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Gujarat Technological UniversityWinter 2024 Examination

GTU 3130006 Probability and Statistics (PS) Winter 2024 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)
Define and give the example of Mutually exclusive events.
3 Marks
(b)

One card is drawn at random from a pack of 52 cards. Find the probability of getting a king or a red card.

4 Marks
(c)

Data on the readership of a certain magazine show that the proportion of male readers under 35 is 0.40 and that over 35 is 0.20. If the proportion of readers under 35 is 0.70, find the probability of subscribers that are female over 35 years. Also, calculate the probability that a randomly selected male subscriber is under 35 years of age.

7 Marks

Question 2

14 MarksMedium
(a)

Three students 𝐴, 𝐵, 𝐶 are in a running race. If 𝑃(𝐴) = 𝑃(𝐵) = 2𝑃(𝐶), then find the probability that 𝐵 or𝐶 wins.

3 Marks
(b)

Let X be a continuous random variable with pdf 𝑓(𝑥) = 𝑘𝑥(1 − 𝑥), 0 ≤ 𝑥 ≤ 1 Find 𝑘 and determine 𝑃(0 ≤ 𝑋 ≤ 0.5).

4 Marks
(c)
The contents of urns I, II and III are as follows:

1 white, 2 red, and 3 black balls 2 white, 3 red and 1 black ball, and 3 white, 1 red and 2 black balls. One urn is chosen at random and two balls are drawn. They happen to be white and red. Find the probability that they came from (i) Urn 𝐼

iiUrn 𝐼𝐼 (iii) Urn 𝐼𝐼𝐼.
7 Marks
OR OPTION
(c)
A random variable X has the following probability distribution:

𝑋 −2 −1 0 1 2 3 𝑃(𝑋 = 𝑥)

01 𝑘 0.2 2𝑘 0.3 3𝑘
iFind the value of 𝑘
iiFind 𝑃(𝑥 ≥ 2)
iii𝑃(−2 < 𝑥 < 2)
7 Marks

Question 3

14 MarksMedium
(a)

Let the continuous random variable X have the probability density function 𝑓(𝑥) = { 1 𝑥4 ; 1 < 𝑥 < ∞ 0; 𝑜𝑡ℎ𝑒𝑤𝑖𝑠𝑒 Find distribution function 𝐹(𝑥).

3 Marks
(b)
A random variable X has the following distribution:

𝑋 −3 −2 −1 0 1 2 𝑃(𝑋 = 𝑥)

005 0.1 0.2 0.3 0.2 0.15 Find (i) 𝐸(𝑋) (ii)𝑉𝑎𝑟(𝑋).
4 Marks
(c)

Compute Karl Pearson’s coefficient of correlation between 𝑋 and 𝑌 for the following data: 𝑥 2 4 5 6 8 11 𝑦 18

7 Marks
OR OPTION
(a)

If the two lines of regression are 2𝑥 − 5𝑦 + 30 = 0 and 10𝑥 − 4𝑦 − 104 = 0 which of these are lines of regression of 𝑥 on 𝑦 and 𝑦 on 𝑥?

3 Marks
(b)
Calculate the first four moments from the following data:

𝑥 5 10 15 20 25 𝑓 6 10 14 6

4 Marks
(c)

Obtain the line of regression of monthly sales (𝑦) on advertisement expenditure (𝑥) and estimate the monthly sales when the company will spend Rs. 50000 on advertisement, if the data on y and x are as follows: 𝑦 (in lac) 74 76 60 68 79 70 71 94 𝑥 (in thousand) 43 44 36 38 47 40 41 54

7 Marks

Question 4

14 MarksMedium
(a)

Explain the term related to testing of hypothesis: (i) One Tailed Test

iiTwo Tailed Test (iii) Critical Region.
3 Marks
(b)

A coinwas tossed 960 times and returned heads 183 times. Test the hypothesis that the coin is unbiased.Usea 0.05 level of significance? (|𝑍0.05| = 1.96)

4 Marks
(c)

In a city A, 20% of a random sample of 900 school boys has a certain slight physical defect. In another city B, 18.5% of a random sample of 1600 school boys has the same defect. Is a difference between the proportions significant at 0.05 level of significance? (|𝑍0.05| = 1.96)

7 Marks
OR OPTION
(a)

The mean breaking strength of a rope is 160 kgs. If 6 bricks (selected from different piles) have a mean breaking strength of 154.3 kgs with a standard deviation of 6.4 kgs. Test the null hypothesis 𝜇 = 160 kgs. against the alternative hypothesis 𝜇 < 160 kgs. at 1% level of significance. [𝑡0.01(𝑣 = 5) = 3.365]

3 Marks
(b)

Examine whether the two samples for which the data are given in the following table could have been drawn from populations with the same SD. Size SD Sample I 100 5 Sample II 200 7 (|𝑍0.05| = 1.96) 𝟎𝟒

4 Marks
(c)
If 𝑋 is a Poisson variate such that 𝑃(𝑋 =
0)and using recurrence formula, find the probabilities at 𝑥 = 1,2,3,4 and 5.
7 Marks

Question 5

14 MarksMedium
(a)

Using least squares approximation method, fit a straight line𝑦 = 𝑎𝑥 + 𝑏 to the following data: 𝑥 1 2 3 4 5 𝑦 14 27 40 55 68

3 Marks
(b)

Using least squares approximation method,fit a curve 𝑦 = 𝑎𝑒𝑏𝑥 to the following data: 𝑥 1 5 7 9 12 𝑦 10 15 12 15 21

4 Marks
(c)

Distribution of height of 1000 students is normal with mean 165 cms and standard deviation 15 cms. How many soldiers are of height (i) less than 138 cms. (ii) more than 198 cms. (iii) between 138 and 198 cms. (𝑃(𝑧 = 1.8) = 0.4641, 𝑃(𝑧 = 2.2) = 0.4841)

7 Marks
OR OPTION
(a)

In a throw of a dice the appearance of a number 4 is taken as a success. If the dice is thrown 5 times, find mean, variance and standard deviation.

3 Marks
(b)
If a random variable has a Poisson distribution such that 𝑃(𝑋 =
1)(iv) 𝑃(1 < 𝑋 < 4).
4 Marks
(c)

Using least squares approximation method, fit a second-degree parabolic curve to the following data: X 1 2 3 4 5 Y 5 12 26 60 97 Also, estimate 𝑦 at 𝑥 = 6.

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 2024, 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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