QUESTION 1

a) There are two independent multiple-choice quizzes where quiz 1 has eight questions and quiz 2 has 15 questions. Each question in the first quiz has four choices and each question in the second quiz has five choices. Suppose a student answers the questions in the quizzes by pure guessing.

i. What is the probability that at most three questions must be answered to obtain the first correct answer in quiz 1? Interpret its value.

ii. What is the probability that less than four correct answers in quiz 2?

iii. In order to get high score, Siti needs to obtain at least six correct answers in quiz 1. Is Siti likely to get the high score in quiz 1?

iv. What is the probability that obtains five correct answers before eighth questions in quiz 2? Interpret its value.

v. Based on your answer in (iv), would you consider this event likely to occur? Explain your reason.

b) Among 25 silver dollars struck in 1903 there are 15 from the Philadelphia mint, 7 from the New Orleans mint, and 3 from the San Francisco mint. If 5 of these silver dollar are picked at random, find the probability of getting i. between 2 and 5 are from the Philadelphia mint.

ii. from 0 to 2 are from the New Orleans mint.

iii. 4 from the Philadelphia mint and 1 from the New Orleans mint.

QUESTION 2

a) Suppose a random variable X whose probability density is given by

i. Find P(1.5 < X < 2.5).

ii. Compute the expected value and variance of X.

iii. Compute the mean and variance of Y where Y = 2 + 3x/5.

iv. Suppose x₁ and x₂ are independent random variables where both are follow the probability density as X, what is the probability that x₁ is at most 1.5 and x2 is at least 2.5? Interpret its value.

b) Suppose that the cumulative density function of a random variable X is as follows:

i. Find the probability density of X.

ii. Compute the average and standard deviation of X.

c) For each of the following moment generating functions, name the associated probability density function with correct parameter. Then, compute the P(X > 5).

i. M_x(t) = (1 - 0.8t)^-1, t ≤ 1.25

ii. M_x(t) = (1 - 7t)^-20

QUESTION 3

a) Two caplets are selected at random from a bottle containing 3 aspirin, 2 sedative and 4 laxative caplets. If X and Y are, respectively, the numbers of aspirin and sedative caplets included among the 2 caplets drawn from the bottle. The following table shows the probability distribution of X and Y.

i. What is the probability that there are less than 2 aspirin and at most 1 sedative caplet?

ii. What is the probability that there are at least 1 aspirin and less than 2 sedative caplets?

iii. Construct the marginal probability distribution of X and Y.

iv. Suppose an aspirin and a sedative is worth RM 0.90 and RM 1.50, respectively. What is the total expected cost needed if randomly two caplets are selected?

b) Given the joint probability density function of

two random variables X and Y.

i. Develop the marginal density function of X.
ii. Develop the conditional density function of Y given X=x.
iii. Are X and Y independent random variables? Justify your answer.
iv. Determine the correlation coefficient between X and Y. Interpret its value.

c) If the random variables X and Y have the means μ_x = 3 and μ_y = 5, the variances σ_x² = 8 and, σ_y² = 12 and cov(X,Y) = 1, find the covariance of U = X + 4Y and V = 3X - Y

QUESTION 4

a) A soft-drink vending machine is set so that the amount of drink dispensed is a random variable with a mean of 200 milliliters and a standard deviation of 15 milliliters.

i. What is the probability that the average amount dispensed in a random sample of size 10 is at least 204 milliliters?

ii. What is the probability that the average amount dispensed in a random sample of size 22 is at least 204 milliliters?

iii. What is the probability that the average amount dispensed in a random sample of size 36 is at least 204 milliliters?

iv. From the answers (i) to (iii), explain your findings.

v. What is the probability that the mean amount dispensed in a random sample of size 36 is less than 0.195 liters?

vi. What is the probability that the mean amount dispensed in a random sample of size 36 is between 198 milliliters and 205 milliliters? Interpret its value.

b) It is found that the lifetimes of television tubes manufactured by a company have a normal distribution with a mean of 2000 hours and a standard deviation of 60 hours. If 10 tubes are selected at random, find the probability that the sample standard deviation will

i. Not exceed 50 hours.

ii. Based on your answer in (i), would you consider this event likely to occur?

Explain your reason.

c) Two samples of sizes 8 and 10 are drawn from two normally distributed populations having variances 21.8 and 36, respectively. Find the probability that the variance of the first sample is more than twice the variance of the second sample.