UNIVERSITY EXAMINATIONS
January/February 2025
DSC4821
STOCHASTIC MODELLING
Duration: 4 hours 100 marks
EXAMINERS
FIRST DR E MUDIMU
SECOND PROF D MATHEBULA
EXTERNAL Prof E RAPOO (UNISA)
_____________________________________________________________________________________________
This paper consists of five pages (including two cover pages).
Instructions:
• Answer ALL the questions. There are 10 questions for a total of 100 marks.
• This is an open-book examination. You may use any books or notes, but you are NOT allowed to consult
sources on the internet or get assistance from other people during the examination.
• Answers may be either handwritten or typed.
• Show ALL your workings in full. Marks will be awarded for intermediate steps and not for final answers
only.
Your answer cover page must include the module code (DSC4821), your student number, your
academic integrity declaration, the date, your initials and surname, and your signature. The
following handwritten declaration, signed and dated, must precede your answers:
ACADEMIC INTEGRITY DECLARATION
1. I declare that this is my own, original work.
2. I did not receive any assistance from another person or the internet during this examination.
, Page 1 of 5
DSC4821
Date: 23 January 2025
INSTRUCTIONS ON THE DAY OF ASSESSMENT:
Ensure you are connected to the internet in order to log into the Invigilator App and
scan this QR code.
If you encounter difficulty in scanning the QR code, you can alternatively enter the Exam
Access Code below the QR code to start the invigilation.
Unless otherwise specified by your institution, note that you can only scan this QR
code once. If your assessment has multiple online sections, tests or attempts, you
should NOT finish the invigilation until your entire assessment has been completed.
Only scan the QR code when the assessment formally commences.
The QR code is only scannable for a limited time and it should therefore be
scanned as soon as possible to start the invigilation.
Once the QR code is scanned, ensure your media volume Is turned up and place your
smartphone next to you. The Invigilator App will notify you with a notification beep
when you are required to action a request, which you should then perform.
We recommend that you keep your smartphone on charge for the duration of the assessment.
If you only have one device you may access your assessment in the application by pressing the ‘Access Exam’ button in the top right
corner of your app.
Keep the Invigilator App open on your cell phone for the full duration of the assessment. You are not allowed to minimise or
leave the app.
Ensure you are connected to the internet in order to commence the invigilation as well as at the end of the assessment. No internet
connection is required during the assessment.
You have to adhere to the assessment time limit communicated to you by your institutionas the time displayed in the Invigilator App could differ
from the time allocated to complete your assessment.
You can click the "Finish Assessment" button in the app if you finish your assessment early.
If you are performing a written or Scan-and-Upload assessment:
The Invigilator App may request you to take a picture of every page of your answer sheet at the end of the assessment. Unless
otherwise specified by your institution, this does NOT replace the normal upload of your script to your institutions online
portal
,After completing invigilation and following all app instructions, you must upload your Invigilation App data. If however there is a delay in
the upload of the app data at the end of the assessment, you should prioritise the upload of your script to your university portal and you
can temporarily minimise the app to do so. Uploading of app data is not time sensitive and you can come back and do it after you have
successfully uploaded your script to the exam portal.
Should you encounter any technical difficulty, please WhatsApp The Invigilator Helpdesk on 073 505 8273.
, DSC4821
4 January/February 2025
[10]
Question 1
Bhakai Natural, an oil exploration company, conducts a speci c test to assess the likelihood of discovering oil in a
speci c region of South Africa.The probability that the test would indicate the presence of oil is 85% if oil is indeed
present in the test region, while the probability of a false positive is 15% if no oil is present in the test region.
Before the test is done, the company believes that the probability of presence of oil in the test region is 45%.
Revise the value of the probability of oil being present in the test region given that the test gives a positive signal.
Question 2 [10]
2.1 Two random variables X and Y have the joint density function
and f(x,y) = 0 otherwise. Find P(X < Y ). [5]
2.2 Let the random vector (X,Y ) have the standard bivariate normal distribution with correlation coe cient ρ =
−1/2. What are the values of a for which the random variables V = aX + Y and
W = X + aY are independent? [5]
Question 3 [10]
Mpho, the buying manager of Blue Printing Company, orders paper supplies from one of three suppliers, S1, S2,
or S3. Orders are placed on two successive days, one order per day. Thus, (S1, S3) might denote that supplier S1
gets the order on the rst day and supplier S3 gets the order on the second day.
3.1 List the sample points in this experiment of ordering paper on two successive days. [3]
3.2 Assume that the suppliers are selected at random each day and assign a probability to each sample point.
[3]
3.3 Let A denote the event that the same vendor gets both orders and let B denote the event that S2 gets at
least one order. Find probabilities P(A), P(B), P(A ∪ B) and P(A ∩ B). [4]
Question 4 [10]
The Wozan Berg manufacturing factory, clears its industrial wastewater once a week. The quantity Y of
wastewater produced per week, measured in thousands of litres, follows the continuous probability density
function
f(y) = 105y4(1 − y)2 for 0 < y < 1 and f(y) = 0 otherwise.
Assume that the capacity of a storage tank is 0.6 expressed in thousands of litres.
4.1 What is the probability that there will be an over ow during a week? [2]
4.2 What is the average quantity of industrial wastewater produced during a week? [3]
4.3 Assume that the storage tank has a capacity of 0.8 expressed in thousands of liters. What is the expected
value of the weekly over ow? [5]
Page 4 of 6
January/February 2025
DSC4821
STOCHASTIC MODELLING
Duration: 4 hours 100 marks
EXAMINERS
FIRST DR E MUDIMU
SECOND PROF D MATHEBULA
EXTERNAL Prof E RAPOO (UNISA)
_____________________________________________________________________________________________
This paper consists of five pages (including two cover pages).
Instructions:
• Answer ALL the questions. There are 10 questions for a total of 100 marks.
• This is an open-book examination. You may use any books or notes, but you are NOT allowed to consult
sources on the internet or get assistance from other people during the examination.
• Answers may be either handwritten or typed.
• Show ALL your workings in full. Marks will be awarded for intermediate steps and not for final answers
only.
Your answer cover page must include the module code (DSC4821), your student number, your
academic integrity declaration, the date, your initials and surname, and your signature. The
following handwritten declaration, signed and dated, must precede your answers:
ACADEMIC INTEGRITY DECLARATION
1. I declare that this is my own, original work.
2. I did not receive any assistance from another person or the internet during this examination.
, Page 1 of 5
DSC4821
Date: 23 January 2025
INSTRUCTIONS ON THE DAY OF ASSESSMENT:
Ensure you are connected to the internet in order to log into the Invigilator App and
scan this QR code.
If you encounter difficulty in scanning the QR code, you can alternatively enter the Exam
Access Code below the QR code to start the invigilation.
Unless otherwise specified by your institution, note that you can only scan this QR
code once. If your assessment has multiple online sections, tests or attempts, you
should NOT finish the invigilation until your entire assessment has been completed.
Only scan the QR code when the assessment formally commences.
The QR code is only scannable for a limited time and it should therefore be
scanned as soon as possible to start the invigilation.
Once the QR code is scanned, ensure your media volume Is turned up and place your
smartphone next to you. The Invigilator App will notify you with a notification beep
when you are required to action a request, which you should then perform.
We recommend that you keep your smartphone on charge for the duration of the assessment.
If you only have one device you may access your assessment in the application by pressing the ‘Access Exam’ button in the top right
corner of your app.
Keep the Invigilator App open on your cell phone for the full duration of the assessment. You are not allowed to minimise or
leave the app.
Ensure you are connected to the internet in order to commence the invigilation as well as at the end of the assessment. No internet
connection is required during the assessment.
You have to adhere to the assessment time limit communicated to you by your institutionas the time displayed in the Invigilator App could differ
from the time allocated to complete your assessment.
You can click the "Finish Assessment" button in the app if you finish your assessment early.
If you are performing a written or Scan-and-Upload assessment:
The Invigilator App may request you to take a picture of every page of your answer sheet at the end of the assessment. Unless
otherwise specified by your institution, this does NOT replace the normal upload of your script to your institutions online
portal
,After completing invigilation and following all app instructions, you must upload your Invigilation App data. If however there is a delay in
the upload of the app data at the end of the assessment, you should prioritise the upload of your script to your university portal and you
can temporarily minimise the app to do so. Uploading of app data is not time sensitive and you can come back and do it after you have
successfully uploaded your script to the exam portal.
Should you encounter any technical difficulty, please WhatsApp The Invigilator Helpdesk on 073 505 8273.
, DSC4821
4 January/February 2025
[10]
Question 1
Bhakai Natural, an oil exploration company, conducts a speci c test to assess the likelihood of discovering oil in a
speci c region of South Africa.The probability that the test would indicate the presence of oil is 85% if oil is indeed
present in the test region, while the probability of a false positive is 15% if no oil is present in the test region.
Before the test is done, the company believes that the probability of presence of oil in the test region is 45%.
Revise the value of the probability of oil being present in the test region given that the test gives a positive signal.
Question 2 [10]
2.1 Two random variables X and Y have the joint density function
and f(x,y) = 0 otherwise. Find P(X < Y ). [5]
2.2 Let the random vector (X,Y ) have the standard bivariate normal distribution with correlation coe cient ρ =
−1/2. What are the values of a for which the random variables V = aX + Y and
W = X + aY are independent? [5]
Question 3 [10]
Mpho, the buying manager of Blue Printing Company, orders paper supplies from one of three suppliers, S1, S2,
or S3. Orders are placed on two successive days, one order per day. Thus, (S1, S3) might denote that supplier S1
gets the order on the rst day and supplier S3 gets the order on the second day.
3.1 List the sample points in this experiment of ordering paper on two successive days. [3]
3.2 Assume that the suppliers are selected at random each day and assign a probability to each sample point.
[3]
3.3 Let A denote the event that the same vendor gets both orders and let B denote the event that S2 gets at
least one order. Find probabilities P(A), P(B), P(A ∪ B) and P(A ∩ B). [4]
Question 4 [10]
The Wozan Berg manufacturing factory, clears its industrial wastewater once a week. The quantity Y of
wastewater produced per week, measured in thousands of litres, follows the continuous probability density
function
f(y) = 105y4(1 − y)2 for 0 < y < 1 and f(y) = 0 otherwise.
Assume that the capacity of a storage tank is 0.6 expressed in thousands of litres.
4.1 What is the probability that there will be an over ow during a week? [2]
4.2 What is the average quantity of industrial wastewater produced during a week? [3]
4.3 Assume that the storage tank has a capacity of 0.8 expressed in thousands of liters. What is the expected
value of the weekly over ow? [5]
Page 4 of 6
