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IGNOU MMT-008 Solved Assignment 2023 | M.Sc. MACS
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IGNOU MMT-008 Assignment Question Paper 2023
- State whether the following statements are True or False. Justify your answer with a short proof or a counter example:
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a) Let
(X,Y) (X, Y)
|
|||||
2 | 3 | 4 | 5 | ||
|
0 | 0 | 0.03 | 0 | 0 |
1 | 0.34 | 0.30 | 0.16 | 0 | |
2 | 0 | 0 | 0.03 | 0.14 |
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a) Let
X∼N_(3)(mu,Sigma) \mathrm{X} \sim \mathrm{N}_{3}(\mu, \Sigma) mu=[5,3,4]^(‘) \mu=[5,3,4]^{\prime}
- a) Consider a Markov chain with transition probability matrix:
- a) A service station has 5 mechanics each of whom can service a scooter in 2 hours on the average. The scooters are registered at a single counter and then sent for servicing to different mechanics. Scooters arrive at a service station at an average rate of 2 scooters per hour. Assuming that the scooter arrivals are Poisson and service times are exponentially distributed, determine:
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a) Let the random vector
X^(‘)=(X_(1),X_(2),X_(3)) X^{\prime}=\left(X_{1}, X_{2}, X_{3}\right) [-2,3,4] [-2,3,4] =([1,1,1],[1,2,3],[1,3,9]) =\left(\begin{array}{lll}1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 9\end{array}\right) Y=b_(0)+b_(1)X+b_(2)X_(2) Y=b_{0}+b_{1} X+b_{2} X_{2} X_(3) \mathrm{X}_{3} [X_(1),X_(2)] \left[\mathrm{X}_{1}, \mathrm{X}_{2}\right]
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a) If the random vector
Z \mathrm{Z} N_(4)(mu,Sigma) \mathrm{N}_{4}(\mu, \Sigma)
ii)
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a) Consider the Markov chain with three states,
S={1,2,3} S=\{1,2,3\}
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a) Let
X=[[X_(1)],[X_(2)]] X=\left[\begin{array}{l}X_{1} \\ X_{2}\end{array}\right] mu=[[0],[1]] \mu=\left[\begin{array}{l}0 \\ 1\end{array}\right] [[1,-1],[-1,2]] \left[\begin{array}{cc}1 & -1 \\ -1 & 2\end{array}\right] Y=AX+b \mathrm{Y}=\mathrm{AX}+\mathrm{b}
ii) Compute
iii) Find the covariance matrix of
iv) Find
A: first coin toss results in a head.
B: second coin toss results in a head.
C: coin 1 (regular) has been selected.
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a) Consider three random variables
X_(1),X_(2),X_(3) X_{1}, X_{2}, X_{3}
MMT-008 Sample Solution 2023
Question:-01
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State whether the following statements are True or False. Justify your answer with a short proof or a counter example:
a) IfP \mathrm{P} lim_(nrarr oo)P^(n) \lim _{\mathrm{n} \rightarrow \infty} \mathrm{P}^{\mathrm{n}}
Counterexample:
Conditions for the Statement to be True:
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Counterexample:
Additional Notes:
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The diagonal elements of a variance-covariance matrix, which represent variances, are always non-negative because variance cannot be negative.
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Off-diagonal elements, which represent covariances, can be negative, zero, or positive, depending on the relationship between the variables involved.
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Justification:
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Mean : The mean of
(X_(1)+X_(2)+X_(3))/(3) \frac{X_1+X_2+X_3}{3} (mu+mu+mu)/(3)=mu \frac{\mu + \mu + \mu}{3} = \mu -
Covariance Matrix : The covariance matrix of
X_(1)+X_(2)+X_(3) X_1+X_2+X_3 Sigma+Sigma+Sigma=3Sigma \Sigma + \Sigma + \Sigma = 3\Sigma X_(1),X_(2),X_(3) X_1, X_2, X_3 (X_(1)+X_(2)+X_(3))/(3) \frac{X_1+X_2+X_3}{3} (1)/(3^(2))(3Sigma)=(1)/(3)Sigma \frac{1}{3^2}(3\Sigma) = \frac{1}{3}\Sigma
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Justification:
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Partial Correlation Coefficients : The partial correlation coefficient measures the strength and direction of the relationship between two variables while controlling for the effect of one or more other variables. Mathematically, it is defined in a way that ensures its value lies between -1 and 1, inclusive. Specifically, it is computed as the correlation between the residuals resulting from the linear regression of each variable against the control variables. Since residuals are uncorrelated with the predicted values, the partial correlation coefficient is constrained to be between -1 and 1.
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Multiple Correlation Coefficients : The multiple correlation coefficient
R R R^(2) R^2 R^(2) R^2 R R R R
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Justification:
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