[SOLVED] CS917 Foundations of Computing – Maths and Stats Assignment Statistics

Department of Computer Science CS917

Foundations of Computing – Maths and Stats

Assignment

This assignment is due at noon on Thursday  19th December, 2024. The submission is on Tabula, and should comprise scanned copies of written work.

The work that you submit should be your own work and please show full working where appropriate, as this is necessary to gain full marks.

Marks for each question are indicated. The total marks you can get is 100.

If you have any questions then do please email meat [email protected].

1    Discrete Mathematics

1. For each of the following formulae, find a logically equivalent formula in which Λ ,  =→   and  ←→   do not occur:  (i) :(p   =→   q)  [5 marks]; (ii) ((p Λ q) _r) [5 marks]; (iii) :((p Λ q)  ←→  r) [5 marks]. Use the truth table to show that the proposed solution is indeed equivalent with the original one. [total 15 marks]

2. Write out paraphrases of the following, using 8, 9 and =  (i) Frodo has a ring [2 marks]; (ii) Sauron does not have any rings [2 marks]; (iii) The One Ring rules all the other rings [2 marks]; (iv) The ring that Frodo has is the One Ring [2 marks]; (v) whoever wears the ring, becomes invisible (i.e., no other ordinary human can see that person) [3 marks]; (vi) Bombadil Tom can see the ring-wearer, hence he is no ordinary human (use the previous statement from (v) to formulate this one) [3 marks].  Auxiliary clauses: has(x, y): x has/is in possession of y; rules(x, y): x rules y; wears(x, y): x wears y; sees(x, y): x can see y; and ordinary(x): x is an ordinary human. [total 14 marks]

3. Write predicate logic formulae which state that the relation expressed by Rx,y  has the following properties: (i) Rx,y  is irreflexive [3 marks]; (ii) Rx,y is intransitive [3 marks]; (iii) Rx,y  is not a partial order [3 marks].  Note that this formulation is a bit diferent from the one in the slides. To make this consistent, think about Rx,y   as Rp  with relation p between x and y (you can assume that both x andy are from the same set A). So you can use Rx,y  as p in your formulations.  Therefore, in your answer, you can use both notations, just be consistent (i.e., if you choose 1 notation, then use the same for all your answers).  [total 9 marks]

4. Determine which of the following functions are injective and which are surjective (please provide explanations as well):

(i) f : Z ! N, where 8n ∈ Z: f(n) = n2024 + 1 [3 marks];

(ii) g : N x N ! N, where 8(n, k) ∈ N x N: g(n, k) = 2n3k5n+k  [3 marks];

(iii) h : P(N) ! P(N), where 8A ∈ P(N): h(A) = N A (recall what P(N) means) [3 marks];

(iv) k : N ! Z, where 8n ∈ N: k(n) = (-1)n   [3 marks]. [total 12 marks].

2 Statistical Analysis

Question 1 has two parts, each is marked out of 5. Both Question 2 and 3 are marked out of 10. Question 4 is marked out of 20.

1. A large database is compiled from files contributed by three sources: Source A, Source B, and Source C, which account for 20%, 50%, and 30% of the total files, respectively.  The percentage of empty files from each source is 4% for Source A, 2.5% for Source B, and 1.5% for Source C. If a file is selected at random and found to be empty, what is the probability that it originated from Source A?

2. A survey was conducted on a large number of individuals to record their dates of birth.  Assume that the sample size is sufflciently large to treat the dates of birth as uniformly distributed across all possible days.

(i) What is the smallest number of randomly selected individuals re- quired such that the probability of at least two of them sharing the same birthday exceeds 50%? Assume there are 365 days in each year.

(ii) Two individuals are selected at random from those born between January 1st , 1961, and December 31th , 2000. Given that at least one of these individuals was born in a leap year, what is the probability that both individuals were born in a leap year?

3. A directory contains 6 high-priority records and 4 low-priority records. These records need to be analysed by employees, and the workload is divided among them.  Four records are randomly assigned to Sam, but the priority of each record is not identifiable from the file names.  Given that the first record Sam analyses is low-priority, what is the probability that all the remaining records assigned to him are of the same priority (all remaining either low-priority or high-priority)?

4. A small company operates two 72-core computers, Computer A and Com- puter B.

(i) On Computer A, it is known that each core is busy 50% of the time on average.  At any given time, what is the probability that 36 or more cores are busy simultaneously?

(ii) On Computer B, a random sample of 8 observations is taken.  The average number of busy cores in this sample is 40.5, with a sample standard deviation of 3.2 cores. One individual claims that the true mean number of busy cores is 36. Based on the sample data, is there sufflcient evidence to reject this claim at the 5% significance level?