JNTU
JNTU B.Tech I Year II Semester MATHEMATICS-III Previous Question Paper (June - 2022 - R16)
Answer any five questions
All questions carry equal marks
Main
For the continuous probability function
Answer
(i) For a probability density function, the total area must be 1: ∫0∞ kx e-x dx = 1. Using the gamma function property ∫0∞ xn e-ax dx = n! / an+1, we have k(1! / 12) = 1, which gives k = 1.
(ii) The mean E[X] = ∫0∞ x f(x) dx = ∫0∞ x2 e-x dx. Applying the gamma integral formula, E[X] = 2! / 13 = 2.
(iii) The variance V[X] = E[X2] - (E[X])2. First, find E[X2] = ∫0∞ x2 f(x) dx = ∫0∞ x3 e-x dx = 3! / 14 = 6. Thus, V[X] = 6 - (2)2 = 6 - 4 = 2.
Derive the mean and variance of random variable X having probability distribution
Answer
For the Poisson distribution f(x) = (λx e-λ) / x!:
Mean E[X] = ∑x=0∞ x (λx e-λ / x!) = e-λ ∑x=1∞ (λx / (x-1)!). Let y = x-1, then E[X] = e-λ λ ∑y=0∞ (λy / y!) = e-λ λ eλ = λ.
To find variance, first calculate E[X(X-1)] = ∑x=0∞ x(x-1) (λx e-λ / x!) = e-λ λ2 ∑x=2∞ (λx-2 / (x-2)!) = λ2 e-λ eλ = λ2. Since E[X2] = E[X(X-1)] + E[X] = λ2 + λ, the variance V[X] = E[X2] - (E[X])2 = (λ2 + λ) - λ2 = λ.
Obtain the moment generating function of the random variable X having the probability density function
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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.
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State Central Limit Theorem and using it approximate
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(i) Find the value of
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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.
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A random sample of size 25 from a normal population has the mean
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A die is tossed 180 times with the following results: x=1, f=28; x=2, f=36; x=3, f=36; x=4, f=30; x=5, f=27; x=6, f=23. Is it a balanced die?
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Suppose we wish to study the effect of computer-aided instruction (CAI) on the performance of college students. Perform analysis of variance and determine whether the differences among the means obtained for the three sections are significant at α=0.05.
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Find the real root of the equation
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Using Gauss Seidel method and solve the equations:
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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
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Evaluate
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Using R-K method of order four, solve
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