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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:2011
QUESTION ONE - (30 MARKS)
a) i) Differentiate between a discrete and continuous random vector. (4 Marks) ii) Let ?? be a 2 × 1 absolutely continuous random vector and denote its components by ??1 ?? ??2 with a joint probability function ?? ?? = ?? ??1,??2 = exp (-??1 - ??2 0 ?? ??1 0,??2 0 derive the marginal probability density function of ??1 ?? ??2. (8 Marks) iii) Define a covariance matrix and give its structure. (3 Marks) iv) Let x be a 2 × 1 random vector and denote its components by ??1 ?? ??2 with covariance matrix of x being ???? ?? = 4 1 1 2 . Compute the variance of the random variable Y define as = 3??1 + 4??2. (7 Marks) b) Differentiate between Lindelberg-lerg central limit theorem and Chebysher’s inequality theorem hence proof by example the chebysher’s inequality (8 Marks)
QUESTION TWO – (20 MARKS)
Let X be a ?? × 1 random vector having a multivariate normal distribution with mean U and covariance v. then ?? = ?? + where is a standard multivariate normal ?? × 1 vector and ?? ?? ?? × ?? invertible matrix such that ?? =
a) Proof that a random vector having a multivariate normal distribution with mean ?? and covariance V is just a linear function of a standard multivariate normal vector. (14 Marks)
2
b) Proof that ?? = ?? ?? ???? ?? = ?? (6 Marks)
QUESTION THREE – (20 MARKS)
a) Let ?? be a constant × ?? and let ?? be a ?? × 1 random vector. The ???????? = ??????(??)?? i. Proof ???????? = ??????(??)?? (5 Marks) ii. Let x be a 3 × 1 random vector and denote it’s components by ??1,??2 ?? ??3. The
covariance matrix of x is ????
3 1 0 1 2 0 1 0 1 compute the following covariance ??????1 + 2??3,3??2 (7 Marks) b) Let ??1 be a random variable having a normal distribution with mean ??, and variance 12. Let ??2 be a random variable, independent of ??1, having a normal distribution with mean ??2 and variance 22Let y be defined as ?? = ??1 × ??2 show that i. = ??1 + ??2 ii. ?????? = 12 + 22 (8 Marks)
QUESTION FOUR – (20 MARKS)
a) Differentiate between a moment generating function (mgf) and a characteristic function of both discrete and continuous random variables. (6 Marks) b) Define a standard multivariate normal vector and hence state and prove the multivariate normal random vector and its joint characteristics. (14 Marks)
QUESTION FIVE – (20 MARKS)
a) Differentiate between uniqueness theorem and the weak law of large numbers and state four central limit theorems for independent sequences. (8 Marks) b) Using Chebysher inequality, find ?? that will gurantee that the probability is 0.95, that deviation x from [??] is no more than ??. (12 Marks)
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