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JNTU B.Tech II PROBABILITY AND STATISTICS Previous Question Paper (March - 2021 - R13/R15)

B.Tech | II | I | R13/R15 | March - 2021

Answer PaperPaper Code: 123AN/113ANMaximum Marks: 75Time: 3 hours

Answer any five questions. All questions carry equal marks.

Main

115 marks

The frequency function of a continuous random variable X is given by

f(x) = A(2x - x^2), 0 ≤ x ≤ 2

. Find variance of X.

7 marks

Answer

To find the variance, we first determine the constant A using the property that the total area under the probability density function is 1:

∫₀² A(2x - x²) dx = 1

Integrating gives:

A [x² - (x³/3)]₀² = A(4 - 8/3) = A(4/3) = 1

Thus, A = 3/4. The mean E(X) is:

E(X) = ∫₀² x(3/4)(2x - x²) dx = (3/4) ∫₀² (2x² - x³) dx = (3/4) [2x³/3 - x⁴/4]₀² = (3/4)(16/3 - 16/4) = (3/4)(16/12) = 1

Next, we find E(X²):

E(X²) = ∫₀² x²(3/4)(2x - x²) dx = (3/4) ∫₀² (2x³ - x⁴) dx = (3/4) [2x⁴/4 - x⁵/5]₀² = (3/4)(8 - 32/5) = (3/4)(8/5) = 6/5 = 1.2

The variance V(X) is:

V(X) = E(X²) - [E(X)]² = 1.2 - (1)² = 0.2

Let

f(x) = 3x^2

, when

0 ≤ x ≤ 1

be the probability density function of a continuous random variable X. Determine a and b such that i)

P(X ≤ a) = P(X > a)

ii)

P(X > b) = 0.05

8 marks

Answer

i) Given P(X ≤ a) = P(X > a), and since P(X ≤ a) + P(X > a) = 1, it follows that P(X ≤ a) = 0.5. We solve for a:

∫₀^a 3x² dx = 0.5

[x³]₀^a = 0.5 ⇒ a³ = 0.5 ⇒ a = (0.5)^1/3 ≈ 0.7937

ii) Given P(X > b) = 0.05, then P(X ≤ b) = 0.95:

∫₀^b 3x² dx = 0.95

[x³]₀^b = 0.95 ⇒ b³ = 0.95 ⇒ b = (0.95)^1/3 ≈ 0.9830

215 marks

Find the moment generating function of

f(x) = (1/8) * 3Cx

x = 0, 1, 2, 3

. and hence find its mean and variance.

7 marks

Answer

If 3 fair coins are tossed and if X is the total number of heads in the outcome. Show that X is a random variable.

8 marks

Answer

The joint probability density function of the random variable X and Y is given by

f(x, y) = K(xy + y^2)

0 ≤ x ≤ 1, 0 ≤ y ≤ 2

. Find

P(Y > 1)

P(X > 1/2, Y < 1)

and

P(X + Y ≤ 1)

15 marks

Answer

415 marks

Construct sampling distribution of means for the population 3, 7, 11, 15 by drawing samples of size two with replacement. Determine i) μ ii) σ iii) Sampling distribution of means iv) μx v) σx.

8 marks

Answer

The mean of certain normal population is equal to the standard error of the mean of the samples of 64 from that distribution. Find the probability that the mean of the sample size 36 will be negative.

7 marks

Answer

515 marks

What is meant by a statistical hypothesis? What are the two types of errors of decision that arise in testing a hypothesis?

7 marks

Answer

A random sample of boots worn by 40 combat soldiers in a desert region showed an average life of 1.08 years with a standard deviation of 0.05 years. Under standard conditions the boots are known to have an average life of 1.28 years. Is there reason to assert at a level of significance of 0.05 that use in the desert causes the mean life of such boots to decrease?

8 marks

Answer

615 marks

Discuss various types of alternative hypothesis with suitable example.

7 marks

Answer

In 1950 in India the mean life expectancy was 50 years. If the life expectancies from a random sample of 11 persons are 58.2, 56.6, 54.2, 50.4, 44.2, 61.9, 57.5, 53.4, 49.7, 55.4, 57.0. Does it confirm the expected view?

8 marks

Answer

715 marks

Discuss basic queuing process.

7 marks

Answer

Workers come to a tool store room to enquiry about the special tools for a particular job. The average time between the arrivals is 60 seconds and the arrivals are assumed to be in Poisson distribution. The average service time is 40 seconds. Find i) average queue length ii) Average length of non-empty queue.

8 marks

Answer

815 marks

A fair die is tossed repeatedly. If Xn denotes the maximum of the number occurring in the first n tosses, find the transition probability matrix P of the Markov chain {Xn}. Find also P^2 and P(X2 = 6).

8 marks

Answer

The transition probability matrix of a Markov chain is given by

[[0.3, 0.7, 0], [0.1, 0.4, 0.5], [0, 0.2, 0.8]]

. Is this matrix irreducible?

7 marks

JNTU B.Tech II PROBABILITY AND STATISTICS Previous Question Paper (March - 2021 - R13/R15)

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