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?
This problem involves finding the population distribution over time using Markov chains. We will solve it numerically. ### Step 1: Define the Transition Matrix The given transition percentages form a transition matrix \( P \): \[ P = \begin{bmatrix} ___ _____ _____ _________ ___ ______ _____ ____ _______ ____ ________ ________.
_________ _________ ______ __________ ________ ______ _____ ____.
____ _______ ____ _____ ___ _____.
________ ____ _________ _________ ____.
__________ ________ ____ ______ ______ _____ ________ _____.
______ _____ _____ _____ ___.
_______ _________ ____ _____ _____ _______ ___ ________ _____.
____ ________ ________ _____ ______ ______ ____.
________ __________ _________ ______ _____ _____ ________ ______.
________ ___ __________ _______ _________ ________ __________.
__________ ____ __________ _______ ____ _______.
________ _________ __________ _________ __________ _________ ______ _____.
_______ ____ ____ ______ ______ ____.
________ _________ _________ ___ ________ ______ __________ __________ ______.
_____ _____ ______ ___ ________ ________ ____ _____ ____ ___ ___ ________.
___ _____ ___ _______ ______ ___ _____ __________ ____ _________ _____.
______ _______ ___ _______ ____ _________ ___ ____ _________ _______ __________ _____.
_____ _______ __________ _________ ___ ____ _______ ______ _________ ________.
_______ ________ _______ ________ ______ _______ __________ ____ ______ ____ ______.
_________ _________ ___ ___ ________.
______ __________ _______ ____ __________ ____ ______ ________ _____ ________ _________ __________.
_________ _____ _________ ___ ___ ____ ____ ____ __________ ______.
_____ __________ __________ ___ _______ ___ ___ ________ _____.
______.
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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.
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.
Let {Xn; }0 n ≥ be a Markov chain with four states; 1, 2, 3, 4 and the following transition probability matrix:
| 1 | 2 | 3 | 4 | |
| 1 | 0 | 0 | 1 | 0 |
| 2 | 1 | 0 | 0 | 0 |
| 3 | 1/2 | 1/2 | 0 | 0 |
| 4 | 1/3 | 1/3 | 1/3 | 0 |
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.
The joint distribution of the random variables X and Y is given by:
| x /y | −1 | 0 | 1 |
| −1 | α | β | α |
| 0 | β | 0 | β |
| 1 | α | β | α |
| 1 | 2 | 3 | 4 | |
| 1 | 0 | 0 | 1 | 0 |
| 2 | 1 | 0 | 0 | 0 |
| 3 | 1/2 | 1/2 | 0 | 0 |
| 4 | 1/3 | 1/3 | 1/3 | 0 |
i) Find the probabilit
b) It is claimed that the functionis the joint distribution function of the random variables X and Y. Then:
ii) calculate the probability
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 |