Distinguish between ‘Age Replacement’ and ‘Block Replacement’ policies. Let the lifetimes Y1 ,Y2 ,...., are independently and identically distributed random variables and follows negative exponential distribution with parameter 5. If lifetimes T > 0 and age replacement policy is to be employed, then:
i) find the mean renewal time and
ii) find the long-run average cost per unit time, given the costs C 4 1 = and C 6 2 = units of money.
Under an age‐replacement policy, each item is preventively replaced when it reaches a specified age T if it has not failed earlier; if it fails before T it is correctively replaced at failure, and the “clock” restarts for the new item. Under a block‐replacement policy, all items (or the single item ___ ___ __________ _____ ___ ______ ____ ___ ___ _____ ______ ___.
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In order to fit the regression line on a data set consisting of 34 pairs of values (z,y), the least square estimates of β1 and β2 are to be computed. From the data set, the following values are obtained:
Obtain the fitted regression line.
What is branching process? Give two real examples of branching process.
If )s(P and P )s( n respectively is the probability generating function (pgf ) of the i.i.d.
random variables and the random variables
Then show that:and
The joint distribution of the random variables X and Y is given by:
| y/x | -1 | 0 | 1 |
| -1 | α | β | α |
| 0 | β | α | β |
| 1 | α | β | α |
Consider the following system considering of two switches I and II between two points A and B:
A signal is sent from the point A to point B and is received at B if both the switches I and II are closed. It is assumed that the probabilities of I and II being closed are 0.8 and 0.6 respectively and that P [II is closed] I is closed] = P [II is closed].
Find:
i) The probability that signal is received at B.
ii) The conditional probability that switch I was open, given that the signal was not received at B,
iii) The conditional probability that switch II was open, given that the signal was not received at B.
State whether the following statements are true or false. Justify your answer with a short proof or a counter example:
i) Although in one-step transition probability matrix, ,P of a Markov chain the sum of each row must necessarily be unity but in higher order transition probability matrices, . This rule is not necessary.
ii) For the joint pdf of random variables (X,Y) given by:
iii) Let the r.v. X follows the negative exponential distribution with parameterThen
iv) One of the examples of non-Poisson queuing system is the (M / G (:)1/ ∞ (FIF0) queuing model.
v) be a random sample from
then maximum likelihood estimators o
A study about a population showed that the mobility of population of a state to a village, town and city is in the following percentages:
| From | To | ||
| Village | Town | City | |
| Village | 60 | 25 | 15 |
| Town | 10 | 70 | 20 |
| City | 10 | 30 | 60 |
A study about a population showed that the mobility of population of a state to a village, town and city is in the following percentages:
| From | To | ||
| Village | Town | City | |
| Village | 60 | 25 | 15 |
| Town | 10 | 70 | 20 |
| City | 10 | 30 | 60 |