Publications by Bryan Persaud
Data 606 HW 9
1 #11 page 363 The price of one share of stock in the Pilsdorff Beer Company (see Exercise 8.2.12) is given by Yn on the nth day of the year. Finn observes that the differences Xn = Yn+1 − Yn appear to be independent random variables with a common distribution having mean µ = 0 and variance σ2 = 1/4. If Y1 = 100, estimate the probability that...
4954 sym R (221 sym/12 pcs)
Data 605 Discussion 9
Chapter 9 Section 9.2 Exercise 5 A die is thrown until the first time the total sum of the face values of the die is 700 or greater. Estimate the probability that, for this to happen, more than 210 tosses are required. less than 190 tosses are required. between 180 and 210 tosses, inclusive, are required. Solution: First lets calculate the tota...
1639 sym R (205 sym/12 pcs)
Data 605 HW 8
11 A company buys 100 lightbulbs, each of which has an exponential lifetime of 1000 hours. What is the expected time for the first of these bulbs to burn out? (See Exercise 10.) Solution: By looking at exercise 10, we know that if \({ X }_{ i }\sim Exp({ \lambda })\) then \(min{ X }_{ i }\sim Exp(\Sigma { \lambda })\), which gives us: \(\frac { 1...
3163 sym R (128 sym/8 pcs)
Data 605 Discussion 8
Chapter 8 Section 8.1 Exercise 3 Write a program to toss a coin 10,000 times. Let Sn be the number of heads in the first n tosses. Have your program print out, after every 1000 tosses, Sn − n/2. On the basis of this simulation, is it correct to say that you can expect heads about half of the time when you toss a coin a large number of times? So...
991 sym R (814 sym/2 pcs)
Data 605 HW 7
1 Let X1, X2, . . . , Xn be n mutually independent random variables, each of which is uniformly distributed on the integers from 1 to k. Let Y denote the minimum of the Xi’s. Find the distribution of Y . Solution: Let \({ k }^{ n }\) equal to the number of values we can have in \({ X }_{ i }\). Then \({ (k-1) }^{ n }\) is equal to all of the va...
2776 sym R (1119 sym/24 pcs)
Data 605 HW 6
1 A box contains 54 red marbles, 9 white marbles, and 75 blue marbles. If a marble is randomly selected from the box, what is the probability that it is red or blue? Express your answer as a fraction or a decimal number rounded to four decimal places. Solution: round((54 + 75) / (54 + 9 + 75), 4) ## [1] 0.9348 The probability that a randomly sele...
5871 sym R (1030 sym/30 pcs)
Data 605 HW 5
Choose independently two numbers B and C at random from the interval [0, 1] with uniform density. Prove that B and C are proper probability distributions. Note that the point (B,C) is then chosen at random in the unit square. Find the probability that (a) B + C < 1/2. Solution: To solve this problem, we need to set B and C as a function f(B,C) a...
4340 sym
Data 605 Discussion 5
Chapter 1 Section 1.2 Exercise 8 A student must choose one of the subjects, art, geology, or psychology, as an elective. She is equally likely to choose art or psychology and twice as likely to choose geology. What are the respective probabilities that she chooses art, geology, and psychology? Solution: The probabilities of art, geology, and psyc...
1165 sym
Data 605 HW 4
Problem Set 1 matrix A: \(\begin{bmatrix} 1 & 2 & 3 \\ -1 & 0 & 4 \end{bmatrix}\) A <- matrix(c(1, -1, 2, 0, 3, 4), nrow = 2) Compute \(x\ =A{ A }^{ T }\) and \(y = { A }^{ T }A\) Solution: X = A %*% t(A) # Multiply matrix A by the transpose of matrix A X ## [,1] [,2] ## [1,] 14 11 ## [2,] 11 17 Y = t(A) %*% A # Multiply the trans...
1172 sym R (3520 sym/39 pcs)
Data 605 Discussion 4
Chapter Representations Section MR Exercise C21 Find a matrix representation of the linear transformation T relative to the bases B and C. \(T:P_{ 2 }\rightarrow C^{ 2 },T(p\left( x \right) )=\begin{bmatrix} p(1) \\ p(3) \end{bmatrix}\) \(B = \left\{ 2 - 5x + x^ 2, 1 + x - x^ 2, x^ 2 \right\}\) \(C = \left\{ \begin{bmatrix} 3 \\ 4 \end{bmatrix}, ...
1309 sym