Jawaharlal Nehru Technological University Hyderabad
Jawaharlal Nehru Technological University Hyderabad B.Tech I MATHEMATICS - III Previous Question Paper (July/August - 2021 - R16)
Answer any five questions. All questions carry equal marks.
Main
If the pdf of a continuous random variable is given by f(x) = e-x, 0 ≤ x < ∞, find the mean and variance of the random variable.
Answer
For a continuous random variable with probability density function f(x) = e-x for x ≥ 0, the mean E(X) is calculated as:
Using integration by parts, E(X) = [-x e-x]0∞ + ∫0∞ e-x dx = 0 + [-e-x]0∞ = 1. Thus, the mean is 1.
To find the variance, we first calculate E(X2):
Using the gamma function property ∫0∞ xn e-ax dx = n! / an+1, we get E(X2) = 2! / 13 = 2. The variance V(X) is given by:
Compute the first four moments about the mean from the following data:
| x | 1 | 2 | 3 | 4 | 5 |
| f | 2 | 3 | 5 | 4 | 1 |
Answer
First, calculate the mean (μ):
The moments about the mean μr are defined as Σ fi(xi - μ)r / N, where N = 15:
- μ1 = Σ fi(xi - 3) / 15 = (2(-2) + 3(-1) + 5(0) + 4(1) + 1(2)) / 15 = 0
- μ2 = Σ fi(xi - 3)2 / 15 = (2(4) + 3(1) + 5(0) + 4(1) + 1(4)) / 15 = 20 / 15 = 1.333
- μ3 = Σ fi(xi - 3)3 / 15 = (2(-8) + 3(-1) + 5(0) + 4(1) + 1(8)) / 15 = -5 / 15 = -0.333
- μ4 = Σ fi(xi - 3)4 / 15 = (2(16) + 3(1) + 5(0) + 4(1) + 1(16)) / 15 = 55 / 15 = 3.667
A continuous random variable X is normally distributed with mean 25 and standard deviation 8. Find the probability that i) 20 ≤ X ≤ 40 and ii) X ≥ 35.
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A random sample of size 100 is taken from an infinite population having the mean 76 and variance 256. Find the probability that mean of sample will be between 75 and 78.
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Determine a 95% confidence interval for the mean of a normal distribution with variance 0.25 using a sample of 100 values with mean 212.3.
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A sample of 64 students have a mean weight 70 kgs. Can this be regarded as a sample from a population with mean weight 56 kgs and standard deviation 25 kgs?
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The following are the values of skills of 2 samples with individuals 5 and 6. Test the difference between the means.
| Sample 1 | 74.1 | 77.7 | 74.4 | 74.0 | 73.8 | |
| Sample 2 | 70.8 | 74.9 | 74.2 | 70.4 | 69.2 | 72.2 |
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Random samples of 500 men and 500 women were asked whether they would like to have a fly over near their residence. 150 men and 200 women are in favor of it. Test the difference between the proportions.
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Perform three iterations of the Gauss-Seidel method to solve the system of equations:
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The equation x ln(x) - 3 = 0 has a root in (4,5). Find the root correct to three decimal places by Newton-Raphson method.
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Evaluate the integral of x dx from 1.0 to 1.6 using Simpson’s 1/3rd rule with h = 0.1.
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Find the approximate value of y(x) at x = 1.2, 1.4, 1.6 for the initial value problem y' = x + y, y(1) = 1, h = 0.2 by Euler’s method.
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Apply Runge-Kutta method of order 4 to find the approximate value of y(1.3) for y' = x2 + y2, y(1) = 2 with h = 0.1.
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