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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)

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