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 level of significance, if:
i) is known,
ii) is unknown.
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From the given summaries with n = 12, the sample means are (\bar X = \frac{96}{12} = 8,\quad \bar Y = \frac{72}{12} = 6). Thus (\bar{\mathbf{x}} = (8,6)'). The hypothesised mean vector is (\boldsymbol{\mu}_0 = (9,7)'), so (\bar{\mathbf{x}} - \boldsymbol{\mu}_0 = (-1,-1)'). i) Σ known The inverse of (\Sigma = _____ ____ _________ __________ ____.
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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.
Let the random vector has mean vector [-2, 3, 4] and variance
covariance matrix . Fit the equation
. Also obtain the multiple correlation coefficient between X3 and [X1, X2].
b) Define ultimate extinction in a branching process. Let
0 < b < c < b + c < 1 and . Then discuss the probability of extinction in different cases for
or E(X1) < 1.
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.
i) 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.
Let the joint probability density function of two discrete random X and Y be given as:
| X | |||||
|---|---|---|---|---|---|
| 2 | 3 | 4 | 5 | ||
| Y | 0 | 0 | 0.03 | 0 | 0 |
| 1 | 0.34 | 0.30 | 0.16 | 0 | |
| 2 | 0 | 0 | 0.03 | 0.14 | |
iii) Find the conditional distribution of X given .
v) Find .
Let the joint probability density function of two discrete random X and Y be given as:
| X | |||||
|---|---|---|---|---|---|
| 2 | 3 | 4 | 5 | ||
| Y | 0 | 0 | 0.03 | 0 | 0 |
| 1 | 0.34 | 0.30 | 0.16 | 0 | |
| 2 | 0 | 0 | 0.03 | 0.14 | |
iii) Find the conditional distribution of X given .
Let , where
and
v) Find .
Suppose life times are i.i.d. uniformly distributed on (0, 3) and
and
. Find:
i)
ii) T which minimizes C(T) and which is the better policy in the long-run in terms of cost.