JNTU
JNTU B.Tech II PROBABILITY AND STATISTICS Previous Question Paper (November/December - 2018 - R15)
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
If probability density function f(x) = Kx^3, 0 ≤ x ≤ 3; 0 elsewhere. Find the value of K.
Answer
For a probability density function, the integral over its range must equal 1. Thus, the integral from 0 to 3 of Kx3 dx = 1.
K(81/4) = 1, which gives K = 4/81.
If X is a continuous random variable whose probability density function is given by f(x) = x, 0 ≤ x ≤ 1; 2-x, 1 ≤ x ≤ 2; 0 elsewhere. Find the moment generating function.
Answer
The moment generating function MX(t) is defined as E[etX].
Evaluating these integrals using integration by parts:
If the two coefficients of regression are 0.4 and 0.9, then find the coefficient of correlation.
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The Joint Probability distribution of X and Y is given in a table. Find the Marginal probabilities for x.
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Define Null hypothesis.
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If the sample number is 500 and the standard deviation is 15, then find the maximum error with 99% confidence.
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Define the Expected queue length.
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If σx = 4, σy = 8, x̄ = 2, ȳ = 3 and r = 0.3, then find the regression line of y on x.
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Define the State space.
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Write the properties of a Transition Probability matrix.
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An insurance agent accepts policies of 5 men all of identical age and in good health. The probability that a man of this age will be alive 30 years is 2/3. Find the probability that in 30 years: (i) all five men (ii) at least one man will be alive.
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In a test on electrical bulbs, it was found that the life of a particular make was normally distributed with an average life of 2040 hours and S.D of 40 hrs. Estimate the number of bulbs likely to burn for more than 2140.
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Given that P(x=2) = 4/5 P(x=6) - 3 P(x=4) for a Poisson variate X. Find: i) P(x>1) ii) P(x<3).
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Suppose the weights of 500 male students are normally distributed with mean μ = 150 with a standard deviation of 15. Find the number of students whose weights are i) Between 140 and 165 ii) More than 170.
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The joint probability density function is given by f(x,y) = K(x+y), 0 < x < 1, 1 < y < 5; 0 elsewhere. Find: a) The value of K b) Marginal probability density function for X c) Marginal probability density function for Y d) P(X + Y > 3).
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Calculate the coefficient of rank correlation for the provided data sets x and y.
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The following figures refer to the observations in independent samples. Analyze whether the samples have been drawn from the populations of equal means.
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A survey of 400 families with 5 children each revealed the following distribution. Is this result consistent with the hypothesis that male births are equally probable?
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Consider a box office ticket window being manned by a single individual. Customers arrive to purchase tickets according to a poisson process. The arrival rate is 30 per hour. The mean service rate is 90 seconds. Find: a) Expected queue length b) Expected waiting time in the system c) Expected waiting time in the queue.
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The milk plant at a city distributes its products by trucks, loaded at the loading dock. It has its own fleet of trucks plus trucks of a private transport company. The trucks arrive at the interval of 20 minutes. The service time is 4 minutes. Find: a) The probability that there are more than or equal to 4 trucks in the queue b) The waiting time of a truck in the queue c) The variance of queue length d) The probability that the waiting time will exceed 10 minutes.
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Define i) Transient state ii) Recurrent state.
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Find the equilibrium vector of [[0.25, 0.75], [0.5, 0.5]].
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If the transition probability matrix of a Marcov chain is [[0.1, 0.2, 0.7], [0.2, 0.2, 0.6], [0.6, 0.1, 0.3]]. Find: a) P(X3=1|X0=0) b) P(X3=0|X0=0) c) P(X3=0|X1=2).
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