The transportation cost of 600 tons of a certain type of material from four factories  B_1, B₂ , B₃ , and B4 to three target stores  T_1,T₂ , and T₃ are given in the following table:

A locality is served by two malls. Each mall has two counters to serve the customers. Both the malls are equally popular and are known to have equal shares of the market. This is evident from the fact that customer’s arrive at each mall’s serving counter at the rate of 12 customers per hour. The average time to serve a customer is 05 minutes. Customers’ arrival is according to a Poisson distribution and the service time is exponential. To provide better service to the customers, the owners of the two malls decide to consolidate into a single larger mall. What is the effect of consolidation on the waiting time of customers?

Let ),t(P measured in kg, be the total mass or biomass of the fish population in a point at time t . Write the continuous model for the population growth using logistic equation. The intrinsic growth rate r and the carrying capacity k are given the values 70.0 per year and 6 80 7. ×10 kg respectively. If the initial biomass is P 25.0 0 = K, find the biomass after 2 years. Also find the time t , for which t(P ) 75.0 1 = K.

A particular component in a machine is replaced instantaneously on failure. The successive component lifetimes are uniformly distributed over the interval ]5,2[ years. Further, planned replacements take place every 3 years.

i) long-terms rate of replacements.
ii) long-terms rate of failures. Compute

Consider three random variables X₁ X₂ ,X₃ , having the covariance matrix

Write the factor model, if number of variables and number of factors are 3 and 1 respectively.

A box contains two coins: a regular coin and one fake two-headed coin. One coin is chosen at random and tossed twice. The following events are defined: (5)
 A: first coin toss results in a head.
 B: second coin toss results in a head.
 C: coin 1 (regular) has been selected.
 Find (P C|A ), (P |B C), (P A ∩ C|)B ), (P A), (P )B and (P A∩ ).B

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: