JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III PROBABILITY AND STATISTICS Previous Question Paper (August - 2022 - R15)
Answer any five questions
All questions carry equal marks
Main
If the discrete Random variable has the distribution:
| x | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| p(x) | k | 2k | 3k | 4k | 5k | 6k | 7k | 8k | 9k |
Find: i) K ii) Mean iii) P(2 ≤ x < 7).
Answer
i) Since the sum of probabilities in a discrete distribution must equal 1, we have: k + 2k + 3k + 4k + 5k + 6k + 7k + 8k + 9k = 1, which simplifies to 45k = 1. Thus, k = 1/45.
ii) The mean (E[X]) is calculated as Σx·p(x):
Substituting k = 1/45, E[X] = 285/45 = 19/3 ≈ 6.33.
iii) P(2 ≤ x < 7) = p(2) + p(3) + p(4) + p(5) + p(6) = (2k + 3k + 4k + 5k + 6k) = 20k.
Substituting k = 1/45, P(2 ≤ x < 7) = 20/45 = 4/9 ≈ 0.444.
A continuous random variable X has the probability density function 𝑓(𝑥) = 2𝑥, for 0 ≤ 𝑥 ≤ 1, and 0 otherwise. Find the cumulative distribution function for X. Also find P(1/4 < x ≤ 3/4).
Answer
The cumulative distribution function F(x) is defined as the integral of the probability density function f(t) from -∞ to x.
For 0 ≤ x ≤ 1:
Thus, the CDF is F(x) = 0 for x < 0, F(x) = x^2 for 0 ≤ x ≤ 1, and F(x) = 1 for x > 1.
To find P(1/4 < X ≤ 3/4), we use the CDF: P(1/4 < X ≤ 3/4) = F(3/4) - F(1/4).
A manufacturer of pins knows that 2% of his product is defective. If he sells pins in boxes of 100 and guarantees that not more than 4 pins will be defective. What is the probability that a box will fail to meet the guaranteed quality?
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Find the probability of getting 1 or 4 or 5 or 6 in throwing a die 5 to 7 times among 9 trials using normal distribution.
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Discuss types of error of statistical hypothesis and give example.
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A normal population has a mean 0.1 and a standard deviation of 2.1. Find the probability that the mean of simple sample of 900 members will be negative.
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The mean income of 100 servants is Rs. 250 with a standard deviation of 20. Test whether it has been drawn from the population whose mean is Rs. 220?
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Under quality improvement programme some teachers are trained by instruction methodology A and some by methodology B. In a random sample of size 10, taken from a large group of teachers exposed to each of these two methods, the following marks are obtained in an appropriate achievement test:
Method A: 65, 69, 73, 71, 75, 66, 71, 68, 68, 74
Method B: 78, 69, 72, 77, 84, 70, 73, 77, 75, 65
Assuming that populations sampled are approximately normally distributed has same variance. Test the claim that method B is more effective at 0.05 LOS.
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Twenty five pairs of value of variates X and Y led to the following results: N = 25, ∑x = 127, ∑y = 100, ∑x2 = 760, ∑y2 = 449, ∑xy = 500. A subsequent scrutiny showed that two pairs of values were copied down as (8, 14) and (8, 6) instead of (8, 12) and (6, 8). Find correct value of r and correct lines of regression.
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Discuss about classification of queuing models.
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Show that for a single service station, Poisson arrivals and exponential service time, the probability that exactly n calling units are in the queuing system is Pn = (1-ρ)ρn, n ≥ 0, where ρ is the traffic intensity.
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A gambler has Rs.1. He bets Rs.0.5 at a time and wins Rs.0.5 with probability ½. He stops playing if he loses Re.1 or wins Rs.2.
What is the transition probability matrix of the related Markov chain?
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What is the probability that he has lost his money at the end of 5 plays?
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What is the probability that the game lasts more than 7 plays?
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