FIT2086 Assignment 1 Solution

Introduction
1. Creating an image of a gorilla riding a motorcyle? [1 mark]
2. Discovering the shopping preferences of customers at Coles? [1 mark]
3. Predicting the amount of panadol that will purchased in the next week? [1 mark]
4. Using genomic and lifestyle factors to determine the chance of an individual contracting diabetes in the next two years? [1 mark]
H = 0 H = 1
W = 0 W = 1 W = 0 W = 1


P = 0 0.176 0.235 P = 0 0.117 0.058
P = 1 0.060 0.117 P = 1 0.059 0.178
Table 1: Table of the joint probabilities of a football team winning (W = 1) or losing (W = 0) when playing at home (H = 1) or away (H = 0) and whether they won (P = 1) or lost (P = 0) their previous game.
It is common in many sporting leagues for teams to alternate playing games at their own venue (i.e., “at home”) and at other team’s home venues (i.e., “away”). It is usually assumed that teams will play better at home, when they have support from their own fans, than when they play away. It is also commonly assumed that teams gain confidence from wins, and lose confidence from losses, so that they are more likely to win after a previous win and lose after a previous loss. Imagine we are working as a data analyst for a football team, and have collected data from the performance of the team. This is a (simple) example of sports analytics, an area of data science which is rapidly growing in importance over the last few years.
Imagine we have obtained information on a large number of previous games the team has played, along with information on whether they were played at home or not. Specifically, we have: whether the team won (W = 1) or lost (W = 0) their game; whether the game was played at home (H = 1) or away (H = 0); and whether the previous game that they played was a win (P = 1) or a loss (P = 0). Using this data we have calculated the joint probabilities P(W,H,P) of these three events. These are shown in Table 1. Please use these probabilities to answer the following questions with appropriate working/justification/explanation:
1. What is the marginal probability of the team winning a game, i.e., what is P(W = 1)? [1 mark]
2. What is the probability that the team will win a game, given that they lost their previous game, regardless of whether they are playing home or away? [1 mark]
3. What is the probability that the team will win a game, given that they won their previous game, regardless of whether they are playing home or away? [1 mark]
4. Do you believe that the team is more likely to win after winning their previous game than if they lost their previous game? [1 mark]
Imagine that we roll a fair six-sided die and a fair four-sided die (i.e., all sides have the same probability). Let X1 and Y1 be the independent random variables representing the outcomes of those events respectively. Let S = 2X1 − Y1 be two times the outcome of the roll of the six-sided die minus the outcome of the roll of the four-sided die. Please answer the following questions with appropriate working/justification.
1. What is the expected value of S, i.e., what is E [S]? [1 mark]
2. What is the variance of S, i.e., what is V [S]? [1 mark]
3. Determine the probability distribution of S, i.e., the probability that S ∈ {−2,...,11}. [1 mark]
4. What is the expected value of S3, i.e., what is ? [1 mark]
Imagine that a continuous random variable X defined on the range (0,s) follows the probability density function
( 2(s − x)
p(X = x|s) = s2 for x ∈ (0,s) .
0 everywhere else
where s > 0 is a parameter that controls the scale of the distribution. Answer the following questions; you must include appropriate working.
2. Determine the expected value of X, i.e., E [X]. [1 mark]
√ h√ i
3. Determine the expected value of X, i.e., E X . [1 mark]
4. Determine the variance of X, i.e., V [X]. [1 mark]
(hint: the answers to Q4.2 through Q4.5 will all be functions of s).
1. Fit a Poisson distribution to the COVID recovery data using maximum likelihood. What is the estimated value of this parameter for this data? [1 mark]
2. Plug the estimated λˆ into the Poisson distribution, and use this to make predictions about future COVID recovery times. Using this model, answer the following questions:
(a) What is the probability of a patient recovering in 10 or less days? [1 mark]
(b) What are the three most likely number of days it takes a patient to recover? [1 mark]
(c) Imagine that five individuals have COVID. What is the probability that these five individuals take a combined total of between 60 to 80 days (inclusive) between them to recover? [1 mark]
(d) What is the probability that three or more of these five patients will recover on or after day 14? [1 mark]