JNTU
JNTU B.Tech II PROBABILITY AND STATISTICS Previous Question Paper (December - 2017 - R13)
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
Part A is compulsory which carries 25 marks.
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 given by MX(t) = (q + pet)n, where q = 1 - p. The first two moments about the origin are obtained by evaluating the derivatives of the MGF at t = 0.
The first moment about the origin is the mean:
The second moment about the origin is:
A random variable X has a probability distribution. Determine (i) k (ii) P(x < 6) (iii) P(0 < x < 5).
Answer
To determine the value of k, we use the property that the sum of all probabilities in a distribution must equal 1. Assuming the distribution is defined for x = 0, 1, 2, 3, 4, 5, 6 with probabilities P(x) = kx, we have:
For the second part, P(x < 6) is the sum of probabilities for x = 0, 1, 2, 3, 4, 5:
For the third part, 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 provided 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 3P(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) = 10xy2, 0 < x < y < 1. Find marginal and conditional probability density functions for X and Y.
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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 provided. Can it be said that two samples came from same normal population?
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Clinic waiting room analysis for M/M/1 queuing model with given arrival rate and service time.
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A toll gate operates on a Poisson distribution with mean frequency 1.2 cars per minute. Find idle time, average number of cars in system/queue and average time spent.
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Define Markov chain, regular, ergodic and Stochastic matrices.
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Check whether the provided 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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Given a three state Markov chain transition probability matrix, prove that the chain is irreducible.
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