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MATH 3080 Final Exam: Probability Questions, Exams of Probability and Statistics

Alliance UniversityProbability and Statistics

The instructions and problems for the final exam of the math 3080 probability course from autumn 2008. The exam covers various topics in probability theory, including conditional probability, random variables, and density functions. Students are required to solve problems using the textbook and without the use of notes or external sources.

Typology: Exams

2012/2013

Uploaded on 02/20/2013

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Introduction to Probability Name:
MATH 3080 (Autumn 2008)
Final Exam
Instructions:
•There are 105 points on the exam and an extra credit problem worth 10 points. 100 points
counts as perfect, but, as always, points over 100 will help on your final grade.
•You may use your class textbook. No notes, calculators, computers, cell phones, consultants,
or any other source material on the exam. Please turn offyour cell phones. You may not use
your cell phone (to send or receive text messages, for example) during the exam.
•You must show your work. No credit for correct answers alone.
•Simplify all answers. That is, your answers should be in terms of fractions (or in problem 2,
a decimal) and there should be no binomial coefficients such as µ12
7left.
Please do not write below this line.
Problem Poi nts
possible
Points
earned
110
210
315
415
517
6 8
710
810
910
10
Extra Credit 10
Tota l
pf3
pf4
pf5
pf8
pf9
pfa

Partial preview of the text

Download MATH 3080 Final Exam: Probability Questions and more Exams Probability and Statistics in PDF only on Docsity!

Introduction to Probability Name: MATH 3080 (Autumn 2008) Final Exam

Instructions:

  • There are 105 points on the exam and an extra credit problem worth 10 points. 100 points counts as perfect, but, as always, points over 100 will help on your final grade.
  • You may use your class textbook. No notes, calculators, computers, cell phones, consultants, or any other source material on the exam. Please turn off your cell phones. You may not use your cell phone (to send or receive text messages, for example) during the exam.
  • You must show your work. No credit for correct answers alone.
  • Simplify all answers. That is, your answers should be in terms of fractions (or in problem 2, a decimal) and there should be no binomial coefficients such as

μ 12 7

left.

Please do not write below this line.

Problem

Points possible

Points earned 1 10

2 10

3 15

4 15

5 17

6 8

7 10

8 10

9 10 10 Extra Credit 10

Total

  1. (10 points) An urn contains 3 red and 3 white balls. 2 balls are selected at random (without replacement) from the urn. Find the probability that at the sample contains one red ball and one white ball.
  2. (10 points) A fair coin, is flipped independently 400 times. The random variable X counts the number of heads in these 400 flips. Using the DeMoivre-Laplace limit theorem, approximate P { 190 ≤ X ≤ 215 }. (You will want to use the table on page 222 of the text. You need not use the continuity correction in this problem, just to make the computations easier.)
  1. (15 points) Two urns (a left urn and a right urn) contain 3 balls each. An urn is selected at random and a ball is picked from that urn. This process is repeated until one of the urns is empty. The random variable X measures the number of balls remaining in the other urn. Find the probability mass function p of X. Then, compute the values of p (k) for each of the values k that X can assume, and fill in the entries of this table with your answer.

p (k)

k

  1. Throughout this problem X is a continuous random variable having density function

f (x) =

cx if 0 < x < 2

0 otherwise

x

y

0 2 Graph of f

(a) (3 points) Find c.

(b) (3 points) Find P {X ≤ 1 }.

(c) (3 points) Find E (X) and var(X).

(d) (8 points) Find the density function fY (x) of the random variable Y = X^2.

  1. (10 points) A stick of length 1 is split at a point X which has density function

f (x) =

3 x^2 if 0 < x < 1

0 otherwise

x

y

Graph of f

Determine the expected length of the piece that contains the point

  1. (10 points) A red card is removed from a regular deck of 52 cards, leaving 25 red cards and 26 black cards. 13 cards are then drawn at random from the deck and found to be the same color. Find the (conditional) probability that all the cards are black.
  1. (Extra Credit: 10 points) Jane and Bob take turns flipping biased coins, independently. Jane

goes first. Jane’s coin turns up heads with probability

and Bob’s turns up heads with

probability

The game ends when either Jane or Bob flip a heads. Let X be the random variable which measures the trial when the game ends.

(a) (This part is worth 5 points) Determine the probability mass function p (x) of X.

(b) (This part is also worth 5 points) How many coin flips do we expect to wait until either Bob or Jane flips a heads? Note: This part is quite computationally intensive! I’ll give quite a bit of partial credit for good progress.