JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II PROBABILITY AND STATISTICS Previous Question Paper (November/December - 2020 - R13)
Answer any five questions. All questions carry equal marks.
Main
Let X denote the minimum of the two numbers that appear when a pair of fair dice is thrown once. Determine the: i) Discrete probability distribution ii) Expectation iii) Variance.
Answer
When two fair dice are thrown, there are 36 equally likely outcomes. Let X be the minimum of the two numbers. The distribution is:
| x | 1 | 2 | 3 | 4 | 5 | 6 |
| P(X=x) | 11/36 | 9/36 | 7/36 | 5/36 | 3/36 | 1/36 |
ii) The expectation E(X) = Σ xP(x) = (1*11 + 2*9 + 3*7 + 4*5 + 5*3 + 6*1) / 36 = (11 + 18 + 21 + 20 + 15 + 6) / 36 = 91/36 ≈ 2.5278.
iii) The variance V(X) = E(X²) - [E(X)]². First, E(X²) = Σ x²P(x) = (1²*11 + 2²*9 + 3²*7 + 4²*5 + 5²*3 + 6²*1) / 36 = (11 + 36 + 63 + 80 + 75 + 36) / 36 = 301/36 ≈ 8.3611. Thus, V(X) = 301/36 - (91/36)² = (10836 - 8281) / 1296 = 2555 / 1296 ≈ 1.9715.
A random variable x has the following probability distribution:
| X | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| P(X=x) | k | 2k | 3k | 4k | 5k | 6k | 7k | 8k |
Find the value of: i) k ii) P(X < 2) iii) P(2 ≤ X ≤ 5).
Answer
i) Since the sum of probabilities must be 1: k + 2k + 3k + 4k + 5k + 6k + 7k + 8k = 1, which simplifies to 36k = 1, so k = 1/36.
ii) P(X < 2) = P(X=1) = k = 1/36.
iii) P(2 ≤ X ≤ 5) = P(X=2) + P(X=3) + P(X=4) + P(X=5) = 2k + 3k + 4k + 5k = 14k = 14/36 = 7/18.
Calculate the rank correlation coefficient for the following data:
| X | 65 | 66 | 67 | 67 | 68 | 69 | 70 | 72 |
| Y | 67 | 68 | 65 | 68 | 72 | 72 | 69 | 71 |
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A marketing professor at a given college is interested in the relationship between hours spent studying and total points earned in a course. Data collected from 10 students who took the course last quarter follow:
| Hours Spent Studying | 45 | 30 | 90 | 60 | 105 | 65 | 90 | 80 | 55 | 75 |
| Total Points Earned | 40 | 35 | 75 | 65 | 90 | 50 | 90 | 80 | 45 | 65 |
Develop an estimated regression equation showing how total points earned is related to hours spent studying.
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Predict total points earned when the student spent 95 hours studying.
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Population consists of five numbers 5, 10, 14, 18, 13 and 24. Consider all possible samples of size two which can be drawn without replacement from the population, then find:
The mean of the population
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The variance of the population
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The mean of the sampling distribution of means
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The standard deviation of the sampling distribution of means.
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A dice is thrown 150 times with the following results:
| Results (turned up) | 1 | 2 | 3 | 4 | 5 | 6 |
| Frequency | 19 | 23 | 28 | 17 | 32 | 31 |
Test the hypothesis that the dice is unbiased or not.
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Customers arrive at an automated coffee-vending machine at a rate of four per minute; following a Poisson distribution. The coffee machine dispenses cups of coffee at a constant time of 10 seconds.
What is the average number of people waiting in line?
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What is the average number in the system?
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How long does the average person wait in line before receiving service?
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Calculate traffic intensity.
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Barber A takes 15 minutes to complete one hair cut. Customers arrive in his shop at an average rate of one every 30 minutes. Barber B takes 25 minutes to complete one hair cut and customers arrive at his shop at an average rate of one every 50 minutes. The arrival processes are Poisson and the service times follow an exponential distribution.
Where would you expect a bigger queue
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Where would you require more time waiting included to complete a haircut.
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A city is divided into three zones I, II, and III. From the records of drivers the following information is available. Out of passengers picked up in zone I, (II) (III) 40 percent, (30 percent) (30 percent) go to a destination in zone I, 40 percent (40 percent) (50 percent) are taken in zone II, 20 percent (30 percent) (20 percent) go to a destination in zone III. Suppose in the beginning of the day 30 percent of taxis are in zone I, 30 percent is in zone II and 40 percent is in zone III. What is the distribution of taxis after all have had one ride, two rides? Also determine the long run distribution of taxis.
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