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.
To solve this problem, we need to derive the renewal function from the given probability density function (PDF). The renewal function \( G(t) \) for a process starting with a new component is defined as: \[ G(t) = \mathbb{E}[N(t)] \] where \( N(t) \) ____ __________ ______ _____ _____ _____ _______ ______ ____ _________ ___ ____.
_____ ___ ________ _________ ________ __________ ________.
_______ _______ ______ __________ _______ _______ ___ ______ ____.
_____ _______ _________ _______ ______ _________ ___.
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______ _________ _______ _________ _______ ____ ______ _____ ______ _____ ____.
____ ______ ________ _____ ____ _____ _____ ______ _____ _____.
__________ ____ ____ ________ ________ ______ ______ _______.
____ _______ __________ ______ ______ ______ _________.
____ _____ _____ _______ ______ ______ _______ _______ ____ ________.
_______ _____ _____ ____ __________ ________ __________ ____ __________.
_______ __________ __________ _________ _______ _______.
__________ ___ ___ _________ ________ __________ ________ ________.
__________ _________ ____ __________ _______ ____ _____ _____ _________ ________ ________.
___ _______ _____ _________ _________ __________ __________ ______ _____ ______ _____ ________.
___ _________ _______ ________ ____.
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_____ ______ ____ __________ _______.
___ _________ ___ ________ ______ ___ ____ _________ ___ ___ ________.
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______ _______ ______ ________ _______ _________.
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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].
Suppose the random variables X1 ,X2 and X3have the covariance matrix
Find all principal components.
Consider the two data sets
for which
and
i) Calculate the linear discriminant function.
ii) Classify the observation as population
or population
with equal priors and equal costs.
A single repairperson looks after the two machines 1 and 2. Each time it is repaired, machine i stays up for an exponential time with rate , where i = 1, 2 . When machine i fails, it requires an exponentially distributed amount of work with rate
to complete its repair. The repairperson will always service machine 1 when it is down. For instance, if machine 1 fails while 2 is being repaired, then the repairperson will immediately stop work on machine 2 and start on
(i) Write down all the states.
(ii) What is the probability that the machine 2 is down.
Suppose that people arrive at a bus stop in accordance with a Poisson process with rate . The bus departs at time t . Let X denote the total amount of waiting time of all those that get on the bust at time t . We want to determine Var ( X ) . Let N (t ) denote the number of arrivals by time t
i) What is E [ X | N (t)] ?
ii) Argue that Var
iii) What is Var ( X ) ?
The joint probability mass function of X and Y , p ( x , y ) , is given by
i)
ii) Find P[x/y=1]
iii) Marginal distributions of X and Y
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