JNTU
JNTU B.Tech II PROBABILITY AND STATISTICS Previous Question Paper (April/May - 2023 - R15)
Note: i) Question paper consists of Part A, Part B. ii) Part A is compulsory, which carries 25 marks. In Part A, Answer all questions. iii) In Part B, Answer any one question from each unit. Each question carries 10 marks and may have a, b as sub questions.
PART - A
Is the function defined by f(x) = {0, x < 2; 1/18(2x+3), 2 ≤ x ≤ 4; 0, x > 4} a probability density function?
Answer
To be a valid probability density function (PDF), a function must satisfy two conditions: f(x) ≥ 0 for all x, and the integral of f(x) over its entire domain must equal 1.
1. Non-negativity: For 2 ≤ x ≤ 4, 1/18(2x+3) is clearly positive.
2. Normalization: We calculate the integral over the interval [2, 4]:
Since both conditions are satisfied, the function is a valid probability density function.
Calculate the expectation and variance of X, if the probability distribution of the random variable X is given by X: -1, 0, 1, 2, 3; f: 0.3, 0.1, 0.1, 0.3, 0.2
Answer
The expectation E(X) is calculated as the sum of x multiplied by its corresponding probability f(x):
To calculate the variance V(X), we first find E(X2):
The variance is then given by V(X) = E(X2) - [E(X)]2:
Define Marginal probability mass function and Marginal probability density function.
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Joint distribution of X and Y is given by f(x, y) = 4xye-(x^2+y^2); x ≥ 0, y ≥ 0. Test whether X and Y are independent.
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Define population, Sampling, Statistic and parameter.
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A coin was tossed 960 times and returned heads 183 times. Test the hypothesis that a coin is unbiased. Use a 0.05 level of significance.
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Define (M/M/1) system.
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Show that average number of units in a M/M/1 system is equal to ρ / (1-ρ).
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Which of the stochastic matrices are regular: I) A = [[1/2, 1/4, 1/4], [0, 1, 0], [1/2, 0, 1/2]] II) B = [[1/2, 1/2, 0], [1/2, 1/2, 0], [1/4, 1/4, 1/2]]
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Define classification of possible states.
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PART - B
If the probability that a man aged 60 will live to be 70 is 0.65, what is the probability that out of 10 men, now 60, at least 7 will live to be 70?
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Find the mean and standard deviation of a normal distribution in which 31% of items are under 45 and 8% are over 64.
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A manufacturer knows that the condensers he makes contain on average 1% defective. He packs them in boxes of 100. What is the probability that a box picked at random will contain 3 or more faulty condensers?
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In a sample of 1000 cases, the mean of a certain test is 14 and standard deviation is 2.5. Assuming the distribution to be normal, find (i) how many students score between 12 and 15 (ii) how many score above 18 (iii) how many score below 18.
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A computer while calculating correlation coefficient between two variables X and Y from 25 pairs of observations obtained: n=25, ΣX=125, ΣX^2=650, ΣY=100, ΣY^2=460, ΣXY=508. Correct two incorrectly copied pairs.
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Ten competitors in a beauty contest are ranked by three judges. Discuss which pair of judges has the nearest approach to common tastes of beauty.
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A cigarette manufacturing firm claims that its brand A line of cigarettes outsells its brand B by 8%. If 42/200 smokers prefer A and 18/100 prefer B, test validity of the 8% claim.
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Nicotine contents in two samples of tobacco: Sample A (24, 27, 26, 21, 25), Sample B (27, 30, 28, 31, 22, 36). Can it be said that two samples came from same normal population?
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A group of 5 patients treated with medicine A (42, 39, 48, 60, 41 kgs) and 7 with medicine B (38, 42, 56, 64, 68, 69, 62 kgs). Does medicine B increase weight significantly?
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Automobile accidents per week (12, 8, 20, 2, 14, 10, 15, 6, 9, 4). Are these frequencies in agreement with belief that accident conditions were same?
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A self service canteen employs one cashier. 8 customers arrive/10 mins on average. Cashier serves 1/min. Determine: (a) Avg customers in system (b) Avg queue length (c) Avg time in system (d) Avg waiting time.
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Maintenance shop inter-arrival time mean 10 mins, service time mean 6 mins. Find (a) Prob. of wait (b) Avg queue length/time (c) Criteria for second booth (d) Prob. wait > 12 mins.
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Three boys A, B, C throwing ball: A throws to B, B throws to C, C throws to B or A (equal probability). Show it is Markov. Find transition matrix and classify states.
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Two state Markov chain: Rain (0) and No rain (1). Transition matrix P = [[0.6, 0.4], [0.2, 0.8]]. Find probability it will rain for 3 days from today assuming it is raining today.
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