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Sma2109:Applied Mathematics Question Paper
Sma2109:Applied Mathematics
Course:Bachelor Of Computer Science
Institution: Meru University Of Science And Technology question papers
Exam Year:2010
QUESTION ONE – (30 MARKS)
(a) Let (??1,??2 ,??) be a p- variate normal with density function given as ?? x = ??1 2 , where = ?? - -1 ?? - is a constant to be determined = 1,2, is the covariance matrix
Determine the (i) Value of so that ?? ?? is a pdf (6 Marks) (ii) Moment generating functions of ?? ?? (7 Marks) (b) State and prove the generalized form of Chebychev’s inequality. (7 Marks) (c) Let three random variables x, y and z have joint p.d.f.
?? ??,, =
? ? ? ? ?
?? ?? ?? 30 10 10
,2
z y x kxyz (i) Show that the constant = 4 9 (5 Marks)
2
(ii) Find the conditional distribution of x given = 1 2 and = 1 (5 Marks)
QUESTION TWO – (20MARKS)
(a) Use the non-generalized form of Chebycher’s inequality to determine the number of times a coin must be tossed so that the probability that the ratio of the number of heads to the number of tosses will be between o.4 and 0.6 is at least 0.9. (6 Marks) (b) Let x be normally distributed with mean and variance 2. Obtain the (i) Probability generating function of x. (4 Marks) (ii) Mean and variance from the probability generating function. (4 Marks) (c) Suppose that ??1,??2 and ??3 are independent random variables with unit variance. Let 1 = ??1 + ??2 + ??3, 2 = ??1 - ??2, 3 = ??1 - ??3 Find the covariance matrix for where = 1,2,3 . (6 Marks)
QUESTION THREE – (20 MARKS)
(a) Let ?? = ??1,??2,??3 have multivariate normal density given by ?? ?? = ?? ????- 1 2 3??1 2 + ??2 2 + 5??3 2 - ??1??2 - 3??2??3
Determine (i) Covariance matrix (5 Marks) (ii) Marginal density of ??1??3 (5 Marks) (iii) Conditional density of ??1 given ??2??3 . (4 Marks) (b) (i) State the weak law of large numbers. (3 Marks) (ii) Prove the weak law of large numbers. (6 Marks)
QUESTION FOUR – (20 MARKS)
(a) Let ?? = ??1,??2 be a random vector P.G.F = ??1??2 = 1 + 1??1 1 + 2 + 2??2 2 + 1 + 2 + 12??1??2 1+2
Find the (i) Mean of ??1 ???? ??2 (6 Marks) (ii) Variance of ??1 (4 Marks) (iii) Covariance if ??1 ???? ??2 (4 Marks) (b) Let x and y have a joint p.d.f. given by
?? ??, ?
? ? ? ??? elsewhere yxyk ,0 10)1(
(i) Calculate the value of k. (3 Marks) (ii) Calculate the ???? ?? 3 4 , 1 2 (4 Marks)
3
QUESTION FIVE – (20 MARKS)
(a) State and probe the central limit theorem (8 marks) (b) Let x and y have the following distribution ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? 2 321 2 12 2 12 1 221 12 1 x xpx p xe p xxf
Show that ? ? 2,1 xxf
is a bivariate density function. (8 Marks) (c) Discuss the importance of the central limit theorem in statistics. (4 Marks)
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