MAE108S22p1 (1)
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University of California, San Diego *
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Course
108
Subject
Aerospace Engineering
Date
Dec 6, 2023
Type
Pages
4
Uploaded by HighnessSandpiper3218
MAE 108: Probability and Statistical Methods for Mechanical
Engineering
Homework 1, Due 8 April 2022 to Gradescope by 10pm
1. Random numbers and histograms. In this problem, you will make four
plots, each with four panels (subplot in Matlab). After making the first
plot, the other three plots follow with small changes to the Matlab com-
mands.
(a) Consider the function rand in Matlab that generates a uniformly dis-
tributed random number. In figure 1, make the following plot.
i. Generate a vector of 1000 uniformly distributed random numbers
for
x
∈
[0
,
1] using the command
>>
xrandu=rand(1000,1);
ii. Plot xrandu as a function of index.
>>
subplot(221), plot(xrandu);
iii. Plot (in subplot(222)) a histogram of xrandu using the function
histogram in Matlab.
Read the documentation of histogram in
the Matlab help.
Label the
x
and
y
axes with the appropriate
description for the default normalization. What is the area under
the histogram for the default normalization?
iv. Plot a histogram (in subplot(223)) where you fix the edges of the
histogram so that you have bin edges at v=[0:.25:1]; How many
bins make up this histogram?
v. Plot (in subplot(224)) a histogram using the normalization option
”pdf”. Label the
x
and
y
axes with the appropriate description.
What is the area under the histogram for the normalization option
”pdf”?
vi. In figure 2, repeat parts (i)-(v) for
>>
xrandu=rand(10000,1);
Provide a brief description of the difference the histograms for 1000
and 10000 random numbers.
(b) In figures 3 and 4, repeat part (a) for a normally distributed random
variable using randn in Matlab. Note that randn has
x
∈
[
−∞
,
∞
]
but your plots might have
x
∈
[
−
5
,
5].
2. Rule of the product MIT OCW 18.05 reading 1b section 2.3.2 states that
if there are
n
ways to perform action 1 and then
m
ways to perform
action 2, then there are
n
·
m
ways to perform action 1 followed by action
2. Find, using the rule of the product, the size of the sample space for
the following experiments, and the elements of the sample space if asked.
(a) toss a coin 4 times and record the result. Here, also define the ele-
ments of the sample space.
(b) toss a coin 9 times and record the result (only give the size of the
sample space)
(c) roll two dice and record the result (first die, second die). Here, also
define the elements of the sample space.
(d) roll five dice, and record the five numbers (first die, second die, third
die, fourth die, fifth die, only give the size of the sample space).
3. Once you know the size of the sample space associated with an experi-
ment, if all outcomes are equally probable, then the probability of occur-
rence of an event is the number of outcomes in the event divided by the
size of the sample space. Find the following probabilities.
(a) assuming a fair coin, what is the probability of tossing 2 heads and 2
tails in four tosses (order does not matter)?
(b) assuming five fair dice, what is the probability of rolling five ones (i.e.
all the dice show 1)?
(c) What is the probability of being dealt four kings in a five card poker
hand?
4. Events: A principal investigator purchases 3 pressure sensors to measure
waves and water levels for a research project. At the end of the 5 year
grant period, let
F
1
,
F
2
and
F
3
denote the events that sensors 1, 2 and 3
have failed (are not operational). Define the following events at the end
of the grant period year in terms of
F
1
,
F
2
and
F
3
and their complements
(a) A= only sensor 1 is operational.
(b) B= exactly two sensors are operational.
(c) C= at least two sensors have failed.
5. A water sample is tested for contamination and salt. The sample is either
contaminated or pure, and either salty or fresh.
(a) What is the sample space for the water sample?
(b) If every outcome is equally likely, what is the probability the event
“a fresh water sample is contaminated”?
(c) If instead the outcome of a water sample being fresh and pure is 3
times a likely as the other outcomes, what is the probability of the
event “a fresh water sample”?
6. In lecture, we determined the probability of having exactly one bulldozer
out of three operational after 6 months to be 3/8, given that the proba-
bility of a single bulldozer being operational after 6 months is 50%.
(a) What is the probability that a single bulldozer is nonoperational after
six months?
(b) New information has the probability of a bulldozer being operational
after 6 months as 35%. What is the new probability that exactly one
bulldozer out of three is nonoperational after 6 months?
7. Two types of impurities in a material sample compromise the strength
of the material under loading. Impurity A contaminates 7% of the sam-
ples and impurity B contaminates 2% of the samples. Assume that the
events of contamination by impurity A and impurity B are statistically
independent.
(a) Determine the probability that a material sample chosen at random
is contaminated by both impurities A and B.
(b) Determine the probability that a material sample chosen at random
for inspection is contaminated.
8. A plant will die if it does not receive any one or more of the follow-
ing:
sunlight (
S
), water (
W
) and fertilizer (
F
) in a given week.
The
probability of the plant not receiving its needs in a given week are
P
(
S
) = 0
.
20
,
P
(
W
) = 0
.
40
P
(
F
) = 0
.
30
Not receiving sunlight, water and fertilizer are statistically independent.
(a) What is the probability of a plant not receiving sunlight and water?
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(b) What is the probability of the plant receiving water and fertilizer?
(c) What is the probability of the plant dying?