JNTU
JNTU B.Tech II PROBABILITY AND STATISTICS Previous Question Paper (April - 2018 - R15)
Note: This question paper contains two parts A and B. Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 Units. Answer any one full question from each unit.
Part A
Determine the mode of a binomial distribution for which the mean is 4 and variance is 3.
Answer
For a binomial distribution, mean μ = np = 4 and variance σ^2 = npq = 3. Thus, q = 3/4 and p = 1 - 3/4 = 1/4. Substituting p, n(1/4) = 4 implies n = 16. The mode is the integer value of (n+1)p. Here, (16+1)(1/4) = 17/4 = 4.25. Since (n+1)p is not an integer, the mode is the integer part, which is 4.
If X is a continuous random variable and k is constant, prove that
Answer
By definition, Var(kX) = E[(kX - E[kX])^2]. Since E[kX] = kE[X], we have Var(kX) = E[(kX - kE[X])^2] = E[k^2(X - E[X])^2]. By the linearity property of expectation, we can factor out the constant k^2, resulting in k^2 E[(X - E[X])^2], which is k^2 Var(X).
Define joint probability mass function.
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Write the properties of Rank correlation coefficient.
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Define the population and sample of a population.
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Write the characteristics of Student t-distribution.
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What is Kendal’s notation in queuing theory?
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If average number of arrivals is 4 per hour and average number of services is 3 per hour, what is the probability that a new arrival need not wait for the service.
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What is chain in stochastic process?
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Define Markov chain.
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Part B
A continuous random variable X has the distribution function F(x) = 0 for x ≤ 1, k(x-1)^2 for 1 ≤ x ≤ 3, 1 for x > 3. Determine: a) f(x) b) k c) Mean.
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Number of monthly breakdowns of a computer is a random variable having Poisson distribution with mean equal to 1.8. Find the probability that the computer will function for a month i) without a breakdown ii) with only one breakdown and iii) with at least one.
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If X is a normal variate with mean 30 and standard deviation of 5. Find the probabilities that i) 26 ≤ X ≤ 40 ii) X ≥ 45 iii) X ≤ 22.
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Calculate the regression equation of Y on X from the following data: X=[48, 33, 40, 9, 16, 16, 65, 24, 16, 57], Y=[13, 13, 24, 6, 15, 4, 20, 9, 16, 19].
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Ten competitors in a musical test were ranked by three judges A, B and C. Using rank correlation method, discuss which pair of judge has the nearest approach to common liking in music.
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A random sample of size 81 was taken whose variance is 20.25 and mean is 32. Construct 98% confidence interval.
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In one sample of 8 observations the sample mean is 84.4 and mean of the other sample of 10 observations is 102.6. Test whether this difference is significant at 5% level. Assume S.D of both sample is same are equal to 15.
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A pair of dice are thrown 360 times and the frequency of the sum obtained is indicated. Would you say that the dice are fair on the basis of the chi-square test at 0.05 level of significance?
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A bank plans to open a single server derive in banking facilities at a particular center. It is estimated that 20 customers will arrive each hour on an average. If on an average, it is required 2 minutes to process a customer’s transaction, determine: a) The proportion of time that the system will be idle. b) On the average how long a customer will have to wait before reaching the server? c) Traffic intensity of bank?
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Average queue length.
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Average length of non empty queue.
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A gambler has Rs. 2. He bets Rs. 1 at a time and wins Rs. 1 with probability 0.5. He stops playing if he loses Rs. 2 or wins Rs. 4. a) What is the transition probability matrix of the related Markov Chain? b) What is the probability that he has lost his money at the end of 5 plays? c) What is the probability that the game lasts more than 7 plays?
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Check whether the following Markov chain is regular and ergodic.
Transition matrix A provided.
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Transition matrix B provided.
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