1301_0506sem1 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE Pmbayuwj Md mrbfl‘c: I STAT1301 S—I—‘ATTSTTGS-jfirN'D—PRO‘BKBEITY‘I December 28, 2005 Time: 2:30 p.m. — 4:30 p.m. Candidates taking examinations that permit the use of calculators may use any cal— culator which fulfils the following criteria: (a) it should be self-contained, silent, battery—operated and pocket-sized and (b) it should have numeral—display facilities only and should be used only for the purposes of calculation. It is the candidate ’3 responsibility to ensure that the calculator operates satisfactorily and the candidate must record the name and type of the calculator on the front page of the examination scripts. Lists of permitted/prohibited calculators will not be made available to candidates for reference, and the onus will be on the candidate to ensure that the calculator used will not be in violation of the criteria listed above. Answer ALL FIVE questions. Marks are shown in square brackets. 1. Consider the random experiment of flipping an unfair coin four times. Assume that at each trial (flip), the probability that the head appears is 2 / 3, the prob- ability that the tail appears is 1/3 and that different trials are independent. Let A and B be two events defined as follows: A = {At least one tail appears}, B 2 {At least three heads appear}. (a) Find the conditional probabilities P(A|B) and P(B|A). [6 marks] (b) Are A and B independent? Give reasons for your answer. [4 marks] Total: [10 marks] 2. (a) Let X and Y be two independent Poisson random variables with pa- rameters A1 = 2 and A2 = 3, respectively (i.e. X N Poisson (2), Y N Poisson (3), and X and Y are independent). (1) Find the probabilities P{X > 3} and P{X = 1, 1 < Y s 3}. [4 marks] (ii) Let z = XY. Obtain P{Z = 0}, P{Z = 3} and P{Z : 4}. [7 marks] S&AS: STAT1301 Statistics and Probability I 2 (iii) Find E (Z) and Var(Z) where Z is defined in (ii). [5 marks] (b) Suppose U ~ N(0, 1) and V N N(—2, 12.25) and that U and V are independent. (i) Find the probabilities P{—2 S V S 3.5}, and P{U > 1, 1.5 < V g 5}. [4 marks] (ii) Let W = 5U + 4V + UV. Find E(W) and Var(W). [Hint You may use the following formula directly (a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc] [6 marks] Total: [26 marks] 3. (a) Let the joint probability mass function of two discrete random variables X and Y be given by x+y 32 ’ p($ay): 56:1)21 y=172a3)4' (i) Find the marginal probability mass functions p x (as) and py (y) of X and Y, respectively. [4 marks] (ii) State, with reasons, whether X and Y are independent. [4 marks] (iii) Find the probabilities P{X > Y} and P{X + Y = 3}. [4 marks] (b) In the early part of the twentieth century, newspaper distributors in many towns in Britain used to hire boys to sell newspapers. Assume the fol- lowing scenario: A newspaper boy would buy his papers for 25 pence each and would sell them for 35 pence each. Unsold papers could be given back to the distributor and the boy could receive 20 pence for each returned paper. Suppose that the number of newspapers required in the area covered by this newspaper boy could be modelled by a Poisson ran- dom variable with mean A = 10. Find the probability mass function of the random variable that represents the daily profit for a boy who used to buy 50 papers each day. [10 marks] Total: [22 marks] 4. Suppose the joint probability density function of the continuous random vari— ables X and Y is given by cxe'zWH) if 0 < a: < +00, 0 < y < +00 flay) = 0, otherwise where c is a positive constant. (a) Show that c = 1. [3 marks] S&AS: STAT1301 Statistics and Probability I 3 (b) Find the marginal probability density functions fX(a:) and fy(y) of X and Y, respectively. [6 marks] (c) State, with reasons, whether X and Y are independent. [3 marks] (d) Find E(X), Var(X) and E(X Y). Does E(Y) exist? Give reasons to support your conclusion. [8 marks] Total: [20 marks] 5. (a) The mean breaking strength of a certain type of cord has been established from considerable experience at 18.3 ounces with a standard deviation of 1.2 ounces. A new machine is purchased to manufacture this type of cord. A sample of 100 pieces obtained from the new machine shows a mean breaking strength of 17.0 ounces. The standard deviation of this type of cord produced by this new machine is assumed to remain unchanged at 1.2 ounces. Use an appropriate test to see whether the products produced are acceptable or not, using the 1% level of significance. State clearly your findings. [6 marks] (b) Two independent random samples, each of size 10, from two independent normal population random variables X and Y, where X N N(p.1, 02) and Y ~ N012, 02) yield the sample means and sample variances as i = 4.8, 3% = 8.64, g = 5.6 and 5% = 7.88. Find a 95 percent confidence interval for [11 — pg. [8 marks] (c) Two surveys were independently conducted to estimate a population mean, a. Denote the estimators by X1 and X2 and the standard de- viations of the estimators by $1 and 52, respectively. Assume that X1 and X2 are both unbiased. For some a and fl, the two estimators can be combined to give a better estimator: X =OZX1 +,BX2 (i) Find the conditions on a and fl that make the combined estimator unbiased. [4 marks] (ii) Find the values of a and fl which minimize the variance of the combined estimator, subject to the condition of unbiasedness. [4 marks] Total: [22 marks] ...
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