JNTU
JNTU B.Tech II PROBABILITY AND STATISTICS Previous Question Paper (December - 2017 - 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. Each question carries 10 marks and may have a, b, c as sub questions.
PART-A
Find the first two moments about origin from Moment generating function of the Binomial distribution?
Answer
The moment generating function (MGF) of a binomial distribution with parameters n and p is MX(t) = (q + pet)n, where q = 1 - p. The moments about the origin are found by taking the derivatives of MX(t) with respect to t and evaluating at t = 0.
The first moment about the origin (mean) is M'X(0):
The second moment about the origin is M''X(0):
A random variable X has the following probability function: Determine (i) k (ii) P(x < 6) (iii) P(0 < x < 5).
Answer
For a discrete probability distribution, the sum of all probabilities must equal 1. Given the values of P(x) for x = 0, 1, 2, 3, 4, 5, 6, 7, the sum is:
64k = 1, therefore k = 1/64.
(ii) P(x < 6) is the sum of probabilities for x = 0, 1, 2, 3, 4, 5:
(iii) P(0 < x < 5) is the sum of probabilities for x = 1, 2, 3, 4:
Show that the coefficient of correlation lies between -1 and 1.
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If E(X) = 5, E(Y) = 15, σX = σY = 2.58 and correlation coefficient is 0.93, find the regression lines of X on Y and Y on X and also determine regression coefficients.
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A population consists of five members 2, 3, 6, 8, 11. Consider all possible samples of size 2 that can be drawn with replacement from this population. Find (i) the mean of the population (ii) the standard deviation of the population.
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Write about (i) Critical Region (ii) Left tailed test (iii) Right tailed test.
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Write any two characteristics of queuing model (M/M/1):(∞/FIFO).
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Explain the different elements of queuing system.
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Calculate the probability of ruin and expected duration of the game, when a = 50, z = 40, p = 0.5.
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Is the given Stochastic matrix regular?
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PART-B
20% of items produced from a factory are defective. Find the probability that in a sample of 5 chosen at random i) none is defective ii) one is defective and iii) P(1 < x < 4).
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If x is a Poisson Variate such that 3 P(x=4) = 1/2 P(x=2) + P(x=0), find: i) mean of x ii) P(x ≤ 2).
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Prove that mean, median and mode of a normal distribution coincide.
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The joint probability density function is given by: f(x,y) = 10xy^2, 0 < x < y < 1. Find: a) Marginal probability density function for X, b) Marginal probability density function for Y, c) Conditional probability density function of X given Y, d) Conditional probability density function of Y given X.
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Calculate the coefficient of rank correlation from the provided data.
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Find the regression line of x on y and y on x for the provided data.
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An ambulance service claims that it takes on average less than 10 minutes to reach its destination in emergency calls. A sample of 36 calls has a mean of 11 minutes and the variance of 16 minutes. Test the significance at 0.05 level.
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A die is thrown 264 times. Show that the die is biased based on the given frequencies.
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The nicotine contents in milligrams in two samples of tobacco were found. Can it be said that two samples came from same normal population?
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Dr. Raju’s outpatient clinic can accommodate six people only in the waiting hall. The patients arrive in a Poisson process at an average rate of 3 per hour and spent on an average of 15 minutes in doctor’s chamber which is exponentially distributed. Find: a) The probability that a patient can directly go into the doctor’s chamber upon his arrival, b) Expected no. of customers waiting for treatment, c) Effective arrival rate, d) The time a patient can expect to spend in clinic.
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A toll gate is operated on a frequency where cars arrive according to a Poisson distribution with mean frequency of 1.2 cars per minute. The time of completing payment follows an exponential distribution with mean of 20 seconds. Find: a) the idle time of the counter, b) average number of cars in the system, c) average number of cars in the queue, d) average time that a car spends in the system, e) average time that a car spends in the queue.
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Define Markov chain, regular, ergodic and Stochastic matrices?
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Check whether the given markov chain is regular and ergodic?
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What do you mean by stochastic process and explain the types of stochastic processes?
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The three state Markov chain is given by the transition probability matrix. Prove that the chain is irreducible.
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