Assignment #1
Important Note: Please upload your solutions (including detailed workings, which are assessed as part of the awarded marks) in a single zip file (titled as SurnameFirstname.zip) to the individual exercise 1 submission folder on the Moodle “Module Assessment Centre” tab before the deadline (12:00, 7th February 2025).
Further important information is provided on the last page of this document.
Question 1 (10 marks)
A and B are two events such that ��(��) = 0.4, ��(��) = 0.3, and ��(����) = 0.1. Draw Venn diagrams and appropriately sketch and/or annotate them to graphically represent the following probabilities:
(a) ��(��|��)
(b) ��(��̅|��̅)
(c) ��(��̅|��̅)
(d) ��(��̅|��)
(e) ��(��|��̅)
(f) ��(��|��̅)
(g) ��(�� ∪ ��|��)
(h) ��(�� ∪ ��|��̅)
Note: Draw separate Venn diagrams for each subpart of the question.
Question 2 (10 marks)
At a given location, let A denote the occurrence of high wind in any single minute and B denote the occurrence of an earthquake in any single minute. Some probabilities related to the occurrences of these events are depicted in the Venn diagram provided in Figure 1.
Figure 1. Venn diagram of the occurrence of high winds (A) and earthquakes (B) in any single minute. The numbers shown denote probabilities.
(a) Building codes do not require the engineer to design buildings for the combined effect of wind and earthquake. Using the information contained in the Venn diagram, provide a brief explanation of why this can be deemed a reasonable approach.
(b) If you have been given enough information to do so, compute the probability of either high wind or an earthquake occurring in a given minute. If necessary, state any assumptions you have to make in your calculation.
(c) If the occurrence of events in successive minutes is mutually independent, what is the probability that there will be no high winds this year? State any assumptions you have to make in your calculation.
Question 3 (5 marks)
An insurance company classifies construction companies as “good”, “average” and “bad” risks. The proportional division of these risks and the corresponding probability of each risk having an accident A in a one-year span are shown in Figure 2.
Figure 2. Division of risks and their associated probability of having an accident in a one-year span, A.
(a) What portion of companies will have an accident in a given year?
(b) If the CEGE Construction Company had no accidents last year, what is the probability that it is a good risk?
Question 4 (25 marks)
A random variable T has the following CDF: ����
(��) = 1 − ��−10��2 , �� ≥ 0
(a) Create a sketch that depicts ��(�� > 0.4)
(b) Create a sketch that depicts ��(0.4 ≤ �� ≤ 0.8)
(c) Provide a brief explanation for why or why not ��(0.4 ≤ �� ≤ 0.8| �� > 0.4) is greater than
��(0.4 ≤ �� ≤ 0.8).
(d) What is the mean value of T?
(e) What is the variance of T?
Hint: Use WolframAlpha to solve parts of this question as required.
Question 5 (20 marks)
A random variable X has a normal distribution with mean ���� = 3 and standard deviation
���� = 1.5.
(a) What is ��(�� > 0)? Submit an excel (or other spreadsheet software) file that shows your workings for this calculation.
(b) What is ��(0 ≤ �� ≤ 4)? Submit an excel (or other spreadsheet software) file that shows your workings for this calculation.
Determine whether the following statements are true or false, referring to the normal probability density functions provided in Figure 3:
(i) (ii)
Figure 3. Normal probability density functions for two normally distributed random variables.
(c) The random variable associated with probability density function (i) has a larger mean than the random variable associated with probability density function (ii).
(d) The random variable associated with probability density function (i) has a larger variance than the random variable associated with probability density function (ii).
Question 6 (20 marks)
Suppose the random variables X and Y have joint density given by
��(��, ��) = {
�� + 3��
, 0 ≤ �� ≤ 3, 0 ≤ �� ≤ 1
9
0, ����ℎ������������
(a) Create a sketch that depicts ��(�� > 0.2)
(b) Create a sketch that depicts ��(0.4 ≤ �� ≤ 0.8)
Some important additional information:
- Statement on the use of AI
This assessment falls into category 2 of three-tiered categorisation of AI use. AI tools can be used to assist you with some intermediate calculations, if desired. It is important that any use of AI is explicitly acknowledged in your submission, per guidelines. Note that AI tools cannot be used where the use of other software tools has been explicitly specified (i.e., Question 5) and any uploading or creation of images/sketches/diagrams on AI tools is expressly forbidden. Any case where the instructions of the previous sentence are believed to have been ignored will undergo the standard academic misconduct procedure.
- Marking criteria
The following criteria will be used for each question:
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Criteria
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Marks Available (%)
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Numerically and theoretically correct and complete
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40
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Clear explanation of the calculation steps and assumptions
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40
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Presentation is in a highly professional format; figures/tables are clear and consistent (where applicable)
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20
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