A study about a population showed that the mobility of population of a state to a village, town and city is in the following percentages:

What is branching process? Give two real examples of branching process.

If )s(P and P )s( n respectively is the probability generating function (pgf ) of the i.i.d.

random variables and the random variables

Then show that:and

It is claimed that the function     is the joint 16

distribution function of the random variables X and Y. Then:

i) determine the corresponding joint probability density function f_x,y and

ii) calculate the probability  

In an investigation related to a specific type of scores of men and women aged 65 to 70, the mean verbal and performance scores for 101 subjects were found to be:

The sample covariance matrix of the scores was

In order to test the null hypothesis that observations came from a population with mean vector   apply a suitable test statistic. You may consider α = 0.01 for the test.

Distinguish between ‘Age Replacement’ and ‘Block Replacement’ policies

Let the lifetimes Y₁ ,Y₂ ,....,  are independently and identically distributed random variables and follows negative exponential distribution with parameter 5. If lifetimes T > 0 and age replacement policy is to be employed, then:

i) find the mean renewal time and

ii) find the long-run average cost per unit time, given the costs C₁ 4  = and C₂ = 6units

Let {X_n; }0 n ≥ be a Markov chain with four states; 1, 2, 3, 4 and the following transition probability matrix:

What is the purpose of principal component analysis? Given the covariance matrix of order 2× ,2 explain how would you extract the principal components. Also, explain how would you find the proportion of total population variance for all the principal components.