JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B. Tech II PROBABILITY AND STATISTICS Previous Question Paper (February - 2024 - R15)
Note: i) Question paper consists of Part A, Part B. ii) Part A is compulsory, which carries 25 marks. In Part A, answer all questions. iii) In Part B, Answer any one question from each unit. Each question carries 10 marks and may have a, b as sub questions.
Part A
Define a random variable and discuss the various types of random variables with examples in each case.
Answer
A random variable is a real-valued function that assigns a unique numerical value to each outcome in the sample space of a random experiment. It maps the set of all possible outcomes to the set of real numbers.
Random variables are classified into two main types:
- Discrete Random Variable: A variable that can take on a countable number of distinct values. For example, the number of heads obtained in three tosses of a coin (0, 1, 2, or 3).
- Continuous Random Variable: A variable that can take on any value within a specified interval or range. For example, the height of students in a class or the time taken to complete a task, which can be measured to any degree of precision.
Discuss the area property of normal distribution in detail.
Answer
The area property of the normal distribution refers to the fact that the total area under the probability density curve is exactly 1, representing the total probability of all possible outcomes. This distribution is symmetric about its mean, μ, meaning 50% of the area lies to the left of the mean and 50% to the right.
Key empirical rules regarding the area include:
- Approximately 68.27% of the area falls within one standard deviation (μ ± σ) of the mean.
- Approximately 95.45% of the area falls within two standard deviations (μ ± 2σ) of the mean.
- Approximately 99.73% of the area falls within three standard deviations (μ ± 3σ) of the mean.
These properties allow for the calculation of probabilities for any interval by finding the area under the curve using the standard normal distribution (Z-table) after converting raw scores into Z-scores using the formula:
Define the joint probability density function.
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Discuss various types of correlation.
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Define Type-I error and type-II error.
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Define the point estimation and interval estimation.
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Describe the probability distribution of queue length.
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Explain the basic probabilistic queuing models.
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What is a transition diagram?
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Find the unique fixed probability vector of the matrix A = [[0.3, 0.7], [0.5, 0.5]].
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Part B
Let X denotes the minimum of the two numbers that appear when a pair of dice is thrown once. Determine the discrete probability distribution, expectation, and variance.
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Define normal distribution and hence find its mean and variance.
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| X\Y | -1 | 0 | 1 |
|---|---|---|---|
| 0 | 3/18 | 2/18 | 3/18 |
| 1 | 1/18 | 1/18 | K/18 |
| 2 | 1/18 | 2/18 | 1/18 |
The probabilities of random variables X and Y are given above. Find the value of K, the joint distribution function, and marginal distribution functions and plot these.
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Find the regression lines of y on x and x on y to the given data set.
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Two horses A and B were tested according to the time (in seconds) to run a particular track. Test whether the two horses have the same running capacity.
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Fit a Poisson distribution to the following data and test for goodness of fit at 0.05 level of significance.
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Queuing system analysis.
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The arrival rate of customers at a railway ticket window follows Poisson distribution with a mean of 50 per hour. The service rate of a window clerk also follows Poisson distribution with a mean of 70 per hour. Find probability of zero customers, average number in system, average number in queue, average waiting time in system, and average waiting time in queue.
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Markov process analysis.
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A student tries to take admission in only three colleges A, B and C. First day he goes to A, second day to B and third day to C. He never goes to some college in two continuous days, but if he goes either B or C, then the next day he is twice as likely to go to college A. Find out how often, in the long run he tries to go to each college?
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