JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

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

B.Tech | II Year | I Semester | R15 | August/September - 2022

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

Answer any five questions. All questions carry equal marks.

Main

115 marks

Two dice are thrown and X is a random variable so that X = Sum (a, b) where a and b are numbers appeared on the two dice. Find the probability distribution of X and hence compute its mean and standard deviation.

8 marks

Answer

When two dice are thrown, there are 36 possible outcomes. The sum X ranges from 2 to 12. The probability distribution P(X=x) is calculated by counting the number of outcomes for each sum:

X	2	3	4	5	6	7	8	9	10	11	12
f(x)	1/36	2/36	3/36	4/36	5/36	6/36	5/36	4/36	3/36	2/36	1/36

The mean E(X) is calculated as:

E(X) = Σ xP(x) = (2(1) + 3(2) + 4(3) + 5(4) + 6(5) + 7(6) + 8(5) + 9(4) + 10(3) + 11(2) + 12(1)) / 36 = 252 / 36 = 7

The variance V(X) = E(X²) - [E(X)]²:

E(X²) = Σ x²P(x) = (4(1) + 9(2) + 16(3) + 25(4) + 36(5) + 49(6) + 64(5) + 81(4) + 100(3) + 121(2) + 144(1)) / 36 = 1974 / 36 ≈ 54.833

V(X) = 54.833 - 7² = 54.833 - 49 = 5.833

The standard deviation σ = √5.833 ≈ 2.415.

If the probability density of a random variable x is given by:

f(x) = k(x+x^3), 0 < x < 1; 0 elsewhere

Find the value k and the probability that the random variable takes on a value: (i) greater than 3/4; (ii) between 1/3 and 2/3.

7 marks

Answer

To find k, we use the condition that the total area under the probability density function must be 1:

∫₀¹ k(x + x³) dx = k[x²/2 + x⁴/4]₀¹ = k(1/2 + 1/4) = 3k/4 = 1 ⇒ k = 4/3

(i) Probability that x > 3/4:

P(X > 3/4) = ∫₃/₄¹ (4/3)(x + x³) dx = (4/3)[x²/2 + x⁴/4]₃/₄¹ = (4/3)[(1/2 + 1/4) - (9/32 + 81/256)] = (4/3)[3/4 - (72/256 + 81/256)] = (4/3)[3/4 - 153/256] = (4/3)[192/256 - 153/256] = (4/3)(39/256) = 13/64 ≈ 0.203

(ii) Probability between 1/3 and 2/3:

P(1/3 < X < 2/3) = ∫₁/₃²/₃ (4/3)(x + x³) dx = (4/3)[x²/2 + x⁴/4]₁/₃²/₃ = (4/3)[(4/18 + 16/324) - (1/18 + 1/324)] = (4/3)[(3/18 + 15/324)] = (4/3)[1/6 + 5/108] = (4/3)[18/108 + 5/108] = (4/3)(23/108) = 23/81 ≈ 0.284

215 marks

The probability that a certain kind of component will survive a shock test is 3/4. Find the probability that exactly 2 of the next 4 components tested survive.

5 marks

Answer

Find Moment Generating Function for normal distribution and hence find its mean and variance.

10 marks

Answer

315 marks

A two-dimensional random variable (X, Y) have a joint probability mass function:

P(X=x, Y=y) = (x+y)/32, for x = 0, 1, 2, 3 and y = 0, 1

Find the marginal distributions of X and Y.

8 marks

Answer

The joint probability density function of a two-dimensional random variable (X, Y) is given by:

f(x, y) = x+y, 0 < x < 1, 0 < y < 1; 0 elsewhere

Find the conditional density functions of Y given X=x and the conditional density functions of X given Y=y.

7 marks

Answer

415 marks

Obtain the rank correlation coefficient for the following data:

x: 48, 60, 72, 62, 56, 40, 39, 52, 30
y: 62, 78, 65, 70, 38, 54, 60, 32, 31

8 marks

Answer

Find the regression line of y on x for the following data:

x: 1, 2, 3, 4, 5
y: 2, 5, 3, 8, 7

7 marks

Answer

515 marks

The means of two large samples of 1000 and 2000 members are 67.5 inches and 68.0 inches respectively. Can the samples be regarded as drawn from the same population having standard deviation 2.5 inches? Test at 5% level of significance.

8 marks

Answer

Random samples of 400 men and 600 women were asked whether they would like to have a flyover near their residence. 200 men and 325 women were in favour of the proposal. Test the hypothesis that proportions of men and women in favour of the proposal are same against that they are not at 5% level.

7 marks

Answer

615 marks

Define the Snedecor’s F-distribution and write its properties.

8 marks

Answer

The Edison Electric Institute has published figures on the number of kilowatt hours used annually by various home appliances. It is claimed that a vacuum cleaner uses an average of 46 kilowatt hours per year. If a random sample of 12 homes included in a planned study indicates that vacuum cleaners use an average of 42 kilowatt hours per year with a standard deviation of 11.9 kilowatt hours, does this suggest at the 0.05 level of significance that vacuum cleaners use, on average, less than 46 kilowatt hours annually? Assume the population of kilowatt hours to be normal.

7 marks

Answer

715 marks

Explain briefly the main characteristics of Queuing system.

8 marks

Answer

A self-service canteen employs one cashier at its counter. 8 customers arrive per every 10 minutes on an average. The cashier can serve on average one per unit. Assuming that the arrivals are Poisson and the service time distribution is exponential. Determine:

The average number of customers in the system
The average queue length
Average time a customer spends in the system
Average waiting time of each customer

7 marks

Answer

815 marks

Explain Stationary and non-stationary random process.

7 marks

Answer

Explain Markov process and Markov chain.

8 marks

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

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