Posted: September 13th, 2017

Question 1: Dependence;

Question 1: Dependence;

Let X be a Binomial random variable defined as the sum of 6 independent
Bernoulli trials. The probability of a Bernoulli taking the value 1 is given
by p. Suppose that prior to the 6 Bernoulli trials, p is chosen to take one
of three possible values with the following probabilities
p Probability
0.2 0.1
0.6 0.7
0.8 0.2
Compute the joint probability distribution of X and p. Are X and p
independent? Provide your reasoning. (3 Marks)
Compute the unconditional mean and variance of X (3 Marks)
Compute the conditional mean of X given each possible value of p. Based
on your calculations, what sign do you expect the covariance between X
and p to be? (4 Marks)
1
Question 2: Road Kill
Consider a 1 kilometre stretch of highway between Sydney and Melbourne.
Animal welfare groups estimate that on average, 12 animals are killed by
traffic for every kilometre of highway between Sydney and Melbourne.
Let X be the number of animals killed in a given 1 kilometre stretch of
highway between Sydney and Melbourne. Write down the probability
distribution function of X. (2 Marks)
Compute the expected value and standard deviation of X (2 Marks)
Now divide the 1 kilometre stretch of highway into meters. What is
the probability of an animal being killed in a given meter? (1 Mark)
Let Y be the number of animals killed within the first 20 meters.
Write down a candidate probability distribution function for Y . Please
give your reasoning behind your choice of distribution function. (2
Marks)
Compute the expected value and the standard deviation of Y . (2
Marks)
Compute the probability that at least 1 animal is killed within the
first 20 meters of the stretch of highway. (1 Mark)
Question 3: Continuous Random Variable
Consider a random variable Z that has the following probability density
function
fZ(z) =
?
?
?
z 0 ? z ? 1
2 ? z 1 < z ? 2
0 otherwise
Provide a visual depiction of this particular density function (1 Mark)
Verify that fZ(z) is a proper probability density function. (2 Marks)
Without computing the integral, determine the mean of Z. Provide
the reasoning behind your answer. (1 Mark)
Without computing the integral, determine the skewness of Z. Provide
the reasoning behind your answer. (1 Mark)
Instruction files

Quantitative Methods 1
Assignment 2: Probability and Random Variables
This assignment has three questions, and is due by 5.00pm on Friday May
1st 2015. It is to be submitted electronically as a .pdf le using the assignment
tool on the subject’s LMS page. Marks depend on your tutor being able to un-
derstand your statements and arguments, so marks may be deducted for poor
presentation or unclear language. Use nothing smaller than 12 point font. If
you wish to write your assignment by hand and scan the le into a .pdf format,
you may, though any illegible content will not be marked. Note that you can
only submit one le, and cannot submit a le larger than around 10 megabytes.
You may work in groups of up to 4 students from the same allocated
tutorial. Groups should nominate one student to submit the assignment for
the whole group, with each student’s name and student number included in the
document. This assignment has a total of 25 marks available and may contribute
up to 10% of your nal mark in this subject.
Question 1: Dependence
Let X be a Binomial random variable dened as the sum of 6 independent
Bernoulli trials. The probability of a Bernoulli taking the value 1 is given
by p. Suppose that prior to the 6 Bernoulli trials, p is chosen to take one
of three possible values with the following probabilities
p Probability
0.2 0.1
0.6 0.7
0.8 0.2
Compute the joint probability distribution of X and p. Are X and p
independent? Provide your reasoning. (3 Marks)
Compute the unconditional mean and variance of X (3 Marks)
Compute the conditional mean of X given each possible value of p. Based
on your calculations, what sign do you expect the covariance between X
and p to be? (4 Marks)
1
Question 2: Road Kill
Consider a 1 kilometre stretch of highway between Sydney and Melbourne.
Animal welfare groups estimate that on average, 12 animals are killed by
trac for every kilometre of highway between Sydney and Melbourne.
{ Let X be the number of animals killed in a given 1 kilometre stretch of
highway between Sydney and Melbourne. Write down the probability
distribution function of X. (2 Marks)
{ Compute the expected value and standard deviation of X (2 Marks)
{ Now divide the 1 kilometre stretch of highway into meters. What is
the probability of an animal being killed in a given meter? (1 Mark)
{ Let Y be the number of animals killed within the rst 20 meters.
Write down a candidate probability distribution function for Y . Please
give your reasoning behind your choice of distribution function. (2
Marks)
{ Compute the expected value and the standard deviation of Y . (2
Marks)
{ Compute the probability that at least 1 animal is killed within the
rst 20 meters of the stretch of highway. (1 Mark)
Question 3: Continuous Random Variable
Consider a random variable Z that has the following probability density
function
fZ(z) =
8<
:
z 0 z 1
2 ?? z 1 < z 2
0 otherwise
{ Provide a visual depiction of this particular density function (1 Mark)
{ Verify that fZ(z) is a proper probability density function. (2 Marks)
{ Without computing the integral, determine the mean of Z. Provide
the reasoning behind your answer. (1 Mark)
{ Without computing the integral, determine the skewness of Z. Pro-
vide the reasoning behind your answer. (1 Mark)
2