practice2d - Probability with Applications-Spring 2008...

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Probability with Applications—Spring 2008 Problem Set 2 for ISYE 2027, Section B (February 26, 2008) Purpose. The second exam will focus on material largely covered in Chapters 3 and 4 of Walpole et al. However, this should be considered a cumulative exam in that prior material will be pre- supposed and may be directly treated. In its final version, this document will contain two practice exams and a list of recommended exercises from the textbook. In the lecture periods prior to the exam, I will solve these practice exams at the board, along with additional exercises as time permits. Please see also the document “Study Guide 2,” which provides a theoretical overview of the new material you are responsible for. Also important are your lecture notes and the document “Homework 2 Solutions.” Notes, books, and electronic devices may not be used during the exam. First practice exam. The following formulas are provided without context, as a memory aid: F ( x ) = P ( X x ) for all x R g ( x ) = y f ( x, y ) or g ( x ) = + -∞ f ( x, y ) dy h ( y ) = x f ( x, y ) or h ( y ) = + -∞ f ( x, y ) dx f ( y | x ) = f ( x, y ) /g ( x ) and f ( x | y ) = f ( x, y ) /h ( y ) E ( X ) = x xf ( x ) or E ( X ) = + -∞ xf ( x ) dx Var( X ) = E [( X - E [ X ]) 2 ] = E ( X 2 ) - ( E ( X )) 2 Cov( X, Y ) = E [( X - E [ X ])( Y - E [ Y ])] = E ( XY ) - E ( X ) E ( Y ) E ( aX + b ) = aE ( X ) + b and Var( aX + b ) = a 2 Var( X ) Var( X + Y ) = Var( X ) + Var( Y ) + 2 Cov( X, Y ) 1. Suppose that the set of possible values for the random variable X is the interval of real numbers x satisfying a x b , for some particular a, b R with a < b . (a) What type of random variable is X : discrete or continuous? (b) Suppose that f ( x ) = c for x in the set of possible values for X . If X is discrete, interpret f as a probability mass function. If X is continuous, interpret f as a probability density function. What value must c take? Your final answer should be a simplified expression involving a and b , and it should not involve a sum or an integral. 2. Suppose we are to toss three fair coins. Assume that any given toss is independent of any collection of other tosses. Define the random variable X to be the number of heads we see. Find F ( x ), the cumulative distribution function of X . You must specify a numerical value for F ( x ) in the form of a simplified fraction (where necessary) for all real numbers x R .
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3. Suppose we are to toss two loaded, six-sided dice; one die is red and the other is green. For each die, the probability of seeing any given even number is twice the probability of seeing any given odd number. Assume that the two tosses are independent. Let X be the number that comes up on the red die, and let Y be the number that comes up on the green die. (a) Identify the following statement as either true or false. For all possible values y of Y , the conditional probability mass function of Y given that X = 5 is equal to the marginal pmf of Y .
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