JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II Year PROBABILITY AND STATISTICS Previous Question Paper (October - 2020 - R15)
Answer any five questions. All questions carry equal marks.
Main
The probabilities of a Poisson variate X taking the values 1 and 2 are equal. Find:
i) Mean μ
ii) P(X ≥ 1)
iii) P(1 < X < 4).
Answer
For a Poisson distribution, the probability mass function is given by P(X = x) = (e-μ * μx) / x!. Given P(X = 1) = P(X = 2):
Dividing both sides by e-μ * μ (assuming μ ≠ 0):
i) The mean μ is 2.
ii) P(X ≥ 1) = 1 - P(X = 0) = 1 - (e-2 * 20) / 0! = 1 - e-2 ≈ 1 - 0.1353 = 0.8647.
iii) P(1 < X < 4) = P(X = 2) + P(X = 3) = (e-2 * 22) / 2! + (e-2 * 23) / 3! = 2e-2 + (4/3)e-2 = (10/3)e-2 ≈ 3.333 * 0.1353 = 0.4511.
Given that the mean heights of students in a class is 158 cms with a standard deviation of 20 cms. Find how many students height lie between 150 cms and 170 cms.
Answer
Given mean μ = 158 and standard deviation σ = 20, we use the normal distribution Z = (X - μ) / σ.
For X = 150: Z1 = (150 - 158) / 20 = -8 / 20 = -0.4.
For X = 170: Z2 = (170 - 158) / 20 = 12 / 20 = 0.6.
We need P(150 < X < 170) = P(-0.4 < Z < 0.6) = P(Z < 0.6) - P(Z < -0.4).
Using standard normal distribution tables: P(Z < 0.6) = 0.7257 and P(Z < -0.4) = 1 - P(Z < 0.4) = 1 - 0.6554 = 0.3446.
Probability = 0.7257 - 0.3446 = 0.3811.
The proportion of students with heights between 150 cm and 170 cm is 38.11%.
Average number of accidents on any day on a national highway is 1.6. Determine the probability that the number of accidents is:
i) At least one
ii) At the most one.
Answer
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The marks obtained by 500 students is normally distributed with mean 65% and Standard deviation 8%. Determine how many get more than 80%.
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Calculate the coefficient of rank correlation from the following data.
| x | 68 | 64 | 75 | 50 | 64 | 80 | 75 | 40 | 55 | 64 |
| y | 62 | 58 | 68 | 45 | 81 | 60 | 68 | 48 | 50 | 70 |
Answer
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The joint probability density function is given by:
Find:
Marginal probability density function for X
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Marginal probability density function for Y.
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A survey of 400 families with 5 children each revealed the following distribution. Is this result with the hypothesis that male births are equally probable?
| No. of boys | 0 | 1 | 2 | 3 | 4 | 5 |
| No. of families | 25 | 60 | 110 | 113 | 62 | 30 |
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The mean height of 90 students in a class is 180 cms Test at 90% level whether the sample had been drawn from a population whose mean is 170 cms and standard deviation 35.
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Among the items produced by a factory out of 500, 15 were defective in another sample out of 400, 20 were defective. Test the significance between the difference of two proportions at 5% level.
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Derive the formula for the waiting time of a customer has to wait before getting service.
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Derive formula for Expected number of customers in the system.
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There are three brands of tooth pastes, 1, 2 and 3. Transition of customers moving from one brand to another brand is given by the following transition probability matrix:
What are the long run market shares?
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