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 | |
ii) Find the marginal distribution of X and Y.
iii) Find the conditional distribution of X given .
iv) Test the independence of variable s X and Y.
Let , where
and
Find the distribution of:
v) Find .
Marginal distribution of X: [ P_X(2) = 0 + 0.34 + 0 = 0.34 ] [ P_X(3) = 0.03 + 0.30 + 0 = 0.33 ] [ P_X(4) = 0 + 0.16 + 0.03 = 0.19 ] [ P_X(5) = 0 + 0 + 0.14 = 0.14 ] So, (P_X = {2:0.34, 3:0.33, 4:0.19, 5:0.14}). Marginal distribution of Y: [ P_Y(0) = 0 + 0.03 + 0 + 0 = 0.03 ] [ P_Y(1) = 0.34 + 0.30 + 0.16 + 0 = 0.80 ] [ P_Y(2) = 0 + 0 + 0.03 + 0.14 = 0.17 ] So, (P_Y = {0:0.03, 1:0.80, 2:0.17}). Conditional distribution of X given (Y=1): ______ ____ ________ ______ _______ ______ _______ _____ _____ ____ ___.
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A particular component in a machine is replaced instantaneously on failure. The successive component lifetimes are uniformly distributed over the interval [2, 5] years. Further, planned replacements take place every 3 years.
Compute
i) long-terms rate of replacements.
ii) long-terms rate of failures.
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 , then find
.
iii) Check whether the chain is irreducible and a periodic.
iv) Find the stationary distribution for the chain.
Let (X, Y) have the joint p.d.f. given by:
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 .
iv) Compute and
.
Let the random vector has mean vector [-2, 3, 4] and variance
covariance matrix . Fit the equation
. Also obtain the multiple correlation coefficient between X3 and [X1, X2].
b) Define ultimate extinction in a branching process. Let
0 < b < c < b + c < 1 and . Then discuss the probability of extinction in different cases for
or E(X1) < 1.
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: ]
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.
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 .