Math Answers

Determine the unique solution of the following differential equations by using Laplace transforms:

y''(t)-6y'(t)+9y(t) = t²e^3t if y'(0) = 6 and y(0) = 2

Determine L^-1{e^s(s²-1/(s²+1)²)}

6. According to a study from 2012, it was found that 19% of people use their browser in private mode. For a new study on privacy policies, you will consider a random sample of 200 people. Let X be the number of people who use their browser in private mode.

(a) What is the expected value (mean) and standard deviation for X?

(b) Use a normal approximation to estimate the probability that you will find 50 or more people using their browser in private mode in your sample.

(c) Use a normal approximation to estimate the probability that you will find more than 35 and up to 45 people (inclusively) using their browser in private mode in your sample.

4. Find the value of the number a such that P(0.43 < Z < a) = 0.1359.

5. Suppose that when playing blackjack with a $5 bet, the average (mean) net earning is −$0.25 with a standard deviation of $4.76. You are planning to play blackjack 40 times in a row.

(a) Calculate the probability that your average earning per game will be above $1 per game.

(b) Calculate the probability that you will have suffered a net loss of money (total earning is negative) after playing 40 times.

3. Professor Brown has been teaching the same university psychology course for many years, and she has been keeping a record of all the grades given to the students. Suppose that her student’s grades follow a normal distribution, with an average of 70 and a standard deviation of 12.

(a) Suppose you will be taking her class next semester. What is the probability that you will get a grade between 70 and 85?

(b) What minimum grade do you need in order to qualify for Professor Brown’s Honour Roll, which is reserved for the top 8% of students?

(c) Suppose this professor will have a group of 25 students next semester. What is the probability that her class average will not be between 65 and 75?

Advanced QM

1. In the past year, Gino’s Pizza Restaurant was profitable on exactly 75% of its business days. During the next 8 business days, what is the probability that Gino’s will be profitable for:

(a) exactly 6 days?

(b) less than 7 days?

2. An insurance company says that 15% of all fires are caused by arson. A random sample of 50 fire insurance claims are under study. Let r be the number of fires that were really started by arson in this sample.

(a) What is the expected number (mean) of arson fires among these 50 fires?

(b) What is the standard deviation of the number of arson fires?

Find all the values of k so that the set{ ( 1,−3,2),(−3,9,−6),(5,−7,k)} form the basis for R³

Estimate the mean weight of 6 students randomly chosen in a university if students 1 weight 53 kg students weight student 2 weight 64 kg student 3 weight 49 kg student 4 weight 59 kg student 5 weight 62 kg and student 6 weight 55 kg

For what values of α are vectors (1,1,2,1), (2,1,2,3), (1,4,2,1) (-1,3,5,α) are linearly inde-

pendent

Let S = {u1,u2,u3} be a basis for the vector space V. Show that T = {w1,w2,w3} is also

a basis for V, where w1 = u1 +u2 +u3, w2 = u2 +u3,w3 = u3.