JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II Year PROBABILITY AND STATISTICS Previous Question Paper (September - 2021 - R13)
Answer any five questions. All questions carry equal marks.
Main
A random variable X has the following probability distribution.
X: 0 1 2 3 4
P(X): 3k 3k K 2k 6k
Find: a) k, b) var(X), c) var(2X-4), d) P(X < 2) and e) P(X > 1).
Answer
a) Since the sum of probabilities must equal 1:
b) First, calculate the mean E(X):
Next, calculate E(X2):
Variance Var(X) = E(X2) - [E(X)]2:
c) Using the property Var(aX + b) = a2Var(X):
d) P(X < 2) = P(X=0) + P(X=1):
e) P(X > 1) = P(X=2) + P(X=3) + P(X=4):
If 2% of the items of a factory are defective and the items are packed in boxes, find the probability that there will be atleast 3 defective items in a box of 100 items.
Answer
Given that the probability of a defective item is p = 0.02 and the number of items in a box is n = 100, we use the Poisson distribution as an approximation since n is large and p is small. The parameter λ is calculated as:
We need to find the probability of having at least 3 defective items, P(X ≥ 3). Using the complement rule:
Using the Poisson formula P(X=k) = (e⁻λ × λᵏ) / k!:
Summing these probabilities: 0.1353 + 0.2707 + 0.2707 = 0.6767. Therefore:
In a Normal distribution, 7% of the items are under 35 and 89% are under 63. Find the mean and variance of the distribution.
Answer
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For the following bivariate probability distribution of X and Y:
| X\Y | 0 | 1 | 2 | 3 |
| 0 | 1/32 | 2/32 | 2/32 | 3/32 |
| 1 | 2/32 | 8/32 | 1/4 | 1/32 |
| 2 | 2/32 | 3/32 | 16/32 | 0 |
Find: a) P(X ≤ 1, Y = 3), b) P(Y ≤ 2), c) P(X ≤ 1, Y ≤ 2), d) FX(2) and e) FY(1).
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Find the coefficient of correlation and the two lines of regression.
X: 10 12 18 24 23 27
Y: 13 18 12 25 30 10
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If samples of size 2 are taken from the population 1, 2, 3, 4, 5, 6 without replacement, then find:
the mean and standard deviation of population.
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the mean and standard deviation of the sampling distribution of means.
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Fit a Poisson distribution to the following data and test for goodness of fit at 5% level of significance.
x: 0 1 2 3 4
f: 214 92 20 3 1
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Cars arrive at a petrol pump with exponential interval times having mean 1/2 minute. The attendant takes an average of 1/5 minute per car to supply petrol and the service time being exponentially distributed. Find:
the average number of cars waiting to be served
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the average number of cars in the system and
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the proportion of time for which the pump attendant is idle.
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A gambler has Rs. 2. He bets 1 at a time and wins Rs. 1 with probability 1/2. He stops playing if loses Rs. 2 or wins Rs. 4. Find:
the tpm of the related Markov chain
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the probability that he has lost his money at the end of 5 plays.
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the probability that the game lasts more than 7 plays.
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