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.
The component lifetimes are uniformly distributed over ([2,5]). Let (X) be the lifetime of a component. Then the probability density function is: [ f_X(x) = \frac{1}{5-2} = \frac{1}{3}, \quad 2 \le x \le 5 ] The mean lifetime is: [ E[X] = \frac{2+5}{2} = \frac{7}{2} = 3.5 \text{ years} ] Planned replacements occur every (T = 3) years. The long-term rate of replacements per year, (R), is given by: [ R = \frac{1}{E[\text{time between replacements}]} ] Here, time between replacements is the minimum of the lifetime (X) and planned replacement ____ ___ _______ ____ _____.
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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
.
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.
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 .
If N1(t), N2(t) are two independent Poisson process with parameters and
respectively, then show that
, where
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: ]
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 .