Let the joint probability density function of two discrete random X and Y be given as:
i) Find the marginal distribution of X and Y.
ii) Find the conditional distribution of X given Y=1.
iii) Test the independence of variable s X and Y.
iv) Find V
i) Marginal Distribution of X and Y:
To find the marginal distribution of X and Y, sum over all possible values of the other variable. The marginal distribution of X (denoted as \( P_X(x) \)) is found by __________ _____ _________ _________ _________ ___ _______ __________ ________.
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Determine the principal components 1 Y2 Y , and Y3 for the covariance matrix:
Also calculate the proportion of total population variance for the first principal component.
Consider a Markov chain with transition probability matrix
i) Whether the chain is irreducible? If irreducible classify the states of a Markov chain i.e., recurrent, transient, periodic and mean recurrence time.
ii) Find the limiting probability vector.
If the random vector Z be N ( , ), 4 µ Σ where:
and
Let ( ,X Y) have the joint p.d.f. given by:
f(x,y)=
i) Find the marginal p.d.f.’s of X and Y.
ii) Test the independence of X and Y.
iii) Find the conditional distribution of X given Y = .y
iv) Compute (E X | Y = )y and (E Y | X = ).x
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 5% level of
significance, if:
i) Σ is known,
ii) Σ is unknown.
Consider the Markov chain with three states, S = { 1,2 }3 following the transition matrix
i) Draw the state transition diagram for this chain
(ii) If P (X1=1) P(X1 =2)=1/4 then find P (X1=3,X2=2,X3=1)
iii) Check whether the chain is irreducible and a periodic.
iv) Find the stationary distribution for the chain.
At a certain filling station, customers arrive in a Poisson process with an average time of 12 per hour. The time interval between service follows exponential distribution and as such the mean time taken to service to a unit is 2 minutes. Evaluate:
i) Probability that there is no customer at the counter.
ii) Probability that there are more than two customers at the counter.
iii) Average number of customers in a queue waiting for service.
iv) Expected waiting time of a customer in the system.
v) Probability that a customer wait for 0.11 minutes in a queue.