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.
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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 , then find
.
iii) Check whether the chain is irreducible and a periodic.
iv) Find the stationary distribution for the chain.
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 .
Let , where
and
v) Find .
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 .
If N1(t), N2(t) are two independent Poisson process with parameters and
respectively, then show that
, where
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.
If the random vector Z be , where:
and
Find r34, r34.21.ii) long-terms rate of failures.
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.