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 |
What will be the proportion of population village, town and city after one year and two years, given that the present proportion of the population in the village, town and city are respectively 0.50, 0.40 and 0.10?
Nice — this is a simple Markov chain (row-vector convention). Let the states be (Village, Town, City). The transition matrix ______ ______ __________ ___ ______ _________ ________ ___ ______ _____.
___ _________ ________ ___ ___ _______.
_________ __________ ___ __________ ___ _____ _________.
___ _____ ______ _______ _________ _________ _______ ___ ________.
__________ ________ ___ _____ _________ _______ ________ __________ ________ ______.
__________ _______ ____ __________ __________ ____.
_______ ____ ________ _______ _____ ____ ____ ____ ________.
_______ __________ __________ ______ ________ _____.
______ ___ __________ _______ _______ _______ ______ ________ _____ ____ _______.
___ _____ _____ _________ ___ ____ _____.
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It is claimed that the function is the joint 16
distribution function of the random variables X and Y. Then:
i) determine the corresponding joint probability density function fx,y and
ii) calculate the probability
What is branching process? Give two real examples of branching process.
If P(s) and Pn(s) respectively is the probability generating function (pgf) of the i.i.d. random variables and the random variables {Xn}, where Xn+1 =
Then show that:
Pn(s) = Pn-1(P(s))
and Pn(s) = Pn-1(P(s)).
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 |
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.
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 C1 4 = and C2 = 6units
A community has two police cars, which operate independently of one another. The probability that a specific car will be available when needed is 0.99.
i) What is the probability that neither car is available when needed?
ii) What is the probability that a car is available when needed?
In order to fit the regression line y = β1 + β2 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.