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Unformatted text preview: THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1801 PROBABILITY AND STATISTICS:
FOUNDATIONS OF ACTUARIAL SCIENCE December 18, 2008 Time: 9:30 a.m.  11:30 a.m. Candidates taking examinations that permit the use of calculators may use any calculator
which fulﬁls the following criteria: {a} it should be selfcontained, silent, batteryoperated
and pocketsized and (b) it should have numeraldisplay facilities only and should be used
only for the purposes of calculation. It is the candidate’s 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 SEVEN questions. Marks are shown in square brackets. 1. Suppose three cities A, B and C in Northern Canada are connected by roads.
There are two roads from A to B and two roads from B to C. In winter, each of
the four roads is blocked by snow with probability p (O < p < 1), independently of
the others. (i) Assume that there are no direct roads connecting A and C without passing
through B. Find the probability that in winter there is an open road from A
to B given that there is no open route from A to C. [5 marks] (ii) If, in addition, there is another direct road from A to C and this road is
blocked in winter with the same probability p, independently of the others,
then ﬁnd, again, in Winter the probability that there is an open road from A
to B given that there is no open route from A to C. [5 marks] [Total: 10 marks] S&AS: STATISOI Probability and Statistics: Foundations of Actuarial Science 2 2. Suppose it is expected that in a future time period, an insurance company may
receive N claims from clients where N is a Poisson random variable with
parameter 3. The quantity of claims, in millions of HK dollars, {X1, X2,X3, . . }
are independent identically distributed continuous random variables whose com—
mon probability density function f(:13) is given by cle'zI + age—4z if a: > 0 f(¢c) = 0 elsewhere, Where c1 + c2 = %. Further assume that the random variable N and the sequence of random variables
{X1,X2,X3, . . .} are also independent. (i) Find the two constants (:1 and Cg. [3 marks] (ii) Let Y be the total amount of claims received in the above mentioned future
time period. Find the expected value of Y. [6 marks] (iii) Find My(t), the moment generating function of Y, where Y is given in (ii)
above. Specify the range of t for which My(t) does exist. [8 marks] [Total: 17 marks] 3. Suppose X is a continuous random variable with probability density function §sin(:c) 0<a3<7r fx($) = 0 elsewhere. (i) Find the cumulative distribution function of X. [4 marks] (ii) Find the cumulative distribution function and the probability density function
of the random variable Y = J)? [6 marks] [Total: 10 marks] S&AS: STAT1801 Probability and Statistics: Foundations of Actuarial Science 3 4. Suppose that the joint probability density function of the two continuous random
variables X and Y is given by 3(ccy2+a:2y) 0<as<1,0<y<1,
f(X,Y)($) y) = O elsewhere. (i) Find the marginal densities of X and Y. [4 marks] (ii) Find the conditional density of Y given X = a: for 0 < w < 1. [4 marks] (iii) Find P{Y > 0.2 I X = 0.9}. [4 marks] > (iv) Find E [Y I X = 0.9]. [4 marks] [Total: 16 marks] 5. Suppose X and Y are two independent, exponentially distributed random variables
with common mean parameter 1. (i) Find the joint probability density function of f(U,V)(u, '0) where
U=X—Y and V=X+Y.
[8 marks] (ii) Derive the marginal probability density functions of U and V, respectively.
Are U and V independent? Justify your answer. [8 marks] [Total: 16 marks] 6. For a discrete random variable X, whose possible values are nonnegative integers
{0, 1, 2, . . .}, the probability generating function G X (t) of X is deﬁned by G X (t) =
E[tX]. (i) Show that E(X) = ’X(1) and
Var“) = 6732(1) + G'xU) — (03((1))2 where G’X (t) and 03205) denote the ﬁrst and second derivatives, respectively,
of OX (15) with respective to t and G'X(1), say, denotes the value of G’X (t) when
t = 1. [4 marks] (ii) Now suppose that the probability mass function of the non—negative integer
valued random variable Y is given by P{Y = k} = 6)ch (k = 0, 1, 2, . . .). Find the probability generating function G'y (t) of Y. [4 marks] S&AS: STAT1801 Prdbability and Statistics: Foundations of Actuarial Science 4 (iii) Use the probability generating function Gy (t) to ﬁnd E(Y) and Var(Y) for
the random variable Y deﬁned in (ii) above. (Note: Marks will not be awarded for a solution, even correct, by any other
method.) [4 marks] [Total: 12 marks] 7. Suppose that the joint probability density function, f (x,y) of the continuous
random variables X and Y is given by f(ac,y) = ﬁe‘ﬁyz—ﬁﬁ—ﬁH—o (—00 < a: < +00, —00 < y < +oo).
(i) Find the two marginal probability density functions fX(:r:) and fy(y) of X
and Y, respectively. [5 marks]
(ii) Are X and Y independent? Justify your answer. [3 marks]
(iii) Find E(XZY). [3 marks] (iv) Find P{—3.5 < Z < 3}, E(Z) and Var(Z) where Z = X + 21". [8 marks] [Total: 19 marks] ...
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 Fall '10
 MrChung
 Statistics, Probability

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