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, P(j); j = 2,3,4,...... 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 parameter . Then
iv) One of the examples of non-Poisson queuing system is the (M/G/1): (∞(F1F0) queuing model.
v) If X1,X2,..., X, be a random sample from Np, (μ, Σ), then maximum likelihood estimators of µ and ∑are:
Let's analyze each statement one by one: --- ### 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, \( P^{(j)} \); \( j = 2,3,4,\dots \), this rule is not necessary. Answer: False. Justification: The sum of each row in a transition probability matrix (whether one-step or higher-order) must always be unity. This is because the rows represent the probabilities of transitioning from a given state to all possible states, and these probabilities must sum to 1. For higher-order transition matrices \( P^{(j)} \), the rule still holds because \( P^{(j)} \) is derived from \( P \) (e.g., \( P^{(2)} = P _________ ____ ________ _________ __________ ________ ______ ___ _________ _________ ______.
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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.
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 |
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 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)).
For the (M/M/K) : (∞ / FIF )0 queuing model with arrival rate λ and service rate per service channel µ, obtain the steady-state probability of n customers in the system, P . n Hence or otherwise, find the probability that an arrival has to wait.
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, P(j); j = 2,3,4,...... 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 parameter . Then
iv) One of the examples of non-Poisson queuing system is the (M/G/1): (∞(F1F0) queuing model.
v) If X1,X2,..., X, be a random sample from Np, (μ, Σ), then maximum likelihood estimators of µ and ∑are:
In an investigation related to a specific type of scores of men and women aged 65 to 70, the mean verbal and performance scores for 101 subjects were found to be:
The sample covariance matrix of the scores was
In order to test the null hypothesis that observations came from a population with mean vector apply a suitable test statistic. You may consider α = 0.01 for the test.