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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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.
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.
Determine the principal components Y1, Y2 and Y3 for the covariance matrix:
Also calculate the proportion of total population variance for the first principal component.
Let be a normal random vector with the mean vector
and covariance matrix
. Suppose
, where
and
.
i) Find .
ii) Compute E(Y).
iii) Find the covariance matrix of Y.
iv) Find .
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.
i) Probability that a customer wait for 0.11 minutes in a queue.
If N1(t), N2(t) are two independent Poisson process with parameters and
respectively, then show that
, where