Math 318 Assignment 9 This assignment is due at the beginning of class on Friday April 3 All problems are from Chapter 4 of Ross and, unless otherwise indicated, have the same numbers in editions 8,9. Some have been reworded to make them slightly less hos
Math 318 Assignment 7 This assignment is due at the beginning of class on March 11 Test 2, to be held during class on March 18, will be based on the material covered in Assignments 57. 1. Recall that (k) = E eikX1 is the characteristic function of a singl
MATH 318: HW #2 KEY
Total marks = [26]. (Questions labelled with [*] are not for marks.)
(1) By the independence of A, B, we have that P (A \ B) = P (A)P (B).
(a) [3] Since A is the disjoint union of A \ B and A \ B c , we have that P (A) =
P (A \ B) + P
MATH 318: HW #1 KEY
Total marks = [30]. (Questions labelled with [*] are not for marks.)
(1) [2] The two marbles can be any of R, G, B. Hence
S = cfw_(R, R), (R, G), (R, B), (G, R), (G, G), (G, B), (B, R), (B, G), (B, B).
Since |S| = 9 and each event has
MATH 318: HOMEWORK #4
DUE FRIDAY, FEBRUARY 5, AT THE START OF CLASS
Late assignments will not be accepted
1. Test #1
Midterm Test #1 (worth 20% of your overall grade) will be held 22:50pm on Wednesday,
February 10, in Buchanan A201. Please arrive on time,
MATH 318: HOMEWORK #5
DUE FRIDAY, FEBRUARY 26, AT THE START OF CLASS
Late assignments will not be accepted
1. Assigned Problems
(1) Let cfw_Nt : t 0 be a Poisson process with rate . Let Sn , n 1, denote the time
of the nth event. Find
(a) E(N5 )
(b) E(S3
MATH 318: HW #3 KEY
Total marks = [30]. (Questions labelled with [*] are not for marks.)
(1) (a) [*] If T Uni(1, 9), then its pdf is f (t) = 1/8 for t 2 [1, 9] (and 0 otherwise).
Hence P (T t) = (9 t)/8, for t 2 [1, 9]. Therefore the probability that the
MATH 318: HOMEWORK #6
DUE FRIDAY, MARCH 4, AT THE START OF CLASS
Late assignments will not be accepted
1. Assigned Problems
P
(1) Let Z1 , Z2 , . . . , Zn be iid standard normal random variables. Put Y = ni=1 Zi2 .
Note: Y is said to have the chi-squared
MATH 318: HOMEWORK #7
DUE FRIDAY, MARCH 11, AT THE START OF CLASS
Late assignments will not be accepted
1. Assigned Problems
(1) Twelve measurements of the percentage of water in a methanol solution yielded
= 0.547 and a sample variance S 2 = (0.032)2 .
MATH 318: HW #4 KEY
Total marks = [30]. (Questions labelled with [*] are not for marks.)
(1) Recall that V Uni(3, 4), so fV (v) = 1 for t 2 [3, 4] (and 0 otherwise). In part
(b), we found fT (t) = (2t) 1/2 for t 2 [9/2, 8] (and 0 otherwise) where T = 12 V
Math 318 The continuity theorem
I quoted this theorem in class while proving the law of large numbers and the
central limit theorem. It is proved in 5.9 of the book Probability and Random
Processes by Grimmett and Stirzaker.
Theorem 1 Suppose that we have
Math 318 Assignment 3
This assignment is due at the beginning of class on January 28.
1. The arrival time of the next job at a server is an exponential random variable with parameter
1
2 . No job has arrived by time 3. What is the probability no job will
MATH 318: HOMEWORK #3
DUE FRIDAY, JANUARY 29, AT THE START OF CLASS
Late assignments will not be accepted
1. Assigned Problems
(1) (a) The time T in hours taken to repair a washing machine has a Uniform(1, 9)
distribution. What is the probability the repa
MATH 318: HOMEWORK #2
DUE FRIDAY, JANUARY 22, AT THE START OF CLASS
Late assignments will not be accepted
1. Assigned Problems
(1) Suppose that A, B are independent events.
(a) Show that A, B c are independent.
(b) Show that Ac , B c are independent. Hint
Math 318 Assignment 8, This assignment is due at the beginning of class on Friday March 27 1. Ross: 4, #8. Let Xn be the number of the coin flipped on day n. Then (Xn ) is a Markov chain with .7 .3 transition matrix P = .6 .4 and initial distribution X0 =
MATH 318 Test #2 Wednesday, March 21, 2007. Dr. D. Brydges and S. Kliem Table 1: Mean and Variances
Distribution Bin (n, p) Geometric (p) Poisson () Uniform (a, b) Exp () Mean np
1 p
Variance np(1 - p)
1-p p2
a+b 2 1
(b-a)2 12 1 2
Table 2: cdf of normal
Math 318 Assignment 6 This assignment is due at the beginning of class on March 4. 1. A "Grow Your Own Tomatoes" kit contains an unknown number of k seeds (mixed into the soil). It is known that each seed will grow into a plant with probability 1/2. Suppo
MATH 318 Test #1 Wednesday, February 14, 2007. Dr. D. Brydges and S. Kliem
Table 1: Mean and Variances
Distribution Bin (n, p) Geometric (p) Poisson () Uniform (a, b) Exp () Mean np
1 p
Variance np(1 - p)
1-p p2
a+b 2 1
(b-a)2 12 1 2
Table 2: cdf of norm
Math 318 Assignment 5 This assignment is due at the beginning of class on Wednesday Feb 25 1. Suppose that X1 , . . . , Xn are independent Gaussian random variables with mean and n standard deviation . Let Yn = n-1 i=1 Xi denote the average of the Xi 's.
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Math 318 Summary Combinations: # of ways to choose a subset of elements from a set of elements ! = ! - ! # of ways to choose a sequence of elements from a set of elements ! - ! = all possible outcomes A function P defined on all events that satisfy the fo
1
The relation between Exponential and Poisson Random Variables.
Recall that X Exp() means that the density of X is e-x . We have a server which is working on a queue of jobs. Let X1 be the time it takes to do the first one, X2 the time for the second, an
MATH 318: HOMEWORK #1
DUE FRIDAY, JANUARY 15, AT THE START OF CLASS
Late assignments will not be accepted
1. Assigned Problems
In all solutions involving permutations and combinations, be sure to briefly explain all
factors arising in your solution.
(1) R
Math 318 Assignment 4,
This assignment is due at the beginning of class on Wednesday Feb. 4.
Test 1, to be held during the usual class hour on Wed, Feb. 11, will be based on the material
covered in Assignments 1-4.
1. A one dimensional box [a, b] R contai