• has all pages clearly readable;

• has all pages cropped to the A4 borders of the original page and is imaged from directly above to avoid excessive ’keystoning’.

These requirements are easy to meet if you use a scanning app on your phone and take some care with your submission - please review it before submitting to double check you have satisfied all of the above requirements.

(4) Late submission within 20 hours after the deadline will be penalised by 5% of the to- tal available marks for every hour or part thereof after the deadline.  After that, the Gradescope submission channel will be closed, and your submission will no longer be accepted. You are strongly encouraged to submit the assignment a few days before the deadline just in case of unexpected technical issues. If you are facing a rather excep- tional/extreme situation that prevents you from submitting on time, please complete the assignment extension request form that is available on Canvas.

(5) Working and reasoning must be given to obtain full credit. Clarity, neatness, and style count.

Q1.  The high-water mark X (in metres) measured from the bottom of a river is a continuous random variable having the cumulative distribution function

8 0;

FX (x) = P(X ≤ x) = < c1     ;

( c1     

(a) Prove that c =  and c2  =  .

if x < 2;

if 2 ≤ x < 4;

if x  4.

(b) Find the pdf of X .

(c) The river bank is three metres high (measured from the bottom of the river) and excessive water fills the wetland next to the river. Find the probability that the wetland receives water from the river.

(d) Compute E (Xk ) for all k 2 R.

(e) Compute V (X).

Q2.  In Victoria, Road Safety Rule 141(2) states that, ‘The rider of a bicycle or an electric scooter must not ride past, or overtake, to the left of a vehicle that is turning left and is giving a left change of direction signal.’Assume that a bike rider is not aware of the rule, and each day on their way to their new workplace, there is a probability of 0.01 of meeting a left turning vehicle, resulting in an incidence of forcing the vehicle to give way to them.

(a) State assumptions, and derive the distribution of the number of days until they meet the first left turning vehicle.

(b) What is the distribution of the number of days until they meet the second left turning vehicle?

(c) What is the probability that there are at most 60 incidence free days till they meet the second left turning vehicle?

(d) What is the probability that on the fifth day of the first week, they meet the second left turning vehicle?

(e) Assume that the rider plans to work in the new job for five years with  1150

working days, find a suitable Poisson approximation to the distribution of the number X of days that they meet left turning vehicles in five years.

(f) Let Y have the Poisson distribution in (e), use the Matlab to compute

1

d =工 |P(X = i) - P(Y = i)|.

i=0

Report the value d and attach the Matlab commands here.

Q3.  Mr X plants 20 seedlings.  After one month, independent of the other seedlings, each

seedling has the probability 0.1 of being dead, the probability 0.4 of showing slow growth and the probability 0.5 of showing satisfactory growth.

(a) What is the probability that after one month exactly 2 seedlings are dead?

(b) What is the probability that after one month exactly 2 seedlings are dead and

exactly 11 seedlings show satisfactory growth?

(c) What is the expected number of seedlings showing satisfactory growth?

(d) What is the probability that no more than 2 seedlings have died and no more than 3 have shown slow growth?

Q4. A professor examined the performance of their students in a math subject, and a simplified version of the model they fit to the scores is tabulated below.

(a) Compute E (X) and E (X2 ) using the formula on lecture slide 143.

(b) Write down the cumulative distribution function FX  of X .

(c) Compute E (X) using the formula on lecture slide 158.