Determine the principal components Y1, Y2 and Y3 for the covariance matrix:
Also calculate the proportion of total population variance for the first principal component.
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Consider three random variables X1, X2, X3 having the covariance matrix
Write the factor model, if number of variables and number of factors are 3 and 1 respectively.
A box contains two coins: a regular coin and one fake two-headed coin. One coin is chosen at random and tossed twice. The following events are defined:
A: first coin toss results in a head.
B: second coin toss results in a head.
C: coin 1 (regular) has been selected.
Find and
.
A random sample of 12 factories was conducted for the pairs of observations on sales (x1) and demands (x2) and the following information was obtained:
The expected mean vector and variance covariance matrix for the factories in the population are:
and
Test whether the sample confirms its truthness of mean vector at level of significance, if:
i) is known,
ii) is unknown.
[You may use: ]
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:
i) Identify the model.
ii) The probability that the system shall be idle.
iii) The probability that there shall be 3 scooters in the service centre.
iv) The expected number of scooters waiting in a queue.
v) The expected number of scooters in the service centre.
vi) The average waiting time in a queue.
Let the joint probability density function of two discrete random X and Y be given as:
| X | |||||
|---|---|---|---|---|---|
| 2 | 3 | 4 | 5 | ||
| Y | 0 | 0 | 0.03 | 0 | 0 |
| 1 | 0.34 | 0.30 | 0.16 | 0 | |
| 2 | 0 | 0 | 0.03 | 0.14 | |
iii) Find the conditional distribution of X given .
v) Find .
Let be a normal random vector with the mean vector
and covariance matrix
. Suppose
, where
and
.
i) Find .
ii) Compute E(Y).
iii) Find the covariance matrix of Y.
iv) Find .
State whether the following statements are True or False. Justify your answer with a short proof or a counter example:
a) If P is a transition matrix of a Markov Chain, then all the rows of are identical.
b) In a variance-covariance matrix all elements are always positive.
c) If X1, X2, X3 are iid from , then
follows
.
d) The partial correlation coefficients and multiple correlation coefficients lie between -1 and 1.
e) For a renewal function .