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Homework #2
STAT351.001
Winter 2024
Due: Sunday, February 4th
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Include all work in a neat and well organized presentation. Grading is based on the quality,
thoroughness, and correctness of the work provided.
You do not need to print out this assignment; you may provide your work and answers on your own
separate paper (had written is sufficient).
Upload your solutions as a single PDF file to the “Homework 2” submission folder in our Canvas course.
Your submission should be titled “(your name) STAT351 Homework 2.”
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Notes:
1. Show all work, points are awarded based on the work provided.
2. When providing probabilities provide them as numbers between 0 and 1 (not as percentages),
and keep at least 3 decimal places.
_____________________________________________________________________________________
1.
You roll a fair (balanced) 6-sided dice two times, record what lands face-up.
(4pt)
Let A be the event: the sum of the face-up values is greater than or equal to 9.
Let B be the event: the difference between face-up values is less than or equal to 2.
Let C be the event: neither dice lands on the value “1”.
Compute the following probabilities.
a.
( | )
b.
( | )
c.
( | )
d.
( | )
________________________________________________________________________
2.
In a particular city, A recent analysis concluded that
(5pt)
12% of all residences have detectable levels of lead in their tap water and
8% of all residences have detectable levels of radon in their basement or sub-foundation.
Additionally, the analysis noted that 5% of all residences have both, detectable levels of
radon, and detectable levels of lead.
Define the following events:
L = a random residence has detectable levels of lead in their tap water.
R = a random residence has detectable levels of radon in their basement or
sub-foundation
Calculate each of the following:
a.
( | ∪ )
c.
( ∩ | ∪ )
b.
( | ∩ )
________________________________________________________________________
3.
Seventy percent of the light aircraft that disappear while in flight in a certain country are
(3pt)
subsequently discovered. Of the aircraft that are discovered, 60% have an emergency
locator, whereas 90% of the aircraft not discovered do not have such a locator. Suppose
a light aircraft has disappeared.
a.
b.
If it has an emergency locator, what is the probability that it will not be
discovered?
If it does not have an emergency locator, what is the probability that it will be
discovered?
________________________________________________________________________
4.
You have three boxes labelled Box #1, Box #2, and Box #3.
(2pt)
Initially each box contains 4 red balls and 4 green balls.
One ball is randomly selected from Box #1 and placed in Box #2
Then one ball is randomly selected from Box #2 and placed in Box #3.
Then one ball is randomly selected from Box #3 and placed in Box #1.
At the conclusion of this process, what is the probability that that each box has the same
number of red and green balls that it started with (i.e. 4 red and 4 green balls)?
________________________________________________________________________
5.
An experiment consists of four trials. Each trial consists of rolling two fair (balanced)
(3pt)
6-sided dice, then recording the sum of the two face-up sides.
Note: the trials are independent.
a.
Calculate the probability that the sum is greater than or equal to 9 on all four
trials.
b.
Calculate the probability that the sum is less than 9 on all four trials.
c.
Calculate the probability that the sum is less than 9 on one or more of the trials.
________________________________________________________________________
6.
Use Matlab (or equivalent) to complete the following.
(3pt)
In Matlab, the symbol, %, allows you to add comments.
Characters entered after % are not considered “code”.
a.
In the first line of your code, type your name, to authenticate that the code is
yours.
>>% John Doe
b.
Generate two separate random samples of 1,000,000 values from a uniform [0, 1]
distribution (use the rand command (below); Note: you also generated Uniform [0,1]
random variables in Activity #1). Call these two vectors something like X1 and X2.
Matlab:
>>N=1000000;
>>X1=rand([1,N]); % generates a vector of length N with elements generated randomly
from a Uniform [0,1] distribution.
Note: the symbol, ;, after the command suppresses output (you don’t want to display all
1000000 values of vector X1.) Also
c.
Plot the normalized histogram with 20 bins for each of the above sequences.
For a continuous random variable, “normalized” means to rescale so that the area
between the x-axis and the probability density function is 1. Hence, the histogram is a
“discrete” estimate of the continuous probability density function (pdf).
The following command works very nicely:
>>bins=20;
>>histogram(X1,bins,’normalization’,’pdf’);
d.
Add these two sequences and produce a normalized histogram of this vector of
sums. i.e. let Y = X1 + X2. Y will be a vector of length N, where [yi] = [x1i + x2i].
Note that since both X1 and X2 values range from 0 to 1, Y values would range
from 0 to 2.
e.
Create a third sequence (length 1,000,000) from the same Uniform distribution as
X1 and X2. Call it X3 if you’d like. Add these together (W = X1+X2+X3) and produce
a normalized histogram of this vector
f.
Discuss your results. Specifically address the range of values of the histograms
and their shapes as you add more and more sequences.
Be sure to label the axes of all your graphs.
g.
Hand in all graphs, your response to f), and your MatLab code. You might want
to copy and paste all into a Word document.
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