Let  X =[ ] be a normal random vector with the mean vector

b) If N₁ ,(t), N₂  (t) are two independent Poisson process with parameters λ1 and λ2 respectively, then show that

Consider the Markov chain with three states, S = { 1,2 }3 following the transition matrix

i) Draw the state transition diagram for this chain

(ii) If P (X₁=1) P(X_{1 =2})=1/4  then find P (X₁₌3,X₂=2,X₃=1)

iii) Check whether the chain is irreducible and a periodic.

iv) Find the stationary distribution for the chain.

Suppose life times X ,X ,..... 1 2 are i.i.d. uniformly distributed on (0,3)and C₁ = 2 and C₂ = .8 .Find:

i) µ ^T

ii) T which minimizes C (T) and which is the better policy in the long-run in terms of cost.

If the random vector Z be N ( , ), 4 µ Σ where:

and 

Find  r₃₄,r_34.21.

If the random vector Z be N ( , ), 4 µ Σ where:

and 

Let the random vector X =(X₁,X₂,X₃) has mean vector [− ,2 ,3,4 ] and variance  covariance matrix = Fit the equation Y = b₀+ b₁ X+ b₂ X ₂ Also obtain the multiple correlation coefficient between X3 and [X ,X₂ ]

A random sample of 12 factories was conducted for the pairs of observations on sales (X₁) and demands ( X₂)  and the following information was obtained:

The expected mean vector and variance covariance matrix for the factories in the population are:

=

and

Test whether the sample confirms its truthness of mean vector at 5% level of
significance, if:
i) Σ is known,
ii) Σ is unknown.

A service station has 5 mechanics each of whom can service a scooter in 2 hours on the average. The scooters are registered at a single counter and then sent for servicing to different mechanics. Scooters arrive at a service station at an average rate of 2 scooters per hour. Assuming that the scooter arrivals are Poisson and service times are exponentially distributed, determine:
 i) Identify the model.
 ii) The probability that the system shall be idle.
 iii) The probability that there shall be 3 scooters in the service centre.
 iv) The expected number of scooters waiting in a queue.
 v) The expected number of scooters in the service centre.
 vi) The average waiting time in a queue.

At a certain filling station, customers arrive in a Poisson process with an average time of 12 per hour. The time interval between service follows exponential distribution and as such the mean time taken to service to a unit is 2 minutes. Evaluate:
 i) Probability that there is no customer at the counter.
 ii) Probability that there are more than two customers at the counter.
 iii) Average number of customers in a queue waiting for service.
 iv) Expected waiting time of a customer in the system.
v) Probability that a customer wait for 0.11 minutes in a queue.

Consider a Markov chain with transition probability matrix

i) Whether the chain is irreducible? If irreducible classify the states of a Markov chain i.e., recurrent, transient, periodic and mean recurrence time.
ii) Find the limiting probability vector.

Determine the principal components 1 Y2 Y , and Y3 for the covariance matrix:

Also calculate the proportion of total population variance for the first principal component.

Let X ~ N₃ ( µ Σ ), µ, Σ where µ =[5,3 ,4] and

Find the distribution of:

Let the joint probability density function of two discrete random X and Y be given as:

i) Find the marginal distribution of X and Y.

ii) Find the conditional distribution of X given Y=1.

iii) Test the independence of variable s X and Y.

iv) Find  V

Let ( ,X Y) have the joint p.d.f. given by:

f(x,y)=

i) Find the marginal p.d.f.’s of X and Y.
ii) Test the independence of X and Y.
iii) Find the conditional distribution of X given Y = .y
iv) Compute (E X | Y = )y and (E Y | X = ).x

State whether the following statements are True or False. Justify your answer with a short proof or a counter example:
 a) If P is a transition matrix of a Markov Chain, then all the rows of lim Pⁿare identical.

b) In a variance-covariance matrix all elements are always positive.

c) If   X_1, X₂ ,X₃ , are iid from N₂ ( µ , ), then 

d) The partial correlation coefficients and multiple correlation coefficients lie between −1 and 1.

e) For a renewal function 

Children’s Errors

Emergent Mathematics

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