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Homework #2. Solution.
Spring 2001
IE 230
Textbook: D.C. Montgomery and G.C. Runger, Applied Statistics and Probability for
Engineers, John Wiley & Sons, New York, 1999. Chapter 3, Sections 3.1-3.2.
Quiz 1, Monday, January 22, 2001. The English and Greek alphabets, the difference between
probability and statistics, set theory (page 1 of the Concise Notes), and textbook
chapters 1 and 2 (page 2 of the Concise Notes).
Chapter 3 is the most important chapter. It is the foundation for the rest of this course and of
IE 330 and of IE 336. Study the concise notes and the textbook. Now is the time to
focus.
Before beginning this homework, or any other, read the relevant textbook sections. Maybe
work some of the textbook’s problems.
1. Venn Diagrams
(a) With a Venn diagram, illustrate two sets, A and B , that are mutually exclusive.
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Draw a rectangle and label it "S" to represent the sample space.
Draw two nonoverlapping circles inside the rectangle and label
them A and B to represent the two mutually exclusive events.
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(b) With a Venn diagram, illustrate A ∩ B ′. (Do not assume that A and B are mutually
exclusive.)
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Draw a rectangle and label it "S" to represent the sample space.
Draw two partially overlapping circles inside the rectangle and label
them A and B . Shade the area inside A that is not inside B .
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2. Consider the random experiment whose procedure is to randomly select a person. The
outcome is the person’s blood type. Let the sample space be S = {O , A , B , AB }.
Consider blood donation. A type "O" person is a universal donor, able to give to all the
other types. A type "AB" person is a universal receiver, able to receive from all other
types. More generally, a person without an "A" cannot receive blood with an "A"; and a
person without a "B" cannot receive blood with a "B".
Define the event RO = "the randomly selected person can receive type O blood". Define
RA , RB , and RAB analogously.
(a) RO = {O , A , B , AB }; that is, RO = S . Write RA , RB , and RAB as sets of outcomes.
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RA = {A , AB }, RB = {B , AB }, RAB = {A , B , AB }
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Homework #2. Solution.
Spring 2001
IE 230
(b) Are RA and RB mutually exclusive? Argue why or why not by carefully using the
definition.
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No. Both contain the outcome AB (that the chosen person has AB blood).
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(c) Write RA ′ by listing its elements. In words, what is RA ′?
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RA ′ = {O , B }, the two blood types not in RA .
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(d) Show that RA ∩ RB and RAB are the same event.
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They are the same event because they both contain the single outcome AB .
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(e) Consider the event of selecting a person with type O blood. This event contains only
one outcome, "O", which can be written {O }. This event can also be written as
RO ∩ (RA ∪ RB )′. This expression is unnecessarily complicated. Find a simpler
way to write {O } in terms of RO , RA , RB , and RAB .
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As noted in Part (a), R 0 = S . Therefore, intersecting with R 0
serves no purpose, because no outcomes are eliminated.
So one simpler expression is (RA ∪ RB )′.
Another reasonable answer is obtained via DeMorgan’s Law,
which says that (RA ∪ RB )′ = RA ′ ∩ RB ′.
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3. (From Problem 3-38.) Use the axioms of probability to show the following:
(a)
A,
B,
and
If
events
P(A ∪ B ∪ C ) = P(A ) + P(B ) + P(C ).
C
are
mutually
exclusive,
then
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P(A ∪ B ∪ C )
= P(A ∪ (B ∪ C ))
= P(A ) + P(B ∪ C )
= P(A ) + P(B ) + P(C )
Same event
Axiom 3
Axiom 3
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(b) If the n equally likely events E 1, E 2, . . . , En partition the sample space S , then
P(Ei ) = 1 / n for each i = 1, 2,..., n .
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Recall: The events partition S if they are mutually exclusive and their union is S .
1
=
=
=
=
P(S )
P(E 1 ∪ E 2 ∪ . . . ∪En )
P(E 1) + P(E 2) + . . . + P(En )
n × P(Ei )
for every i = 1, 2,..., n .
Axiom 1
Same event
Mutually exclusive events
Equally likely events
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Schmeiser
Homework #2. Solution.
Spring 2001
IE 230
4. (Do the following analysis using a computer spread sheet. E-mail the spread sheet to
"[email protected]". In the subject line, say "HW2, #4 your.name". The reply will
be electronic, so mail from the account where you want your reply.)
(a) In the first row or two or three, provide some heading information, including your
name, date, problem number, and a brief description of the meaning of the data.
(b) In Column A enter the integers 1 through 10. Above these integers enter the column
heading Observation #.
(c) Flip a coin until it comes up heads; let x 1 be the number of flips. Enter this first
observation in Column B next to the "1". Repeat nine more times, so that you have
data x 1, x 2, . . . , x 10. Record xi next to the integer i . Above these data enter the
column heading x_i.
(d) Below the data in Column B, compute the sample average. Label it.
(e) Below the sample average in Column B, compute the sample standard deviation.
Label it.
(f) In column C, place the values i /(n +1) for i = 1, 2,..., n , where n = 10 is the sample size.
(g) Sort the original data and place the order statistics x (1), . . . , x (n ) into Column D.
Highlight in red the sample median. Highlight in blue the first and third sample
quartiles. Highlight in green x (1) and x (n ).
(h) Below the sample average in Column B, compute the sample range. Label it.
(i) Create a scatter plot from Columns C and D to obtain an empirical cumulative
distribution plot (similar to Figure 2-7 in the textbook). Label the plot and both
axes.
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Schmeiser