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EE 126 Fall 2006 Midterm #1: Probability Theory and Random Variables, Exams of Probability and Statistics

Alagappa UniversityProbability and Statistics

The instructions and problems for the midterm #1 exam of ee 126, a fall 2006 course focused on probability theory and random variables. The exam consists of three problems that involve computing joint and conditional probability mass functions, determining expected values and variances, and analyzing waiting times for buses.

Typology: Exams

2012/2013

Uploaded on 03/22/2013

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EE 126 Fall 2006 Midterm #1
Thursday October 6, 7–8:30pm
• You have 90 minutes to complete the quiz.
• Write your solutions in the exam booklet. We will not consider any work not in the exam
booklet.
• This quiz has three problems that are in no particular order of difficulty.
• You may give an answer in the form of an arithmetic expression (sums, products, ratios, factori-
als) of numbers that could be evaluated using a calculator.
• A correct answer does not guarantee full credit and a wrong answer does not guarantee loss of
credit. You should concisely indicate your reasoning and show all relevant work. The grade on
each problem is based on our judgment of your level of understanding as reflected by what you
have written.
• This is a closed-book exam except for one single-sided, handwritten, 8.5 × 11 formula sheet
plus a calculator.
• Be neat! If we can’t read it, we can’t grade it.
• At the end of the quiz, turn in your solutions along with this quiz (this piece of paper).
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Download EE 126 Fall 2006 Midterm #1: Probability Theory and Random Variables and more Exams Probability and Statistics in PDF only on Docsity!

EE 126 Fall 2006 Midterm

Thursday October 6, 7–8:30pm

• You have 90 minutes to complete the quiz.

  • Write your solutions in the exam booklet. We will not consider any work not in the exam booklet.
  • This quiz has three problems that are in no particular order of difficulty.
  • You may give an answer in the form of an arithmetic expression (sums, products, ratios, factori als) of numbers that could be evaluated using a calculator.
  • A correct answer does not guarantee full credit and a wrong answer does not guarantee loss of credit. You should concisely indicate your reasoning and show all relevant work. The grade on each problem is based on our judgment of your level of understanding as reflected by what you have written.
  • This is a closedbook e xam except for one singles ided, handwritten, 8.5 × 11 formula sheet plus a calculator.
  • Be neat! If we can’t read it, we can’t grade it.
  • At the end of the quiz, turn in your solutions along with this quiz (this piece of paper).

Consider the following game: first a coin with P(heads) = q is tossed once. If the coin comes up tails, then you roll a 4sided die ; otherwise, you roll a 6sided die. You win the amount of money (in dollars $) corresponding to the given die roll. Let X be an indicator random variable for the coin toss (X = 0 if toss is tails; X = 1 if toss is heads), and let Y be the random variable corresponding to the amount of money that you win. (a) (3pt) Compute the joint PMF pX,Y. (It will be a function of q). (b) (4pt) Compute the conditional PMF pX | Y , again as a function of q. Supposing that it is known that (on some trial of this game) you made 2$ or less, determine the probability that the initial coin toss was heads, as a function of q. (c) (3pt) Assume that you have have to pay 3$ each time that you play this game. Determine, as a function of q, how much money you will win or lose on average. For what value of q do you break even?

John can either walk to school (which takes 25 min), or take the bus (the bus takes 10 min). However, the buses don’t have a fixed schedule. Instead, there is probability p that a bus will arrive on each even numbered minute (e.g., t = 0, 2, 4,.. .). If John goes to the bus stop, then he always arrives at some oddn umbered minute (e.g., t = 1, 3, 5.. .). Buses never arrive at an oddnu mbered minute. (a) (2 pt) Let X be a random variable associated with the time between two consecutive buses. Find the expected value E[X]. (b) (3 pt) What is the expected time it takes to get to school if John goes by bus (including both the waiting time at the bus stop, and driving time)? Now suppose that John has no idea what p is, so that his strategy is to flip a fair coin: if the coin is heads, he walks, if the coin is tails, he waits for the bus. (c) (3 pt) Letting Y be the total time it takes to get to school, find the PMF of Y and compute E[Y ]. (d) (3 pt) We are interested in the variance of Y. John’s friend Bob gives the following argument: “Let v1 be the variance of the time needed to go to school if John walks, and v2 the variance of the time needed if he waits for the bus. Because John has equal chances of walking or taking the bus, the variance of Y is just the average of v1 and v2 ”. Is Bob right? Explain why or why not. (In doing so, you are not required to find the variance of X). For the following two parts, suppose that John always decides to take the bus. (e) (4 pt) Let Znext be a discrete random variable corresponding to the time (in minutes) that elapses from John’s arrival at the bus stop until the next bus comes, and Zlast a random variable associated with the time by which John missed the last bus. Compute the expected values E[Znext ] and E[Zlast ]. (f) (3 pt) One might that expect E[X] = E[Znext ] + E[Zlast ] (see part (a) for the definition of X). Explain why this is not true.