## Mathematical Foundations for Computing, Probability & Statistics Computer Science & Allied Engg. branches-21MATCS41 Set-1 Solved Model Question Paper

**Module 1**

or

2.A] Test the validity of the arguments using rules of inference.

2.B] Find whether the following arguments are valid or not for which the universe is the set of all triangles. In triangle XYZ, there is no pair of angles of equal measure. If the triangle has two sides of equal length, then it is isosceles. If the triangle is isosceles, then it has two angles of equal measure. Therefore Triangle XYZ has no two sides of equal length.

2.C]

**Module 2**

3.C] **Prove that in every graph the number of vertices of odd degree is even**.

or

4.B] **compute gof and show that gof is invertible**

4.C] **Define Graph isomorphism. Determine whether the following graphs are isomorphic or not**.

**Module 3**

5.A] **Ten competitors in a beauty contest are ranked by two judges A and B in the following order:**

ID No. of competitors | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

Judge A | 1 | 6 | 5 | 10 | 3 | 2 | 4 | 9 | 7 | 8 |

Judge B | 6 | 4 | 9 | 8 | 1 | 2 | 3 | 10 | 5 | 7 |

**Calculate the rank correlation coefficient.**

5.B] In a partially destroyed laboratory record, the lines of regression of y on x and x on y are available as 4𝑥 − 5𝑦 + 33 = 0 𝑎𝑛𝑑 20𝑥 − 9𝑦 = 107. Calculate 𝑥̅ 𝑎𝑛𝑑 𝑦̅ and the coefficient of correlation between x and y.

5.C] An experiment gave the following values:

v(ft/min) | 350 | 400 | 500 | 600 |

t(min.) | 61 | 26 | 7 | 26 |

It is known that v and t are connected by the relation v=at^{b} . Find the best possible values of a and b.

or

6.A] **The following table gives the heights of fathers(x) and sons (y):**

x | 65 | 66 | 67 | 67 | 68 | 69 | 70 | 72 |

y | 67 | 68 | 65 | 68 | 72 | 72 | 69 | 71 |

**Find the lines of regression and Calculate the coefficient of correlation.**

6.B] **Fit a parabola y=ax ^{2} + bx + c for the data**

x | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 |

y | 1.1 | 1.3 | 1.6 | 2.0 | 2.7 | 3.4 | 4.1 |

**Module 4**

7.A]** A random variable 𝑋 has the following probability function:**

x | -2 | -1 | 0 | 1 | 2 | 3 |

P(x) | 0.1 | k | 0.2 | 2k | 0.3 | k |

**Find the value of k and calculate the mean and varianc**e

7.B] Find the mean and standard deviation of the Binomial distribution.

or

8.A]

8.B] 2% of fuses manufactured by a firm are found to be defective. Find the probability that a box containing 200 fuses contains (i) no defective fuses (ii) 3 or more defective fuses (iii) at least one defective fuse.

**Module 5**

9.A] **The joint distribution of two random variables X and Y is as follows**

x\y | -4 | 2 | 7 |

1 | 1/8 | 1/4 | 1/8 |

5 | 1/4 | 1/8 | 1/8 |

**Compute the following. (i) E(X) and E(Y) (ii) E(XY) (iii) 𝜎 _{𝑋} and 𝜎_{𝑌} (iv) COV(X,Y) (v) 𝜌(𝑋, 𝑌)**

or

10.A] **Explain the terms: (i) Null hypothesis (ii) Confidence intervals (iii) Type-I and Type-II errors**.

x | 1 | 2 | 3 | 4 | 5 | 6 |

y | 40 | 32 | 28 | 58 | 54 | 60 |