JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II Year PROBABILITY AND STATISTICS Previous Question Paper (March - 2022 - R15)

B.Tech | II Year | II-I | R15 | March - 2022

Answer PaperPaper Code: 123ANMaximum Marks: 75Time: 3 Hours

Answer any five questions. All questions carry equal marks.

Main

115 marks

A continuous Random variable has the p.d.f

f(x) = Ke^-λx for x ≥ 0, and 0 otherwise

. Determine: i) K ii) The mean iii) variance.

8 marks

Answer

i) To find K, we use the property of the probability density function that the total area under the curve must be 1:

∫[0 to ∞] K e^(-λx) dx = 1

Integrating the expression:

K [-1/λ e^(-λx)][0 to ∞] = 1

K (0 - (-1/λ)) = 1 => K/λ = 1 => K = λ

ii) The mean (E[X]) is defined as:

E[X] = ∫[0 to ∞] x f(x) dx = ∫[0 to ∞] x λ e^(-λx) dx

Using integration by parts:

E[X] = λ [(-x/λ e^(-λx)) - (1/λ^2 e^(-λx))][0 to ∞] = 1/λ

iii) To find the variance, we first find E[X^2]:

E[X^2] = ∫[0 to ∞] x^2 λ e^(-λx) dx = 2/λ^2

The variance is given by:

Var(X) = E[X^2] - (E[X])^2 = 2/λ^2 - (1/λ)^2 = 1/λ^2

Assume that 50% of the Engineers are good in Mathematics. Find the probability that among 9: i) Exactly 5 ii) At least 6 iii) At the most 5 are good in Mathematics.

7 marks

Answer

This is a binomial distribution problem with n = 9 and p = 0.5 (q = 1 - p = 0.5). The probability mass function is P(X = x) = 9Cx (0.5)^x (0.5)^(9-x) = 9Cx (0.5)^9.

i) Exactly 5 are good in Mathematics:

P(X = 5) = 9C5 (0.5)^9 = 126 / 512 ≈ 0.2461

ii) At least 6 are good in Mathematics:

P(X ≥ 6) = P(6) + P(7) + P(8) + P(9)

P(X ≥ 6) = (9C6 + 9C7 + 9C8 + 9C9) * (0.5)^9 = (84 + 36 + 9 + 1) / 512 = 130 / 512 ≈ 0.2539

iii) At the most 5 are good in Mathematics:

P(X ≤ 5) = 1 - P(X > 5) = 1 - (P(6) + P(7) + P(8) + P(9))

P(X ≤ 5) = 1 - (130 / 512) = 382 / 512 ≈ 0.7461

215 marks

If the variance of a Poisson variate is 2, find the probability that: i) P(X = 0) ii) P(1 < X < 4) iii) P(0 < X < 3)

7 marks

Answer

Suppose the weights of 1000 male students are normally distributed with mean

μ = 80

with a standard deviation of 10. Find the number of students whose weights are: i) Between 90 and 100 ii) More than 105.

8 marks

Answer

Calculate the coefficient of correlation and the two lines of regression between the two variables x and y.

x	14	16	17	18	19	20	21	22	23
y	84	78	70	75	66	67	62	58	60

15 marks

Answer

415 marks

Fit an equation of the form

y = ab^x

to the following data: x (2, 3, 4, 5, 6), y (144, 172.8, 207.4, 248.8, 298.5)

7 marks

Answer

The equations of two regression lines obtained in a correlation analysis are 3x + 12y = 19, 3y + 9x = 46 Find: i) Coefficient of Correlation ii) The means of x and y iii) The ratio of the coefficient of variability of x to that of y.

8 marks

Answer

515 marks

The S.D of a population is 140.5 and the sample size 50. Find the maximum error with 95% confidence.

7 marks

Answer

Random samples of 400 men and 600 women in a locality were asked whether they would like to have a bus stop near their residence 200 men and 325 women were in favour of the proposal. Test the significance between the difference of two proportions at 5% level.

8 marks

Answer

To determine whether really a relationship between an employer’s performance in the company in training program and his/her ultimate success in the job taking a sample of 400 cases, from its very extensive files and obtains the results shown in the following table:

	Below Average	Average	Above average
Poor	23	60	29
Average	28	79	60
Very good	9	49	63

15 marks

Answer

A P.C repair man finds that the time spent on jobs has an exponential distribution with mean 30 minutes. If the sets are repaired in the order, in which they come in, and if the arrival of sets is approximately Poisson with an average of 10 per 8 hours day. Find:

The expected idle time per day
The probability that there is no unit in the system.
Average number of units in the queue.
The average waiting time in queue
The probability that a unit will have to wait for more than 10 minutes.

15 marks

Answer

In a presidential election at the end of June 40% of the voters registered for liberal, 45% for conservative and 15% for independent. Over one month, of the people those who registered for liberal 80% were retained, 15% changed to conservative and 5% to independent. Of the people those who registered for conservative 70% were retained, 20% changed to liberal and 10% to independent. Of the people those who registered for independent, 60% were retained, 20% changed to liberal and 20% to conservative.

15 marks

Write transition probability matrix

7 marks

Answer

Find the % of the voters in each category at the end July and August.

8 marks

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II Year PROBABILITY AND STATISTICS Previous Question Paper (March - 2022 - R15)

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