If N1(t), N2(t) are two independent Poisson process with parameters and
respectively, then show that
, where
We are asked to find the conditional probability [ P(N_1(t) = k \mid N_1(t) + N_2(t) = n) ] where (N_1(t)) and (N_2(t)) are independent Poisson processes with rates (\lambda_1) and (\lambda_2). Start with the definition of conditional probability: [ P(N_1(t) = k \mid N_1(t) + N_2(t) = ______ __________ _________ ____ __________ ____.
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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 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
.
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 .
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.
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.
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.