JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech I Year MATHEMATICS - III Previous Question Paper (June - 2022 - R16)

B.Tech | I Year | II Semester | R16 | June - 2022

Answer PaperPaper Code: 132ACMaximum Marks: 75Time: 3 Hours

Answer any five questions. All questions carry equal marks.

Main

115 marks

For the continuous probability function

f(x) = kx e^-x

when

x ≥ 0

, find (i) k (ii) Mean (iii) Variance.

9 marks

Answer

(i) Since f(x) is a probability density function, the total area under the curve must be 1:

∫₀^∞ kx e^-x dx = 1

Using the Gamma function integral formula ∫₀^∞ x^n-1 e^-ax dx = Γ(n) / aⁿ, where n=2 and a=1:

k Γ(2) / 1² = 1 ⇒ k(1!) = 1 ⇒ k = 1

(ii) The Mean (E[X]) is given by:

E[X] = ∫₀^∞ x f(x) dx = ∫₀^∞ x² e^-x dx

Using the same formula with n=3 and a=1:

E[X] = Γ(3) / 1³ = 2! = 2

(iii) The Variance is given by Var(X) = E[X²] - (E[X])². First, calculate E[X²]:

E[X²] = ∫₀^∞ x² f(x) dx = ∫₀^∞ x³ e^-x dx = Γ(4) = 3! = 6

Thus, Var(X) = 6 - (2)² = 6 - 4 = 2.

Derive the mean and variance of random variable X having probability distribution

f(x) = (λ^x e^-λ) / x!

x = 0, 1, 2, ..., λ > 0

6 marks

Answer

For the Poisson distribution f(x) = (λ^x e^-λ) / x!:

Mean E[X] = ∑_x=0^∞ x (λ^x e^-λ) / x! = ∑_x=1^∞ (λ^x e^-λ) / (x-1)!

Let y = x-1, then x = y+1:

E[X] = e^-λ ∑_y=0^∞ λ^y+1 / y! = λ e^-λ ∑_y=0^∞ λ^y / y! = λ e^-λ e^λ = λ

To find Variance, calculate E[X(X-1)]:

E[X(X-1)] = ∑_x=0^∞ x(x-1) (λ^x e^-λ) / x! = ∑_x=2^∞ (λ^x e^-λ) / (x-2)! = λ² e^-λ ∑_x=2^∞ λ^x-2 / (x-2)! = λ²

Since E[X²] = E[X(X-1)] + E[X] = λ² + λ, the Variance is:

Var(X) = E[X²] - (E[X])² = λ² + λ - (λ)² = λ

215 marks

Obtain the moment generating function of the random variable X having the probability density function

f(x) = x, 0 ≤ x ≤ 1

;

f(x) = 2-x, 1 ≤ x ≤ 2

;

f(x) = 0, elsewhere

8 marks

Answer

Suppose that the temperature is normally distributed with expectation 50 °C and variance 4 °C. What is the probability that the temperature T will be between 48 °C and 58 °C.

7 marks

Answer

315 marks

State Central Limit Theorem and using it approximate

P(0.03 ≤ X ≤ 0.15)

, where X is the mean of a random sample size n=15 from a distribution whose PDF is

f(x) = 3/2 x², -1 ≤ x ≤ 1

8 marks

Answer

(i) Find the value of

P(0.025 ≤ T ≤ 0.05)

(ii) Find ‘k’ such that

P(k ≤ T ≤ 1.761) = 0.045

7 marks

Answer

415 marks

A sample of 400 items is taken from a population whose standard deviation is 15. The mean of the sample is 25. Test whether the sample has come from a population with mean 26.8. Also calculate that 98% confidence limits of the population mean.

7 marks

Answer

A random sample of size 25 from a normal population has the mean x = 47.5 and the standard deviation s = 8.4. Does this information support or refute the claim that the mean of the population is μ = 42.1?

8 marks

Answer

515 marks

A die is tossed 180 times with the following results: x=1,2,3,4,5,6; f=28, 36, 36, 30, 27, 23. Is it a balanced die?

7 marks

Answer

Suppose we wish to study the effect of computer-aided instruction (CAI) on the performance of college students. We randomly divide the incoming class into three sections. Perform analysis of variance and determine whether the differences among the means obtained for the three sections are significant at α = 0.05.

8 marks

Answer

615 marks

Find the real root of the equation

x⁴ + 2x³ - x - 1 = 0

by Regula Falsi method correct to three decimal places.

7 marks

Answer

Using Gauss Seidel method and solve the equations:

4x + y + z = 7; 8x + 5y + 3z = 21; 2x + y + 6z = 10

8 marks

Answer

Using least squares method fit a quadratic curve to the following data: x: -2, -1, 0, 1, 2; f(x): 13, 6, 3, 4, 9.

15 marks

Answer

815 marks

Evaluate

∫₀¹ (1 / (1+x+x³)) dx

using (i) Trapezoidal rule (ii) Simpson’s 3/8th rule.

7 marks

Answer

Using R-K method of order four, solve

y'' - xy' - y = 0

y(0) = 1, y'(0) = 0

to find y(0.2) and y'(0.2).

8 marks

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech I Year MATHEMATICS - III Previous Question Paper (June - 2022 - R16)

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