B.Tech./B.Tech. (Working Professional) III Semester
Examination, June 2025
Grading System (GS) / Working Professional
Max Marks:
70 | Time: 3 Hours
Note:
i) Attempt any five questions.
ii) All questions carry equal marks.
a) Determine (i) $P(B/A)$ (ii) $P(A/B^c)$ if A and B are events with $P(A) = \frac{1}{3}, P(B) = \frac{1}{4}, P(A \cup B) = \frac{1}{2}$. (Unit 1)
b) A random variable X has the following probability function.
| $X_i$ | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| $P(X_i)$ | K | 0.1 | K | 0.2 | 2K | 0.4 | 2K |
Find:
i) K
ii) Mean
iii) Variance
a) Define Moments. Explain types of moments. (Unit 1)
b) Find the mean and standard deviation of a normal distribution in which 7% of items are under 35 and 89% are under 63. (Unit 2)
a) The density function of a random variable X is $f(x) = e^{-x}$, $x \ge 0$. Find $E(x)$, $E(x^2)$ and variance of X. (Unit 2)
b) Define gamma distribution. Find the mean and Variance of gamma distribution. (Unit 2)
a) Two factories produce identical clocks. The production of the first factory consists of 10,000 clocks of which 100 are defective. The second factory produces 20,000 clocks of which 300 are defective. What is the probability that a particular defective clock was produced in the first factory. (Unit 3)
b) Explain bivariate distributions and their properties. (Unit 3)
a) If the variance of a Poisson variate is 3, then find $P(x=0)$, $P(0 < x \le 3)$ and $P(1 \le x < 4)$. (Unit 4)
b) Fit a straight line to the form $y = a + bx$ for the following data. (Unit 5)
| X | 0 | 5 | 10 | 15 | 20 | 25 |
| Y | 12 | 15 | 17 | 22 | 24 | 30 |
a) 20% of items produced from a factory are defective. Find the probability that in a sample of 5 chosen at random:
i) none is defective
ii) one is defective
iii) $P(1 < x < 4)$
(Unit 4)
b) Calculate the coefficient of rank correlation from the below data. (Unit 4)
| X | 68 | 64 | 75 | 50 | 64 | 80 | 75 | 40 | 55 | 64 |
| Y | 62 | 58 | 68 | 45 | 81 | 60 | 68 | 48 | 50 | 70 |
a) The average marks scored by 32 boys is 72 with a standard deviation of 8. While that for 36 girls is 70 with a standard deviation of 6, does this indicate that the boys perform better than girls at level of significance 0.05? Explain. (Unit 5)
b) Experience has shown that 20% of a manufactured product is of the top quality. In one day's production of 400 articles only 50 are of top quality. Test the hypothesis at 0.05 level. (Unit 5)
a) Explain partial and multiple correlation coefficients. (Unit 4)
b) Define chi-square distribution. Explain its properties. (Unit 6)
