COURSE NAME: ENGINEERING DATA ANALYSIS
COURSE DESCRIPTION: This course is designed for undergraduate students with emphasis
on problem solving related to societal issues that engineers are
called upon to solve.
The table below was lifted from the course outline, the section next to that of week 3.
DISCRETE PROBABILTY DISTRIBUTIONS
RANDOM VARIABLES AND THEIR PROBABILTY DISTRIBUTIONS
EXPECTED VALUES OF RANDOM VARIABLES
THE BINOMIAL DISTRIBUTION
CONTINUOUS PROBABILITY DISTRIBUTION
CONTINUOUS RANDOM VARIABLES AND THEIR PROBABILTY DISTRIBUTION
--------------------------------------Before anything else, as for my reference, I would like to express my deepest gratitude to the
Coconino Community College for a very defined deliberation of this topic.
Probability distribution as defined is an assignment of probabilities to
the values of the random variable. When we speak of random variable, it
represents a quantitative (numerical) variable that is measured or observed in
an experiment. Therefore, you can actually find a probability associated with
that variable.
These random variables which are results from counting or measuring are
generally categorized as quantitative variables. There are two types of
quantitative variables. These are discrete and continuous.
Discrete data can only take on particular values in a range. Continuous
data can take on any value in a range. Discrete data usually arises from counting
while continuous data usually arises from measuring.
Discrete data can be expressed in different methods namely: General
Probability Distribution, Cumulative Distribution, Binomial and Poisson
Distribution
Below are examples to delineate discrete from continuous variable.

Discrete variable
Fleas in a colony
The number of students in a class
Continuous Variable
Height of a plant
Time in a race
(You cannot have half of a student.)
(You can have fraction of a second.)
The results of rolling 2 dices
A dog's weight
The number of test questions you
answered correctly.
Your Age
(You can actually be 25 years, 10
months, 2 days, 5 hours, 4 seconds, 4
milliseconds,
8
nanoseconds,
99
picoseconds…and so on.)
Shoe size in a mall

(As asses, you cannot have the size
same as the length of your feet. You
will have to choose then from the
standard
sizes
provided
by
the
center.)
The number of arrangements
(However, if it is bounded for
instance: ”A person’s age in years,”
it is now considered as discrete.)
Temperature of room in a day
(Temperature may vary even in a
change of second)
As evident, continuous variables are those data gathered mostly from measuring
the height, temperature, time and weight.
For a discrete probability distribution function:
The mean or expected value is
The variance is
The standard deviation is
where x = the value of the random variable
P(x) = the probability corresponding to a particular x value.
0 ≤P(x)≤1 and ∑P(x) = 1
To determine the rarity of cases, we have to assess by setting a fix
boundary of 0.05
 If P(the
consider
 If P(the
consider
variable has a value of x or more) < 0.05, then you can
the x an unusually high value.
variable has a value of x or less) < 0.05, then you can
this an unusually low value.
Following are some examples to give you an overview of discrete data and how it
is processed to come up with its probability distribution, bearing values and
the explanation of these values.
Pr#1) Given the probability distribution below. (Please see solution)
a.)
b.)
c.)
d.)
e.)
f.)
State the random variable.
Find the mean.
Find the variance.
Find the standard deviation.
Is it unusual for a household to have six people in the family.
Is it unusual for a household to have four people in the family?
*Calculating the expected value
Pr#2) In a lottery called Pick 3, a player pays $1 and then picks a three-digit
number. If those three numbers are picked in that specific order the
person wins $500. What is the expected value in this game? (Please see
solution.)
*Binomial Distribution
A binomial distribution is a special discrete distribution.
Binomial distributions are from binomial experiments wherein you have a fixed
number of independent trials with only two outcomes.
To consider an experiment as binomial, it should have guided by these rules.
 There should be fixed number of trials so you have the boundaries or
limitation on your events.
 Each trial is an independent event which means that every time you repeat
the trial there will be no effect to next trial.
 There can only be two outcomes, a success or failure.
where p = probability of success
q = probability of failure,
q = 1− p
For instance, in tossing a coin.
 You state the number of trials.
 Every time you toss the coin, it has no effect on the next toss.
 The outcome will either be a failure/success of getting a head.
Other examples of binomial experiments are as follows:
 Toss a fair coin ten times, and find the probability of getting
two heads.
 Question twenty people in class, and look for the probability of
more than half of them are women?
 Shoot five arrows at a target
Find the probability of hitting it five times?
As follows, is the Binomial Formula for the probability of r successes in n
trials.
where nCr is the number of combinations of n things taking r at a
time. It is read “n choose r”.
As you may recall from the week3 handouts,
is a Combination Formula.
If you list all possible values of x in a Binomial distribution, you get the
Binomial Probability Distribution
Mean
Variance
Standard Deviation
where
p = the probability of success
q=1-p = probability of failure
Let us now see how are we going to use this Binomial Formula through examples
and check whether we arrive at the same values if we use the solution
discussed from week 3 handouts.
Pr#3.) Suppose you are given a 3 question multiple-choice test. Each question
has 4 responses and only one is correct. Suppose you want to find the
probability that you can just guess at the answers and get 2 questions
right. (Please see solution.)
a.)
b.)
c.)
d.)
What is the random variable?
Is this a binomial experiment?
What is the probability of getting 2 questions right?
What is the probability of getting zero right
one right and all three right?
Pr#4.) When looking at a person’s eye color, it turns out that 1% of people
in the world has green eyes. Consider a group of 20 people. (Please
see solution.)
a.) State the random variable
b.) Argue that this is a binomial experiment
Find the probability that
c.) None have green eyes.
d.) Nine have green eyes.
e.) At most three have green eyes.
f.) At least four have green eyes.
g.) In Europe, four people out of twenty have green eyes.
Is this unusual?
What does that tell you?
h.) Write the probability distribution.
i.) Find the mean
j.) Find the variance
k.) Find the standard deviation
EDA11(WEEK4(ASSIGNMENTS)
DISCRETE PROBABILTY DISTRIBUTIONS
PR#1.) Eyeglassomatic manufactures eyeglasses for different retailers. The
number of days it takes to fix defects in an eyeglass and the its
probability.
Table #5.1.8: Number of Days to Fix Defects
Number
days
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
of
Probabilities%
24.9
10.8
9.1
12.3
13.3
11.4
7.0
4.6
1.9
1.3
1.0
0.8
0.6
0.4
0.2
0.2
0.1
0.1
a.)
b.)
c.)
d.)
e.)
State the random variable.
Find the mean number of days to fix defects.
Find the variance for the number of days to fix defects.
Find the standard deviation for the number of days to fix defects.
Find probability that a lens will take at least 16 days to
make
a fix the defect.
f.) Is it unusual for a lens to take 16 days to fix a defect?
g.) If it does take 16 days for eyeglasses to be repaired, what would
you think?
PR#2.) Suppose you have an experiment where you flip
then count the number of heads.
a.) State the random variable.
b.) Write the probability distribution for the
c.) Find the mean number of heads.
d.) Find the variance for the number of heads.
e.) Find the standard deviation for the number
f.) Find the probability of having two or more
a coin three times. You
number of heads.
of heads.
number of heads.
g.) Is it unusual for to flip two heads?
PR#3.) The lottery has a game called Pick 4 where a player pays Php1 and picks
a four-digit number. If the four numbers come up in the order you picked,
then you win Php2,500. What is your expected value?
PR#4.) An LG Dishwasher, which costs Php 800, has a 20% chance of needing to be
replaced in the first 2 years of purchase. A two-year extended warrantee
costs Php 112.10 on a dishwasher. What is the expected value of the
extended warranty assuming it is replaced in the first 2 years?
PR#5.) According to the Center for Disease Control (CDC), about 1 in 88 children
in the U.S. have been diagnosed with autism ("CDC-data and statistics,"
2013). Suppose you consider a group of 10 children
a.)
b.)
c.)
d.)
e.)
f.)
g.)
State the random variable.
Argue that this is a binomial experiment.
None have autism.
Seven have autism.
At least five have autism
At most two have autism.
Suppose five children out of ten have autism.
Is this unusual? What does this tell you?
PR#6.) Approximately 10% of all people are left-handed. Consider a grouping of
fifteen people.
a.) State the random variable.
b.) Argue that this is a binomial experiment
Find the probability that
c.) None are left-handed.
d.) Seven are left-handed.
e.) At least two are left-handed.
f.) At most three are left-handed.
g.) At least seven are left-handed.
h.) Seven of the last 15 U.S. Presidents were left-handed.
Is this unusual? What does that tell you?
PR#7.) According to an article in the American Heart Association’s publication
Circulation, 24% of patients who had been hospitalized for an acute
myocardial infarction did not fill their cardiac medication by the
seventh day of being discharged (Ho, Bryson & Rumsfeld, 2009). Suppose
there are twelve people who have been hospitalized for an acute
myocardial infarction.
a.) State the random variable.
b.) Argue that this is a binomial experiment
Find the probability that
c.) All filled their cardiac medication.
d.) Seven did not fill their cardiac medication.
e.) None filled their cardiac medication.
f.) At most two did not fill their cardiac medication.
g.) At least three did not fill their cardiac medication.
h.) At least ten did not fill their cardiac medication.
i.) Suppose of the next twelve patients discharged, ten did not fill their
cardiac medication, would this be unusual? What does this tell you?
PR#8.) Suppose a random variable, x, arises from a binomial experiment.
n = 6, and p = 0.13.
a.) Write the probability distribution.
b.) Find the mean.
c.) Find the variance.
d.) Find the standard deviation.
Suppose
PR#9.) Suppose a random variable, x, arises from a binomial experiment.
n = 10, and p = 0.81.
a.) Write the probability distribution.
b.) Find the mean.
c.) Find the variance.
d.) Find the standard deviation.
Suppose
PR#10.) Suppose a random variable, x, arises from a binomial experiment. Suppose
n = 7, and p = 0.50.
a.) Write the probability distribution.
b.) Find the mean.
c.) Find the variance.
d.) Find the standard deviation