MATH 180A Introduction to Probability Homework 2 (due 10/09/06)
1. After asking many individuals across the nation, a researcher comes with the following conclusion: 65% like dogs 55% like cats 25% like both (a) What is the proportion of people tha
180A HW 3 Solutions
October 25, 2006
1) On a multiple-choice quiz with 3 possible answers for each of the 10 questions, what is the expected number of correct answers a student would get just by guessing? Answer Let X denote the number of correct an
MATH 180A Introduction to Probability Review for Midterm 1 (NOT due)
1. If 8 castles (i.e. rooks) are randomly placed on a chessboard, what are the chances that no rook can capture any of the others? In other words, what is the probability that NO
180A HW 1 Solutions
October 4, 2006
1) A coin is tossed four times resulting in a sequence of heads and tails. Let A: exactly two heads B: heads and tails alternate C: first two tosses are headed. (a) Which events, if any, are mutually exclusive? (b
MATH 180A Introduction to Probability Review for Midterm 2 (NOT due)
1. At the mayor election in a very large city, 51% of the people support candidate A, while 49% of the people support B. Use the normal approximation to the binomial distribution t
180A HW 6 Solutions
November 19, 2006
1) Consider an urn with 1 red, 2 green and 3 blue balls. 5 balls are drawn from that urn without replacement. Let X be the number of red balls picked and Y the number of green balls picked. Find the joint distri
MATH 180A Introduction to Probability Homework 6 (due 11/20/06)
1. Consider an urn with 1 red, 2 green and 3 blue balls. 5 balls are drawn from that urn without replacement. Let X be the number of red balls picked and Y the number of green balls pi
180 Midterm 1 Solutions
October 17, 2006
1) Three dice are rolled. Let A: all numbers are even B: all numbers are equal C: there is at least one 6 Compute P (A B), P (B C), P (A B C ), and P (A B C c ). Answer Note that P (A B) = P (A) + P (B)
180 Midterm 2 Review Solutions
1) At the mayoral election in a very large city, 51% of the people support candidate A, while 49% support B. Use the normal approximation to the binomial distribution to find a numerical expression for the probability t
180 Midterm 1 Review Solutions
October 10, 2006
1) If 8 castles (i.e. rooks) are randomly placed on a chessboard, what are the chances that no rook can capture any of the others? In other words, what is the probability that no row or column contains
180 Final Solutions
December 4, 2006
1) Consider 0, 2 , f (x) = 91 x0 0<xa , a<x x2
What value(s) of a R make(s) f a probability density function? Answer Note that if f is a PDF
a
1 =
0
2 dx + 9
a a 0
1 dx x2
= =
2 a x + 9 0 2 1 a+ 9 a
MATH 180A Introduction to Probability Review for the Final (NOT due)
Please review homework sets 1-7, midterms 1 & 2, midterm reviews 1 & 2. The following problems focus on what was covered since midterm 2. 1. Two balls are drawn with replacement fr
180A HW 6 Solutions
November 30, 2006
1) Suppose X has the exponential distribution with parameter and Y has the exponential distribution with parameter . Assume X and Y are independent. Let Z = X/(Y + 1). (a) Compute E[Z]. (b) Compute the PDF of Z
MATH 180A Introduction to Probability Homework 7 (due Wedenesday, 11/29/06)
1. Suppose X has the exponential distribution with parameter and Y has the exponential distribution with parameter . Assume X and Y are independent. Let Z = X/(Y + 1). (a)
MATH 180A Introduction to Probability Homework 5 (due 11/06/06)
1. Suppose X has the uniform distribution on (0, 1). Compute the probability density function and expected value of (a) X , where > 0; (b) log(X); (c) exp(X); (d) sin(2X). 2. A coin
MATH 180A Introduction to Probability Homework 4 (due 10/30/06)
1. Jimmy rides his old motorcycle every day of the year. The motorcycle starts with probability 1/2 each time Jimmy presses on the starter. Compute the probability that in a given year
MATH 180A Introduction to Probability Homework 3 (due 10/25/06)
1. On a multiple-choice quiz with 3 possible answers for each of the 10 questions, what is the expected number of correct answers a student would get just by guessing? 2. Two marbles a
MATH 180A Introduction to Probability Homework 1 (due 10/02/06)
1. A coin is tossed four times. Let A: exactly two heads B: heads and tails alternate C: first two tosses are heads (a) Which events, if any, are mutually exclusive? (b) Which events,
180A HW 5 Solutions
1) Suppose X has the uniform distribution on (0, 1). Compute the probability density function and expected value of: (a)X , where > 0 (b) log(X); (c)exp(X); (d) sin(2X) answer Let Y = X . Then, clearly: FY (y) = P (X y) = P
180 HW 4 Solutions
1) Jimmy rides his old motorcycle every day of the year. The motorcycle starts with probability 1/2 each time Jimmy presses on the starter. Compute the probability that in a given year (365 days) Jimmy has to press the starter at
Math 180A
Homework 2 Solutions
October 8, 2006
Problem 1
a) P (A B)C ) = 1 - P (A B) = 1 - (P (A) + P (B) - P (A C) = 1 - (.65 + .55 - .25) = .05 b) P (A B C ) + P (B AC ) = P (A) - P (A B) + P (B) - P (A B) = .65 - .25 + .55 - .25 = .70 c)