School of Business and Economics

(10 marks)


Solution

(a)

(b)

(c)

Q2. Consider a family with four children. The binomial random variable X will represent the number of children who are male.

(a) define the sample space for this experiment (having four children and counting the number of boys);

(b) given that the gender of each child is independent from the genders of their siblings, the probability of a child being male is ½, find the probability function, f(x);

Note: to find probability function for binomial random variable, you use the following rule

(c) find the distribution function, F(x) for the random variable X and graph this distribution.

(10 marks)


Answer

(a) The sample space for this experiment (having four children and counting the number of boys) is the set S = {0, 1, 2, 3, 4}.

(b)















f(0) = P(X = 0) = 1/16
f(1) = P(X = 1) = 1/4,
f(2) = P(X = 2) = 3/8,
f(3) = P(X = 3) = 1/4,
f(4) = P(X = 4) = 1/16.

(c)

Q3. (a) You choose a month at random and then a day of the week at random. What is the probability that you will get a Saturday in May? Illustrate your answer with a tree.
(b) What is the probability of getting exactly 7 heads when 12 coins are tossed?
(10 marks)


Solution
(a) The probability of getting May is 1/12 and the probability of Saturday is 1/7, so the probability of a Saturday in May is (1/12)*(1/7) = 0.012.

		Saturday, (1/7)	Saturday in May, (1/12)(1/7) = 0.012
	May, (1/12)
		Not Saturday, (6/7)	May, not Saturday, (1/12)(6/7) = 0.071
		Saturday, (1/7)	Not May, Saturday, (11/12)(1/7) = 0.131
	Not May, 11/12
		Not Saturday, (6/7)	Not May, Not Saturday, (11/12)(6/7) = 0.786

(b) or 0.19

Q4. Show that , if .
(10 marks)


Solution


; ; ;


. Therefore,
Q5. (a) What is the approximate yield to maturity (YTM) of a bond that is currently selling for $1,200 in the market place? The annual bond’s maturity value is $1,000. It has 10 years remaining until maturity and it pays a 10% coupon. Note: Approximation method can be used to find YTM: .

(b) Find the market price of a bond maturing in 10 years, paying coupon rate 10%, yield to maturity is 7% and face value is $1000. Note: To find the bond price we can use the following mathematical expression: .

(c) Show that the following is true:

(10 marks)


Solution

(a) YTM = [100 + (1000 – 1200)/10] / [(1000 + 1200)/2] = 0.072727 or 7.27%

(b) P = 100*(1/0.07 – 1/(0.07*(1 + 0.07)^10)) + 1000/(1 + 0.07)^10 = 1210.707
(c)

[END OF SECTION A]

Section B
Answer ALL question from the following questions;
Q6. (a) Find the critical points. (b) Test whether the function is at a relative maximum or minimum, given and (c) Find the value of z.