Let be a random vector with mean vector
and variance-covariance matrix
Find the means and covariance matrix for the linear combinations
or
in terms of and
To solve this problem, we need to find the means and the covariance matrix for the linear combinations \( Z_1 = X_1 - X_2 \) and \( Z_2 = X_1 + X_2 \). ### 1. Mean Vector for \( Z \) The random vector \( Z = \begin{bmatrix} Z_1 \\ Z_2 \end{bmatrix} \) can be expressed as: \[ Z = C \cdot X = \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} X_1 \\ X_2 \end{bmatrix} \] where \( C = \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix} \) is the transformation matrix. The mean of \( Z \) is calculated as: \[ \mu_Z = C \cdot \mu_X \] where \( \mu_X = \begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix} \) is the mean vector of the random vector _____ __________ _____ ___ _____ __________ _____ ___.
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Consider the two data sets
for which
and
i) Calculate the linear discriminant function.
ii) Classify the observation as population
or population
with equal priors and equal costs.
A machine component is found to have life-time distribution with probability density function
Show that the renewal function is given by
if the process starts with a new component.
A single repairperson looks after the two machines 1 and 2. Each time it is repaired, machine i stays up for an exponential time with rate , where i = 1, 2 . When machine i fails, it requires an exponentially distributed amount of work with rate
to complete its repair. The repairperson will always service machine 1 when it is down. For instance, if machine 1 fails while 2 is being repaired, then the repairperson will immediately stop work on machine 2 and start on
(i) Write down all the states.
(ii) What is the probability that the machine 2 is down.
Which of the following statements are true or false. Give reasons
a) is the quadratic form of a positive definite matrix
b) In a bivariate normal distribution then x and y are independent
c) The relation of accessibility in states is transitive.
d) For any pair of discrete random variables X and
e) If {X (t) : t 0 } is a Poisson process, then
, where a is a constant is also a Poisson process.
Suppose the random variables X1 ,X2 and X3have the covariance matrix
Find all principal components.
Suppose the joint density function of X and Y is given by
Consider a birth and death process with birth rate and death rates
.
i) Determine the expected time to go from state 0 to state 4.
ii) Determine the expected time to go from state 2 to state 5.
iii) Determine the variances in parts (i) and (ii)