LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOV 2006
ST 3500 – STATISTICAL MATHEMATICS – II
(Also equivalent to STA 502)
Date & Time : 02112006/9.0012.00 Dept. No. Max. : 100 Marks
SECTION A
Answer ALL questions. Each carries 2 marks [10×2=20]
 Define Hermitian matrix and give an example of 3×3 Hermitian matrix
 Write the formula for finding the determinant by using partioning of matrices
 Define Riemann integral
 Evaluate
 Define repeated limits and give an example
 Define improper integral of I^{st }kind
 What are the order and degree of the differential equation ?
 Define continuity of functions of two variables
 Evaluate
 If X, Y are random variables with joint distribution function F(x,y), express Pr[k_{1}< X ≤ k_{2 , }m_{1 }< Y ≤ m_{2}] in terms of
SECTION B
Answer any FIVE questions (5×8 =40)
 State and prove the first fundamental theorem of integral calculus
 Find the inverse of the matrix A =by using 2×2 partitioning
 Discuss the convergence of following improper integrals:
[a] [b]
 Define Gamma distribution and hence derive its mean and variance
 Solve the differential eqation
 Investigate the existence of the repeated limits and double limit at the origin of the
function f(x,y) =
 Investigate for extreme values of f(x,y) = (yx)^{4} + (x2)^{2}, x, y Î R.
18 . Define Beta distribution of 2^{nd } kind. Find its mean and variance by stating the
conditions for their existence.
SECTION C
Answer any TWO questions (2 x 20 =40)
 [a] Find the rank of the matrix A=
[b] Find the characteristic roots of the following matrix. Also find the inverse of A
using CayleyHamilton theorem, where
A=
 [a] Test if converges absolutely
[b] Compute mean, mode and variance for the following p.d.f
 a] Find maximum or minimum of f(x,y) = (xy)^{2 }+ 2x – 4xy, x, y Î
b] Show that the mixed derivatives of the following function at the origin are
different:
22.a] Let f(x,y) = be the joint p.d.f of (x,y).
Find the coefficient of correlation between X and Y.
[b] Change the order of integration and evaluate
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