Consider a birth and death process with birth rate and death rates
.
i) Determine the expected time to go from state 0 to state 4.
ii) Determine the expected time to go from state 2 to state 5.
iii) Determine the variances in parts (i) and (ii)
This problem involves solving a birth and death process where the rates of birth (\(\lambda_i\)) and death (\(\mu_i\)) are defined as: - Birth rate: \(\lambda_i = (i+1)\lambda\) for \(i \geq 0\), - Death rate: \(\mu_i = i\mu\) for \(i \geq 0\). The general formula to compute the expected time to transition from state \(i\) to state \(j\) in a birth-death process is given by: _____ ____ ________ ___ ____ _______ ___ _______ ___ __________ _____ _____.
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Suppose the random variables X1 ,X2 and X3have the covariance matrix
Find all principal components.
Let X have covariance matrix
i) Determine p and V 1/2
ii) Multiply your matrices to check the relation V 1/2 P V 1/2
A machine component is found to have life-time distribution with probability density function
Show that the renewal function is given by
if the process starts with a new component.
Twenty-five portfolio managers were evaluated in terms of their performance. Suppose Y represents the rate of return achieved over a period of time; Z1 is the manager’s attitude toward risk measured on a five-point scale from “very conservative” to “very risky”; and Z2 is years of experience in the investment business. The observed correlation coefficients between pairs of variables are
i) Interpret the sample correlation coefficients and
ii) Calculate the partial correlation coefficient and interpret this quantity with respect to the interpretation provided for
in Part (i)
Let be a random vector with mean vector
and variance-covariance matrix
Find the means and covariance matrix for the linear combinations
or
in terms of and
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i) What is E [ X | N (t)] ?
ii) Argue that Var
iii) What is Var ( X ) ?
For a series of dependent trials the probability of success on any trial is where k is equal to the number of successes on the previous two trials. Compute lim
[success on the n th trial].