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 .
Since (X) is a bivariate normal random vector with mean (\mu=(0,1)^T) and covariance matrix [ \Sigma=\begin{bmatrix}1 & -1\ -1 & 2\end{bmatrix}, ] the marginal distribution _____ _______ ____ ____ __________ ____ __________ _____ _________ ____ ______.
_________ ___ _______ ___ __________ ________ ______ ______ ____ ____ ____ ___.
_______ ___ __________ _______ _________ _______ ______ _________ _____.
_________ _____ ______ _____ _____ _________ ______ ____ ____ ____.
_____ ______ ___ _________ __________ _________ ________ __________ ________ __________ ____.
_________ ___ ____ ____ __________ ____ ___ ________ ______.
________ __________ ____ ___ ____ ________ ___.
___ ___ ______ ___ ____ __________ ______ _________.
____ ______ ____ _________ _____ ______.
__________ ______ ____ _____ ___ __________ _____.
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______ _________.
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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.
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.
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.
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 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.
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:
A: first coin toss results in a head.
B: second coin toss results in a head.
C: coin 1 (regular) has been selected.
Find and
.
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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