JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech I Year MATHEMATICS - III Previous Question Paper (November/December - 2020 - R16)

B.Tech | I Year | II Semester | R16 | November/December - 2020

Answer PaperPaper Code: 132ACMaximum Marks: 75Time: 2 hours

Answer any five questions

All questions carry equal marks

Main

Derive the formula for finding the mean of the binomial distribution.

15 marks

Answer

A binomial distribution is defined by the probability mass function for a random variable X, which represents the number of successes in n independent Bernoulli trials, each with probability of success p and failure q = 1 - p:

P(X = x) = ⁿC_x p^x q^n-x, for x = 0, 1, 2, ..., n

The mean (expected value) of a discrete random variable is defined as:

E(X) = Σ_x=0ⁿ x P(X = x)

Substituting the binomial probability formula:

E(X) = Σ_x=0ⁿ x [n! / (x!(n-x)!)] p^x q^n-x

Since the term for x = 0 is zero, we start the summation from x = 1:

E(X) = Σ_x=1ⁿ x [n! / (x!(n-1)!)] p^x q^n-x

Using the identity x/x! = 1/(x-1)! and factoring out np:

E(X) = np Σ_x=1ⁿ [(n-1)! / ((x-1)!(n-x)!)] p^x-1 q^n-x

Let k = x - 1 and m = n - 1. As x goes from 1 to n, k goes from 0 to m:

E(X) = np Σ_k=0^m [m! / (k!(m-k)!)] p^k q^m-k

By the Binomial Theorem, the summation Σ_k=0^m ^mC_k p^k q^m-k is equal to (p + q)^m. Since p + q = 1:

E(X) = np(p + q)^n-1 = np(1)^n-1 = np

Thus, the mean of the binomial distribution is E(X) = np.

Find the moment generating function of

f(x) = (3Cx)/8

x = 0, 1, 2, 3

and hence find its mean and variance.

15 marks

Answer

The probability mass function is given by f(x) = (3Cx)/8 for x = 0, 1, 2, 3. Calculating the values: f(0) = 1/8, f(1) = 3/8, f(2) = 3/8, and f(3) = 1/8.

The moment generating function M_X(t) is defined as E[e^tX] = Σ e^tx f(x):

M_X(t) = (1/8)e⁰ + (3/8)e^t + (3/8)e^2t + (1/8)e^3t = (1/8)(1 + 3e^t + 3e^2t + e^3t)

This can be simplified using the binomial expansion (1 + e^t)³:

M_X(t) = (1/8)(1 + e^t)³

To find the mean and variance, we use the derivatives of M_X(t) at t = 0:

Mean E[X] = M_X'(0):

M_X'(t) = (3/8)(1 + e^t)²(e^t)

E[X] = (3/8)(1 + 1)²(1) = (3/8)(4) = 1.5

Second moment E[X²] = M_X''(0):

M_X''(t) = (3/8)[(1 + e^t)²(e^t) + (e^t)(2(1 + e^t)e^t)]

M_X''(0) = (3/8)[(4)(1) + (1)(2)(2)(1)] = (3/8)(8) = 3

Variance Var(X) = E[X²] - (E[X])²:

Var(X) = 3 - (1.5)² = 3 - 2.25 = 0.75

315 marks

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

Distinguish between the point estimation and interval estimation.

8 marks

Answer

415 marks

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?

7 marks

Answer

If the sample of size 36 is taken and the mean and standard deviation of samples are 50 and 6. Find the probability that x lies between 48 and 51.

8 marks

Answer

515 marks

A random sample of 500 pineapples was taken from a large consignment and 65 were found to be bad. Show that the S.E. of the proportion of bad ones in a sample of this size is 0.015 and deduce that the percentage of bad pineapples in the consignment almost certainly lies between 8.5 and 17.5.

7 marks

Answer

Discuss various types of alternative hypothesis with suitable example.

8 marks

Answer

615 marks

A machine runs on an average of 125 hours/year. A random sample of 49 machines has an annual average use of 126.9 hours with standard deviation 8.4 hours. Does this suggest believing that machines are used on the average more than 125 hours annually at 5% level of significance?

7 marks

Answer

The average marks scored by 32 boys is 72 with a standard deviation of 8. While that for 36 girls is 70 with a standard deviation of 6. Does this indicate that the boys perform better than girls at level of significance 0.05?

8 marks

Answer

Find the root of the equation

3x - 1 + sin(x) = 0

by iteration method.

15 marks

Answer

Solve

dy/dx = x + y^2

y(0) = 1

for

y(0.1)

using Runge kutta method.

15 marks

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech I Year MATHEMATICS - III Previous Question Paper (November/December - 2020 - R16)

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