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 : 02-11-2006/9.00-12.00 Dept. No. Max. : 100 Marks
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
- Define repeated limits and give an example
- Define improper integral of Ist kind
- What are the order and degree of the differential equation ?
- Define continuity of functions of two variables
- If X, Y are random variables with joint distribution function F(x,y), express Pr[k1< X ≤ k2 , m1 < Y ≤ m2] in terms of
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:
- 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) = (y-x)4 + (x-2)2, x, y Î R.
18 . Define Beta distribution of 2nd kind. Find its mean and variance by stating the
conditions for their existence.
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 Cayley-Hamilton theorem, where
- [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) = (x-y)2 + 2x – 4xy, x, y Î
b] Show that the mixed derivatives of the following function at the origin are
22.a] Let f(x,y) = be the joint p.d.f of (x,y).
Find the co-efficient of correlation between X and Y.
[b] Change the order of integration and evaluate
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