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Define 22, t and ‏ع‎ distributions.

Describe spectral decomposition of a symmetric matrix (4 Marks)

State Cayley - Hamilton theorem. Also using Cayley - Hamilton theorem, compute A” in terms of
powers of A. (4 Marks)

Let ‏بلا‎ be the mean of a random sample of size ‏م‎ from a population with mean ‏لز‎ and variance ൪.
Show that X, converges stochastically to யூ, if o*is finite. (5 Marks)

Derive the confidence interval for the difference of means of two normal populations, based on small
samples of sizes n, and n, taken from the populations by assuming the variances of the populations.

are same. (5 Marks)
Let X have the distribution 2) = 2(1- 2)! ५ x= 0,1; 0 < 4 < 1. Construct the SPRT for testing

ಗೃ: 25 A, againstHH,: A= A,. (5 Marks)
For a resolvable 8180 with parameters ४, b, ந k and A, show that b2v+r—1. (5 Marks)
a) Define joint, marginal and conditional p.d.f. of multivariate distributions.

b) What is lack of memory property ? Give discrete and continuous distributions having lack of
memory property.
யி Define multivariate normal distribution. ⋅ (5 Marks)

In the general linear model Y=XB+U, with usual notations and if U follows N, (0, (ஸீ |). derive

an unbiased estimator for 07. (5 Marks)