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.
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A particular component in a machine is replaced instantaneously on failure. The successive component lifetimes are uniformly distributed over the interval [2, 5] years. Further, planned replacements take place every 3 years.
Compute
i) long-terms rate of replacements.
ii) long-terms rate of failures.
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 .
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.
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.
If N1(t), N2(t) are two independent Poisson process with parameters and
respectively, then show that
, where
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.
[You may use: ]
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.