UNIVERSITY EXAMINATIONS
January/February 2024
STA4811
Probability and Stochastic Processes
100 Marks
Duration: 3 Hours
EXAMINERS:
FIRST: Prof G Kabera
SECOND: Mr TP Mohlala
EXTERNAL: Dr AJ van Zyl
_____________________________________________________________________________________________
This paper consists of 5 pages.
INSTRUCTIONS
1. This is an open book examination. You may use your textbook, your course material or any other printed
material to answer the questions.
2. You must upload your answers in one PDF file.
3. Answer ALL the questions.
4. Show all relevant computations and steps.
[TURN OVER]
, 2 STA4811
JAN/FEBR 2024
ADDITIONAL STUDENT INSTRUCTIONS
1. Students must upload their answer scripts in a single PDF file on the official myExams platform (answer
scripts must not be password protected or uploaded as "read-only" files). The university only marks one
submission received from a student. Priority is granted to scripts received via the myExams platform.
2. NO e-mailed scripts will be accepted.
3. Students are advised to preview submissions (answer scripts) to ensure legibility and also to ensure that the
correct answer script file has been uploaded.
4. Students are permitted to resubmit their answer scripts, but only within the stipulated submission duration,
should they regard the initial submission to be unsatisfactory. Resubmissions after the closing time of the
examination are not permitted.
5. Uncollated answer scripts and scripts with an incorrect file format will not be considered.
6. Wrong answer scripts that are uploaded, that is, any file other than the exam answer script (eg uploading
the exam question paper instead of the exam answer script) will not be marked and no opportunity will be
granted for resubmission.
7. A mark awarded for an incomplete submission will be the student’s final mark. No opportunity for resub-
mission will be granted.
8. A mark awarded for an illegible scanned submission will be the student’s final mark. No opportunity for
resubmission will be granted.
9. Only the last answer file uploaded within the stipulated submission duration period will be marked.
10. Submissions will only be accepted from registered student accounts.
11. Students who have not utilised invigilation or proctoring tools will be deemed to have transgressed Unisa’s
examination rules and will have their marks withheld. For examinations which use the IRIS invigilation
system, IRIS must be recording throughout the duration of the examination until the submission of the
examination script.
12. Students must complete the online declaration of their work when submitting their examination script. Stu-
dents suspected of dishonest conduct during the examinations will be subjected to disciplinary processes.
Students may not communicate with any other person or request assistance from any other person during
their examinations. The use of Telegram, WhatsApp or any other instant messaging services with any other
person (except when asking for technical assistance via official channels of the SCSC or the Invigilator
WhatsApp line) are strictly prohibited. Plagiarism is a violation of academic integrity and students who
plagiarise, copy from published work or use Artificial Intelligence Software (eg ChatGPT) or online sources
(eg course material) will be in violation of the Policy on Academic Integrity and the Student Disciplinary
Code and may be referred to a disciplinary hearing. Unisa has a zero tolerance for plagiarism and/or any
other forms of academic dishonesty.
[TURN OVER]
, 3 STA4811
JAN/FEBR 2024
13. Listening to audio (music) and making use of audio-to-text software is strictly prohibited during your exam-
ination session unless such usage of the software is related to a student’s assistive device which has been so
declared. Failure to do so will be a transgression of Unisa’s examination rules and the student’s marks will
be withheld.
14. Students are provided 30 minutes to submit their answer scripts after the official examination time. Students
who experience technical challenges should report the challenges immediately to the SCSC telephonically
on 080 000 1870 and choose option 4 for exams, or their College exam support centres (refer to the Get help
during the examinations by contacting the Student Communication Service Centre [unisa.ac.za]) within 30
minutes. Queries received after 30 minutes of the official examination duration time will not be responded
to. Submissions made after the official examination time will be rejected according to the examination regu-
lations and will not be marked. Only communication received from your myLife account will be considered.
15. Non-adherence to the processes for uploading examination responses will not qualify the student for any
special concessions or future assessments.
16. Queries that are beyond Unisa’s control include the following:
a. Personal network or service provider issues
b. Load shedding/limited space on personal computer
c. Crashed computer
d. Non-functioning cameras or web cameras
e. Using work computers that block access to the myExams site (employer firewall challenges)
f. Unlicensed software (eg license expires during exams)
Postgraduate students experiencing the above challenges are advised to apply for an aegrotat and submit
supporting evidence within ten days of the examination session. Students will not be able to apply for an
aegrotat for a third examination opportunity.
Postgraduate/undergraduate students experiencing the above challenges in their second examination oppor-
tunity will have to reregister for the affected module.
[TURN OVER]
, 4 STA4811
JAN/FEBR 2024
QUESTION 1
A manuscript is sent to a typing company consisting of typists A, B and C. If it is typed by typist A, then the
number of errors made follows a Poisson distribution with mean 3; if it is typed by typist B, then the number of
errors made follows a Poisson distribution with mean 3:5; and if it is typed by typist C, then the number of errors
made follows a Poisson distribution with mean 3:8. Let X denote the number of errors in the typed manuscript.
Assume that the job is given to typists A, B and C with probabilities 0:5, 0:3 and 0:2, respectively. Calculate:
(1.1) E[X ]. (3)
(1.2) Var.X /: (6)
[9]
QUESTION 2
Customers arrive at a small shop in accordance with a Poisson process with the rate of 12 per hour. The amount of
money paid by each customer is a random variable with a mean of R150 and a standard deviation of R100. The
shop is open 10 hours per day. Approximate the probability that the daily earnings of the shop are greater than
R16000. [12]
QUESTION 3
A Markov chain fX n ; n 0g with three states 0; 1; 2 has a transition probability matrix given by
0 1 2
01 1
1
0 2 2
0
B C
P D 1 @ 12 0 1
2 A:
2 21 1
2
0
1
If PfX 0 D 0g D 6
and PfX 0 D 1g D 14 , calculate E[X 3 ]. [17]
QUESTION 4
Consider two machines, both of which have an exponential lifetime with a mean of 5 years. There is a single
repairman that can service each machine with a mean time of 2 months. Assume that a state is the number of
machines that are working.
(4.1) Justify why the process can be considered as a birth and death process. (2)
(4.2) Write down the states of the process. (3)
(4.3) Write down the birth and death rates. (4)
(4.4) Set up the Kolmogorov backward equations. You do not have to solve them. (20)
[29]
[TURN OVER]
January/February 2024
STA4811
Probability and Stochastic Processes
100 Marks
Duration: 3 Hours
EXAMINERS:
FIRST: Prof G Kabera
SECOND: Mr TP Mohlala
EXTERNAL: Dr AJ van Zyl
_____________________________________________________________________________________________
This paper consists of 5 pages.
INSTRUCTIONS
1. This is an open book examination. You may use your textbook, your course material or any other printed
material to answer the questions.
2. You must upload your answers in one PDF file.
3. Answer ALL the questions.
4. Show all relevant computations and steps.
[TURN OVER]
, 2 STA4811
JAN/FEBR 2024
ADDITIONAL STUDENT INSTRUCTIONS
1. Students must upload their answer scripts in a single PDF file on the official myExams platform (answer
scripts must not be password protected or uploaded as "read-only" files). The university only marks one
submission received from a student. Priority is granted to scripts received via the myExams platform.
2. NO e-mailed scripts will be accepted.
3. Students are advised to preview submissions (answer scripts) to ensure legibility and also to ensure that the
correct answer script file has been uploaded.
4. Students are permitted to resubmit their answer scripts, but only within the stipulated submission duration,
should they regard the initial submission to be unsatisfactory. Resubmissions after the closing time of the
examination are not permitted.
5. Uncollated answer scripts and scripts with an incorrect file format will not be considered.
6. Wrong answer scripts that are uploaded, that is, any file other than the exam answer script (eg uploading
the exam question paper instead of the exam answer script) will not be marked and no opportunity will be
granted for resubmission.
7. A mark awarded for an incomplete submission will be the student’s final mark. No opportunity for resub-
mission will be granted.
8. A mark awarded for an illegible scanned submission will be the student’s final mark. No opportunity for
resubmission will be granted.
9. Only the last answer file uploaded within the stipulated submission duration period will be marked.
10. Submissions will only be accepted from registered student accounts.
11. Students who have not utilised invigilation or proctoring tools will be deemed to have transgressed Unisa’s
examination rules and will have their marks withheld. For examinations which use the IRIS invigilation
system, IRIS must be recording throughout the duration of the examination until the submission of the
examination script.
12. Students must complete the online declaration of their work when submitting their examination script. Stu-
dents suspected of dishonest conduct during the examinations will be subjected to disciplinary processes.
Students may not communicate with any other person or request assistance from any other person during
their examinations. The use of Telegram, WhatsApp or any other instant messaging services with any other
person (except when asking for technical assistance via official channels of the SCSC or the Invigilator
WhatsApp line) are strictly prohibited. Plagiarism is a violation of academic integrity and students who
plagiarise, copy from published work or use Artificial Intelligence Software (eg ChatGPT) or online sources
(eg course material) will be in violation of the Policy on Academic Integrity and the Student Disciplinary
Code and may be referred to a disciplinary hearing. Unisa has a zero tolerance for plagiarism and/or any
other forms of academic dishonesty.
[TURN OVER]
, 3 STA4811
JAN/FEBR 2024
13. Listening to audio (music) and making use of audio-to-text software is strictly prohibited during your exam-
ination session unless such usage of the software is related to a student’s assistive device which has been so
declared. Failure to do so will be a transgression of Unisa’s examination rules and the student’s marks will
be withheld.
14. Students are provided 30 minutes to submit their answer scripts after the official examination time. Students
who experience technical challenges should report the challenges immediately to the SCSC telephonically
on 080 000 1870 and choose option 4 for exams, or their College exam support centres (refer to the Get help
during the examinations by contacting the Student Communication Service Centre [unisa.ac.za]) within 30
minutes. Queries received after 30 minutes of the official examination duration time will not be responded
to. Submissions made after the official examination time will be rejected according to the examination regu-
lations and will not be marked. Only communication received from your myLife account will be considered.
15. Non-adherence to the processes for uploading examination responses will not qualify the student for any
special concessions or future assessments.
16. Queries that are beyond Unisa’s control include the following:
a. Personal network or service provider issues
b. Load shedding/limited space on personal computer
c. Crashed computer
d. Non-functioning cameras or web cameras
e. Using work computers that block access to the myExams site (employer firewall challenges)
f. Unlicensed software (eg license expires during exams)
Postgraduate students experiencing the above challenges are advised to apply for an aegrotat and submit
supporting evidence within ten days of the examination session. Students will not be able to apply for an
aegrotat for a third examination opportunity.
Postgraduate/undergraduate students experiencing the above challenges in their second examination oppor-
tunity will have to reregister for the affected module.
[TURN OVER]
, 4 STA4811
JAN/FEBR 2024
QUESTION 1
A manuscript is sent to a typing company consisting of typists A, B and C. If it is typed by typist A, then the
number of errors made follows a Poisson distribution with mean 3; if it is typed by typist B, then the number of
errors made follows a Poisson distribution with mean 3:5; and if it is typed by typist C, then the number of errors
made follows a Poisson distribution with mean 3:8. Let X denote the number of errors in the typed manuscript.
Assume that the job is given to typists A, B and C with probabilities 0:5, 0:3 and 0:2, respectively. Calculate:
(1.1) E[X ]. (3)
(1.2) Var.X /: (6)
[9]
QUESTION 2
Customers arrive at a small shop in accordance with a Poisson process with the rate of 12 per hour. The amount of
money paid by each customer is a random variable with a mean of R150 and a standard deviation of R100. The
shop is open 10 hours per day. Approximate the probability that the daily earnings of the shop are greater than
R16000. [12]
QUESTION 3
A Markov chain fX n ; n 0g with three states 0; 1; 2 has a transition probability matrix given by
0 1 2
01 1
1
0 2 2
0
B C
P D 1 @ 12 0 1
2 A:
2 21 1
2
0
1
If PfX 0 D 0g D 6
and PfX 0 D 1g D 14 , calculate E[X 3 ]. [17]
QUESTION 4
Consider two machines, both of which have an exponential lifetime with a mean of 5 years. There is a single
repairman that can service each machine with a mean time of 2 months. Assume that a state is the number of
machines that are working.
(4.1) Justify why the process can be considered as a birth and death process. (2)
(4.2) Write down the states of the process. (3)
(4.3) Write down the birth and death rates. (4)
(4.4) Set up the Kolmogorov backward equations. You do not have to solve them. (20)
[29]
[TURN OVER]
