State whether the following statements are True or False. Justify your answer with a short proof or a counter example:
a) If P is a transition matrix of a Markov Chain, then all the rows of are identical.
b) In a variance-covariance matrix all elements are always positive.
c) If X1, X2, X3 are iid from , then
follows
.
d) The partial correlation coefficients and multiple correlation coefficients lie between -1 and 1.
e) For a renewal function .
a) False. A transition matrix of a Markov chain has rows that represent conditional probability distributions of moving from a given state to other states. Each row must sum to 1, but the rows need not be identical. For example, in a two-state Markov __________ _______ ____ _______ ________ _________ _______ _______ ________.
____ _________ ________ ______ _______ _____ ____ ____ _____ _____ __________.
___ _______ _______ _________ _______ ___ _________ ________ ________ ______.
______ _________ ____ __________ ___ ________.
______ ___ _____ ____ _______ _____ __________ _______ _______ _________ ____ _____.
_________ _____ ___ _________ _______ ____ _________ ______ _______ ____ ______.
____ _____ _______ _________ ____ _____ _____.
___ __________ _______ ___ _________ ____.
_____ _____ __________ __________ __________ ___ _____ ____.
_______ ______ ____ ______ _________ _______ ______.
_____ _____ ______ _____ _________ _______ ___ ________.
______ _______ _________ __________ ______ __________.
_______ ___ ______ ____ __________ ______ ____ ___ _________ _________ ___.
________ __________ _____ ____ ___ ______ ________.
__________ ____ ___ ___ _______ _________ _________ _________ ____.
_____ _______ _________ _____ ___ _______ ___ __________ _________.
_________ _________ _____ _____ ______ ______ ________ _________ ________ _________.
_________ _____ _______ _______ ____ __________ ____ _____ ___ _______ _____.
_____ ________ __________ _______ _____ ____ ________ _______.
________ _______ _____ ___ __________ __________ ______ _______ _______ ___ ______.
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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
.
Let (X, Y) have the joint p.d.f. given by:
i) Find the marginal p.d.f.'s of X and Y.
ii) Test the independence of X and Y.
iii) Find the conditional distribution of X given .
iv) Compute 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.
[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.
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.
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 .
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.