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Gujarat Technological UniversityWinter 2025 Examination

GTU 3130006 Probability and Statistics (PS) Winter 2025 Paper Solution & PDF

B.E. · Computer Engineering · Semester 3 · Subject Code: 3130006
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Total Marks70 MarksExternal theory exam
Passing Marks23 Marks33% minimum cutoff
Exam Duration2.5 Hours10:30 AM – 1:00 PM
Paper Structure5 QuestionsWith internal OR choices
Jump toQ1Q2Q3Q4Q5

Question 1

14 MarksMedium
(a)

A can hit a target 4 times in 5 shots, B, 3 times in 4 shots; and C, 2 times in 3 shots. They fire a volley (it means one shot each). Find the probability that target will be hit.

3 Marks
(b)

Three companies A, B and C supply 30%, 50% and 20% of the notebooks to a school. Past experience shows that 8%, 5% and 10% of the notebooks produced by these companies are defective. If a notebook was found to be defective, what is the probability that the notebook was supplied by A?

4 Marks
(c)

A random variable X has the following probability function X=x -2 -1 0 1 2 3 P(X=x) 0.1 K 0.2 2K 0.3 3K

iFind K (ii) Find P(X<2)
iiiDetermine the distribution function F(x) of X.
7 Marks

Question 2

14 MarksMedium
(a)

Four balls are drawn simultaneously from a bag containing 6 white, 5 black and 7 red balls. Find E(X), where X denotes number of white drawn.

3 Marks
(b)

It is known that the mean diameters of rivets produced by two firms A and B are practically the same, but the standard deviation may differ. For 22 rivets produced by firm A , the standard deviation is 2.9mm, while for 16 rivets manufactured by firm B, the standard deviation is 3.8mm. Compute the statistic you would use to test whether the products of firm A have the same variability as those of firm B. Using the value F0.05 (15,21) = 2.20

4 Marks
(c)

Fit a curve of the formx ab y = to the following data x : 1 2 3 4 5 6 7 y : 87 97 113 129 202 195 193

7 Marks
OR OPTION
(c)

Fit the second degree parabola using the least square method to the following data x : 1 2 3 4 y : 1.7 1.8 2.3 3.2

7 Marks

Question 3

14 MarksMedium
(a)

Find the mean and median for the given data

CI. 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100 f 4 6 5 10 20 22 24 6 2
3 Marks
(b)

The first four moments of a distribution about the value 5 of the variate are 2, 20, 40, and 50. Find mean and variance.

4 Marks
(c)

Calculate the correlation coefficient between x and y using the following data x 1 2 3 4 5 6 7 8 9 y 9 8 10 12 11 13 14 16 15

7 Marks
OR OPTION
(a)

State Chebyshev’s inequality. A fair dice is tossed 120 times. Use Chebyshev’s inequality to find a lower bound for the probability of getting 80 to 120 sixes.

3 Marks
(b)

Calculate the rank correlation coefficient for given data X 85 74 85 50 65 78 74 60 74 90 Y 78 91 78 58 60 72 80 55 68 70

4 Marks
(c)

The equation of two regression lines obtains in a correlation analysis of 60 observations are 3608 786 1000 24 6 5 − = + = x y and y x . What is the correlation coefficient? Show that the ratio of coefficient of variability ofx to that ofy is 5/24. What is the ratio of variance of y and x .

7 Marks

Question 4

14 MarksMedium
(a)

A speaks truth in 75% cases and B in 80% cases. Find the probability that they are likely to contradict each other in stating the same fact.

3 Marks
(b)

The incidence of occupational disease in an industry is such that the workmen have a 20% chance of suffering from it. What is the probability that out of 6 workmen 4 or more will suffer from disease?

4 Marks
(c)

The following mistakes per page were observed in a book. No. of Mistake per page 0 1 2 3 4 Total No of Pages 211 90 14 5 0 320 Fit a Poisson distribution for the given data.

7 Marks
OR OPTION
(a)

Suppose on an average 1 house in 1000 in a certain district has a fire during a year. If there are 2000 houses in that district, what is the probability that exactly 5 houses will have a fire during the year?

3 Marks
(b)

In a normal distribution 31% of the items are under 45 and 8% are over 64. Find the parameters of the distribution.

4 Marks
(c)

In a sample of 10 pens. If 10% of pens produced by a company are defective. Find out the probability that

iNo pen must be defective.
iiOne must be defective.
iiiAt least two must be defective.
7 Marks

Question 5

14 MarksMedium
(a)

A sample of 100 students is taken from a large population. The mean height of the students in this sample is 160 cm. Can it be reasonably regard that, in the population, the mean height is 165 cm and the SD is 10 cm? (use 5% level of significance)

3 Marks
(b)

Below are given the gain in weights (in lbs) of pigs fed on two diets A and B. Gain in weight Diet A : 25, 32, 30, 34, 24, 14, 32, 24, 30, 31, 35, 25 Diet B : 44, 34, 22, 10, 47, 31, 40, 30, 32, 35, 18, 21, 35, 29, 22 Test if the two diets differ significantly as regards their effect on increase in weight. (using t0.05 at 25 d. f. = 2.07)

4 Marks
(c)

From the following table, test the hypothesis that the flower colour is independent of flatness of leaf Flat leaves Curled leaves Total White flowers 99 36 135 Red flowers 20 5 25 Total 119 41 160 If the value of2  for 1 d. f at 5% level of significance is 3.841.

7 Marks
OR OPTION
(a)

Two large population, there are 30 % and 25 % respectively of blue eyed people. Is this difference likely to be hidden in the sample of 1200 and 900 respectively from the two populations? (Test at 5% level of significance).

3 Marks
(b)

A certain stimulus administered to each of the 12 patients resulted in the following increase of blood pressure: 5, 2, 8, -1, 3, 0, -2, 1, 5, 0, 4 and 6 Can it be concluded that the stimulus will, in general be accompanied by an increase in blood pressure? (Given that for 5% level of significancet for 11 d. f is 1.796)

4 Marks
(c)

12 dice were thrown 4096 times and a throw of 6 was considered as a success; the observed frequencies were given below : No. of successes : 0 1 2 3 4 5 6 7 & above Frequencies 447 1145 1181 796 380 115 24 8 Find the value of chi-square on the basis of the hypothesis that the dice were unbiased and hence show that the data are consistent with the hypothesis so far as the −2  test is concerned. The value of2 χ for 7 degrees of freedom at 5% level of significance is 14.067.

7 Marks
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About this Examination Paper & Attribution

Official Gujarat Technological University (GTU) examination paper and step-by-step solutions for Probability and Statistics (PS) (Winter 2025, B.E. · Computer Engineering, Sem 3). Features complete 70-mark regular & remedial examination pattern, official marking distribution across all 5 questions, and direct 1-click official PDF download.

Transcribed for student exam preparation from Gujarat Technological University official examination archives. Questions, syllabus guidelines, and curriculum marking schemes remain the intellectual property of Gujarat Technological University.

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