Thapar University 2006 M.C.A Statistics & Combinatorics - Question Paper

1(a)

A random variable X follows the discrete probability distribution: {(xj,p,): (0, 1/3), (1, 1/6), (2, 1/2)}. Write a procedure to generate 50 random deviates forX Mention all the intermediate steps, if any.

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1(b)

Prove that, if A c 5, then P (A) < P(5).

Suppose that a sample space is given by S = {5: - < 5 < } and A c S is a set for which the integral, Jexp (- \x I) dx exists, (Integral is taken over the set A). Decide whether this integral can be taken as a probability density function for a random variable whose domain is S or not. If not, modify this integral suitably so that it can be used as a probability density function.

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2(a)

A shipment of 8 similar PCs to a retail outlet consists of 3 defective ones. If our School purchases 2 of these computers randomly, find the probability distribution for the number of defective PCs.

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2(b)

The joint probability density function of (X, y) is given by: \4xy, 0<x<l;0<<l

[0, elsewhere.

Find (i) P(0 < X< 0.5 and V* < Y< 4) and (ii) P(X< Y).

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2(c)

The life span in hours of the RAMs is a random variable with cumulative distribution function

F(x) = {'* *^{/5 X>}

[0, elsewhere.

Find (i) the probability density function for the life span and (ii) the probability that the life span of RAMs exceeds 70 hr.

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3(a)

Suppose that (X, F) is uniformly distributed over the triangle formed by the vertices (0, 0), (1, 3) and (-1, 3). Find the variance of X and variance of Y. Also find the coefficient of correlation between X and Y.

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3(b)

Define the sample variance and show that it is an unbiased estimate of population variance.

5

4(a)

Describe the Binomial process precisely. Show that Poisson distribution is a limiting case of the Binomial distribution clearly defining the limits that are forced.

If the random variable X has a Poisson distribution and if ?{X = 2) = 2*P(A' = 1 )/3. Find ?{X= 2) and P(*=3).

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4(b)

Define the Standard Normal distribution and the Chi-Square distribution. Show that, if X is a standard normal variate then is a Chi-Square variate with parameter 1. Establish the intermediate result, if any.

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5(a)

Find the norma! equations in order to fit a quadratic curve to the data given by {(X_ti Ki) i = 1, 2, 3, ..., w). Fit a curve of the form of y = a.b^x to the data {(.X, V): (0, 1), (1,2), (2, 5), (3, 10), (4, 17), (5, 26), (6, 30)}.

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5(b)

State the central limit theorem.

4

6(a)

State the principle of inclusion and exclusion giving a suitable example.

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6(b)

Define the generating function. Find the generating functions for the following sequences.

(i) {1,0, 1,0,...} (ii) {1,1,1/2!, 1/3!,...}

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6(c)

Define a linear recurrence relation and find the general solution of the recurrence relation, a_n - 6a_n__} + 9a_₂ = 0, a₀ = 5, = 12.

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