JNTU
JNTU B.Tech I Year MATHEMATICS - III Previous Question Paper (August - 2018 - R16)
Note: This question paper contains two parts A and B. Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 Units. Answer any one full question from each unit. Each question carries 10 marks and may have a, b, c as sub questions.
Part A
Find the mean and variance of the probability distribution having pdf,
Answer
For the probability density function f(x) = e^(-x) where x > 0, the mean (E[X]) is calculated as the integral from 0 to infinity of x * e^(-x) dx. Using integration by parts, this yields 1. The variance (V[X]) is calculated as E[X^2] - (E[X])^2. E[X^2] is the integral from 0 to infinity of x^2 * e^(-x) dx, which equals 2. Therefore, the variance is 2 - (1)^2 = 1.
A fair coin is tossed until a head or consecutive five tails occurs. Find the discrete probability distribution.
Answer
Let X be the random variable representing the number of tosses until either a head (H) appears or five consecutive tails (TTTTT) occur. The possible values for X are 1, 2, 3, 4, and 5.
We calculate the probability P(X = x) for each outcome:
- P(X = 1): The first toss is H. P(X = 1) = 1/2.
- P(X = 2): The first toss is T, second is H. P(X = 2) = (1/2) * (1/2) = 1/4.
- P(X = 3): The sequence is TTH. P(X = 3) = (1/2)^3 = 1/8.
- P(X = 4): The sequence is TTTH. P(X = 4) = (1/2)^4 = 1/16.
- P(X = 5): The sequence is TTTTH or TTTTT. P(X = 5) = (1/2)^5 + (1/2)^5 = 1/32 + 1/32 = 2/32 = 1/16.
The probability distribution is given by P(X=1)=1/2, P(X=2)=1/4, P(X=3)=1/8, P(X=4)=1/16, and P(X=5)=1/16. Note that the sum of probabilities is 1/2 + 1/4 + 1/8 + 1/16 + 1/16 = 1.
Write the conditions of validity of
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Construct sampling distribution of means for the population 3, 7, 11, 15 by drawing samples of size two without replacement. Determine i)
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Discuss types of errors of the test of hypothesis.
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Give the graphical interpretation of the bisection method.
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Write the iterative formula for finding
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Explain briefly about method of least square.
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Derive Trapezoidal rule for computing integral
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What are the limitations of Taylor’s series method?
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Part B
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.
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Probability density function of a random variable X is
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A die is cast until 6 appears. What is the probability that it must be cast more than 5 times?
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The marks obtained in mathematics by 1000 students is normally distributed with mean 78% and standard deviation 11%. Determine: i) What was the highest mark obtained by the lowest 25% students? ii) Within what limit did the middle 90% of the student lie?
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Explain why the larger variance is placed in the numerator of the statistic F. Discuss the application of F-test in testing if two variances are homogenous.
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A sample of 11 rats from a central population had an average blood viscosity of 3.92 with a standard deviation of 0.61. Estimate the 95% confidence limits for the mean blood viscosity of the population.
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The mean voltage of a battery is 15 and standard deviation 0.2. Find the probability that four such batteries connected in series will have a combined voltage of 60.8 or more volts.
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Discuss critical region and level of significance with example.
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Suppose the diameter of motor shafts in a lot have a mean of 0.249 inches and standard deviation if 0.003 inches. The inner diameter of bearings in another lot have a mean of 0.255 inches and standard deviation of 0.002 inches. If a shaft and bearing are selected at random, find the probability that the shaft will not fit inside the bearing. Assume that both dimensions are normally distributed.
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A sample of 400 items is taken from a normal population whose mean is 4 and variance 4. If the sample mean is 4.45, can the samples be regarded as a simple sample?
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In a sample of 600 students of a certain college 400 are found to use ball pens. In another college from a sample of 900 students 450 were found to use ball pens. Test whether two colleges are significantly different with respect to the habit of using ball pens?
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Estimate y at x = 5 by fitting a least squares curve of the form
| x | 3.6 | 4.8 | 6.0 | 7.2 | 8.4 | 9.6 | 10.8 |
| y | 0.83 | 0.31 | 0.17 | 0.10 | 0.07 | 0.05 | 0.04 |
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Show that the Gauss-Seidel methods diverge for solving the system of equations 2x + 3y + z = -1; 3x + 2y + 2z = 1; x + 2y + 2z = 6.
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Find the successive approximate solution of the differential equation y' = y, y(0) = 1 by Picard’s method and compare it with exact solution.
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Use Runge-Kutta method of order four to find y when x = 0.6 in steps of 0.2 given that
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