fifteen married couples are at a dance lesson Assume that male-female) dance pairs are assigned at random a) What is the number of possible assignments? b) What is the

### Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam

Hypothesis Testing 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 3 2. Hypothesis Testing... 3 3. Hypothesis Tests Concerning the Mean... 10 4. Hypothesis Tests

### Introduction to Monte-Carlo Methods

Introduction to Monte-Carlo Methods Bernard Lapeyre Halmstad January 2007 Monte-Carlo methods are extensively used in financial institutions to compute European options prices to evaluate sensitivities

### EE 302 Division 1. Homework 5 Solutions.

EE 32 Division. Homework 5 Solutions. Problem. A fair four-sided die (with faces labeled,, 2, 3) is thrown once to determine how many times a fair coin is to be flipped: if N is the number that results

### Homework n o 7 Math 505a

Homework n o 7 Math 505a Two players (player A and player B) play a board game. The rule is the following: both player start at position 0. Then, at every round, a dice is thrown: If the result is different

### Chapter 1. Probability, Random Variables and Expectations. 1.1 Axiomatic Probability

Chapter 1 Probability, Random Variables and Expectations Note: The primary reference for these notes is Mittelhammer (1999. Other treatments of probability theory include Gallant (1997, Casella & Berger

### 9740/02 October/November MATHEMATICS (H2) Paper 2 Suggested Solutions. Ensure calculator s mode is in Radians. (iii) Refer to part (i)

GCE Level October/November 8 Suggested Solutions Mathematics H (974/) version. MTHEMTICS (H) Paper Suggested Solutions. Topic : Functions and Graphs (i) 974/ October/November 8 (iii) (iv) f(x) g(x)

### Bivariate Unit Normal. Lecture 19: Joint Distributions. Bivariate Unit Normal - Normalizing Constant. Bivariate Unit Normal, cont.

Lecture 19: Joint Distributions Statistics 14 If X and Y have independent unit normal distributions then their joint distribution f (x, y) is given by f (x, y) φ(x)φ(y) c e 1 (x +y ) Colin Rundel April,

### Seminar paper Statistics

Seminar paper Statistics The seminar paper must contain: - the title page - the characterization of the data (origin, reason why you have chosen this analysis,...) - the list of the data (in the table)

### Continuous random variables

Continuous random variables So far we have been concentrating on discrete random variables, whose distributions are not continuous. Now we deal with the so-called continuous random variables. A random

### Introduction to Probability

Motoya Machida January 7, 6 This material is designed to provide a foundation in the mathematics of probability. We begin with the basic concepts of probability, such as events, random variables, independence,

### Sufficient Statistics and Exponential Family. 1 Statistics and Sufficient Statistics. Math 541: Statistical Theory II. Lecturer: Songfeng Zheng

Math 541: Statistical Theory II Lecturer: Songfeng Zheng Sufficient Statistics and Exponential Family 1 Statistics and Sufficient Statistics Suppose we have a random sample X 1,, X n taken from a distribution

### Ching-Han Hsu, BMES, National Tsing Hua University c 2014 by Ching-Han Hsu, Ph.D., BMIR Lab

Lecture 2 Probability BMIR Lecture Series in Probability and Statistics Ching-Han Hsu, BMES, National Tsing Hua University c 2014 by Ching-Han Hsu, Ph.D., BMIR Lab 2.1 1 Sample Spaces and Events Random

### Consider a system that consists of a finite number of equivalent states. The chance that a given state will occur is given by the equation.

Probability and the Chi-Square Test written by J. D. Hendrix Learning Objectives Upon completing the exercise, each student should be able: to determine the chance that a given state will occur in a system

### Lecture 8: Continuous random variables, expectation and variance

Lecture 8: Continuous random variables, expectation and variance Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2012 Lejla Batina Version: spring 2012 Wiskunde

### 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

### 1 Maximum likelihood estimators

Maximum likelihood estimators maxlik.tex and maxlik.pdf, March, 2003 Simplyput,ifweknowtheformoff X (x; θ) and have a sample from f X (x; θ), not necessarily random, the ml estimator of θ, θ ml,isthatθ

### 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 <

### Chi Square for Contingency Tables

2 x 2 Case Chi Square for Contingency Tables A test for p 1 = p 2 We have learned a confidence interval for p 1 p 2, the difference in the population proportions. We want a hypothesis testing procedure

### 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

### Chi-square test Testing for independeny The r x c contingency tables square test

Chi-square test Testing for independeny The r x c contingency tables square test 1 The chi-square distribution HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided

### Chapter 11: Two Variable Regression Analysis

Department of Mathematics Izmir University of Economics Week 14-15 2014-2015 In this chapter, we will focus on linear models and extend our analysis to relationships between variables, the definitions

### Lesson 5 Chapter 4: Jointly Distributed Random Variables

Lesson 5 Chapter 4: Jointly Distributed Random Variables Department of Statistics The Pennsylvania State University 1 Marginal and Conditional Probability Mass Functions The Regression Function Independence

### Chapters 5. Multivariate Probability Distributions

Chapters 5. Multivariate Probability Distributions Random vectors are collection of random variables defined on the same sample space. Whenever a collection of random variables are mentioned, they are

### 3 Multiple Discrete Random Variables

3 Multiple Discrete Random Variables 3.1 Joint densities Suppose we have a probability space (Ω, F,P) and now we have two discrete random variables X and Y on it. They have probability mass functions f

### Topic 8 The Expected Value

Topic 8 The Expected Value Functions of Random Variables 1 / 12 Outline Names for Eg(X ) Variance and Standard Deviation Independence Covariance and Correlation 2 / 12 Names for Eg(X ) If g(x) = x, then

### Solution Using the geometric series a/(1 r) = x=1. x=1. Problem For each of the following distributions, compute

Math 472 Homework Assignment 1 Problem 1.9.2. Let p(x) 1/2 x, x 1, 2, 3,..., zero elsewhere, be the pmf of the random variable X. Find the mgf, the mean, and the variance of X. Solution 1.9.2. Using the

### Answers to some even exercises

Answers to some even eercises Problem - P (X = ) = P (white ball chosen) = /8 and P (X = ) = P (red ball chosen) = 7/8 E(X) = (P (X = ) + P (X = ) = /8 + 7/8 = /8 = /9 E(X ) = ( ) (P (X = ) + P (X = )

### Statistics revision. Dr. Inna Namestnikova. Statistics revision p. 1/8

Statistics revision Dr. Inna Namestnikova inna.namestnikova@brunel.ac.uk Statistics revision p. 1/8 Introduction Statistics is the science of collecting, analyzing and drawing conclusions from data. Statistics