Extending the variance of the sum rule. For mathematical convenience we first extend the sum rule to three random variables with zero

expectation. Next we further extend the rule to three random variables with

nonzero expectation. By the same line of reasoning we extend the rule to n

random variables.

a. Let X, Y and Z be random variables with expectation 0. Show that

Var(X + Y + Z) = Var(X) + Var(Y ) + Var(Z)

+ 2Cov(X, Y ) + 2Cov(X, Z) + 2Cov(Y,Z).

Hint: directly apply that for real numbers y1,...,yn

(y1 + + yn)

2 = y2

1 + + y2

n + 2y1y2 + 2y1y3 + + 2yn?1yn.

b. Now show a for X, Y , and Z with nonzero expectation.

Hint: you might use the rules on pages 98 and 141 about variance and

covariance under a change of units.

c. Derive a general variance of the sum rule, i.e., show that if X1, X2,...,Xn

are random variables, then

Var(X1 + X2 + + Xn)

= Var(X1) + +Var(Xn)

+2Cov(X1, X2) + 2Cov(X1, X3) + + 2Cov(X1, Xn)

+ 2Cov(X2, X3) + + 2Cov(X2, Xn)

...

+ 2Cov(Xn?1, Xn).

d. Show that if the variances are all equal to ?2 and the covariances are all

equal to some constant ?, then

Var(X1 + X2 + + Xn) = n?2 + n(n ? 1)?.

10.18 Consider a vase containing balls numbered 1, 2,...,N. We draw

n balls without replacement from the vase. Each ball is selected with equal

probability, i.e., in the first draw each ball has probability 1/N, in the second

draw each of the N ? 1 remaining balls has probability 1/(N ? 1), and so

on. For i = 1, 2,...,n, let Xi denote the number on the ball in the ith draw.

From Exercise 9.18 we know that the variance of Xi equals

Var(Xi) = 1

12(N ? 1)(N + 1).150 10 Covariance and correlation

Show that

Cov(X1, X2) = ? 1

12(N + 1).

Before you do the exercise: why do you think the covariance is negative?

Hint: use Var(X1 + X2 + + XN ) = 0 (why?), and apply Exercise 10.17.

10.19 Derive the alternative expression for the covariance: Cov(X, Y ) =

E[XY ] ? E[X]E[Y ].

Hint: work out (X ? E[X])(Y ? E[Y ]) and use linearity of expectations.

10.20 Determine ?

U, U2

when U has a U(0, a) distribution. Here a is a

positive number.

Question 3. [This question is based on Problem 43, p. 116 in the textbook] (a) Let {A1,A2} be a partition of a sample space and let B be any event. State and prove the Law of Total Probability as it applies to the partition {A1, A2} and the event B. There are two coins in a box; one coin is a fair coin, and the other is a biased coin that comes up heads 75 percent of the time. Fiona chooses a coin at random and tosses it. (b) What is the probability that the chosen coin ips heads? (Hint: The Law of Total Probability.) (c) Given that the chosen coin ips heads, what is the conditional probability that it was the biased coin? (Hint: Bayes' Theorem.) (d) Given that the chosen coin ips tails, what is the conditional probability that it was the biased coin? Table Q1 gives the probabilities that a certain traffic light around Parit Raja will malfunction 0, 1, 2, 3, 4, 5, or 6 times on any one day. Compute the basic descriptive statistics of this probability distribution. Table Q1: Probability distribution function of traffic light malfunctions Number of malfunctions, x 2 3 4 5 6 Probability, f (x) 0.17 0.29 0.27 0.16 0.07 0.03 0.01Q. 3. (5 marks) [CLO 2] At the Business College of UMS, 800 students are enrolled in the Bachelor of Business Administration (BBA). Among them, 200 students enrolled in an Introductory Statistics course. Of these 200 students, 50 are also enrolled in an Introductory Accounting course. There are an additional 250 business students enrolled in accounting but not enrolled in statistics. If a business student is selected at random, a) What is the probability that the student is enrolled in both statistics and accounting? b) If a business student is selected at random and found to be enrolled in statistics, what is the probability that the student is also enrolled in accounting? c) If a business student is selected at random, what is the probability that the student is not enrolled in statistics?Fifteen percent of all students at a large university are absent on Mondays. If a random sample of 12 names is called on a Monday, what is the probability that four students are absent? (Use normal approximation to the binomial distribution to answer this question) 5a) A University found that 20% of the students withdraw without completing the introductory statistics course. Assume that 20 students registered for the statistics course. . has un a) Compute the probability that two or fewer students will withdraw. b) Compute the expected number of withdrawals. 5b). In a large university, 20% of the students are business majors. A random sample of 100 students is selected, and their majors are recorded. What is the probability that the sample contains between 12 and 14 business majors