Question 1: (5 Total Marks)

At present, my wife, who is a school teacher, is COVID-19 free. However, each school day there is a

10% chance she will meet a COVID-infected individual and have her chance of staying uninfected

decrease by a random proportion, 𝑃𝑖

. Thus, over a 50 day school term, her chances of still being

uninfected is given by 𝐷 = 𝑃1 Γ— β‹― Γ— 𝑃𝑁, where 𝑁 ~ 𝐡𝑖𝑛(50,0.1). Assume the 𝑃𝑖

’s are independent

(of each other and 𝑁) and have a beta distribution with parameters 𝛼 = 10 and 𝛽 = 1. Finally, define

𝑆 = βˆ’ ln(𝐷) = βˆ’ ln(𝑃1 Γ— β‹― Γ— 𝑃𝑁) = βˆ’βˆ‘ ln(𝑃𝑖

)

𝑁

𝑖=1

= βˆ‘ ln(1/𝑃𝑖

)

𝑁

𝑖=1

= βˆ‘ 𝑋𝑖

𝑁

𝑖=1

where we have defined 𝑋𝑖 = ln(1/𝑃𝑖

).

[NOTE: It can be shown (and you may use without proof) 𝑋𝑖 = ln(1/𝑃𝑖

) ~ 𝐸π‘₯𝑝(10); see Q15 of Tutorial 5.]

a) Find the mgf of 𝑆. [2 marks]

b) Find the mean and standard deviation of 𝐷. [2 marks]

c) We know 𝐸(𝑆) = 𝐸(𝑁)𝐸(𝑋𝑖

), but does 𝐸(𝐷) = {𝐸(𝑃𝑖

)}

𝐸(𝑁)

? [1 mark]

Question 2: (3 Total Marks)

In my first year of teaching (at a university not named Bond), I had a class with an official enrolment

of 350 students, but lectures were scheduled in a room with a maximum capacity of 160. When I

queried the timetabling department, they told me the chance a typical student goes to a lecture (at this

university not named Bond) is 40%.

a) Use a normal approximation without the continuity correction to estimate the probability my room

was too small to hold the audience that showed up to my lecture on a typical day. Recalculate your

estimate, this time using the continuity correction. [2 marks]

b) During the mid-semester exam period, students were even less likely to attend lectures. In these

circumstances, do you think the normal approximation for the chance the room is too small will be

better or worse than it was for part (a)? Briefly justify your answer. [1 mark]

Question 3: (4 Total Marks)

I taught my daughter to drive and she was a bit heavy on the brakes to start. During drives to and from

her school, there are 16 locations requiring braking (e.g., roundabouts, stop signs, slip lanes etc.).

Further, a school term has 50 days, meaning 100 total drives back and forth. Assume the wear on the

brake pads from each braking instance has a mean of 0.009mm and standard deviation of 0.025mm.

a) If the lining of my brake pads is 16.5mm thick at the start of a term, what is the approximate

chance they last out the term (assuming my daughter misses no days of school)? [2 marks]

b) In fact, wear is uneven between front and rear pads. Suppose total wear on the rear pads during a

single trip is normal with mean 0.16mm and standard deviation 0.12mm, while total wear on the

front pads is normal with mean 0.128mm and standard deviation 0.08mm. Further, assume the

correlation between wear on the pads is 0.8. If the rear pad was worn down by 0.192mm during

today’s morning drive, what is the probability the front pad wear was less than 0.16mm? [2 marks]

Question 4: (3 Total Marks)

Consider independent random variables, 𝑋1 and 𝑋2, each having a π‘Šπ‘’π‘–π‘π‘’π‘™π‘™(0.5, πœƒ) distribution. For

this distribution it is readily seen that the pdf and CDF, respectively, are:

𝑓(π‘₯) =

πœƒ

2√π‘₯

𝑒

βˆ’πœƒβˆšπ‘₯

and 𝐹(π‘₯) = 1 βˆ’ 𝑒

βˆ’πœƒβˆšπ‘₯

for π‘₯ > 0.

Define 𝑀 = min{𝑋1,𝑋2

}, and let 𝑇1 = 2βˆšπ‘€ and 𝑇2 = (βˆšπ‘‹1 + βˆšπ‘‹2)/2.

a) It can be shown 𝐸(βˆšπ‘‹1) = 𝐸(βˆšπ‘‹2) = 1/πœƒ and 𝐸(βˆšπ‘€) = 1/(2πœƒ). Use this information to show

both 𝑇1 and 𝑇2 are unbiased estimators of 𝜏(πœƒ) = 1/πœƒ. [1 mark]

b) Show that π‘‰π‘Žπ‘Ÿ(𝑇2

) =

1

2

πœƒ

βˆ’2

. Will π‘‰π‘Žπ‘Ÿ(𝑇1

) be bigger or smaller? Justify your answer. [NOTE: You

do not need to actually calculate π‘‰π‘Žπ‘Ÿ(𝑇1

) to answer this question.] [2 marks]