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Homework #5. Due: Wednesday, September 29, 1999.
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 3.8, and
Chapter 4, Sections 4.1-4.5. Page 5 of the concise notes.
1. Consider an experiment in which a fair six-sided die is tossed once. The usual sample
space is S = {1, 2,..., 6}. Suppose we play a game in which I pay you X dollars,
where X = "the square of the outcome" in S. That is, if you toss a "4" then I will
pay you $16.
(a) State a sample space other than S for this experiment. (In the following parts,
ignore your answer to this part.)
(b) State the pmf of X. (Recall that the pmf is defined for every real number.) Plot
the pmf. (By hand is fine, but label and scale both axes.)
(c) State the cdf of X. (Recall that the pmf is defined for every real number.) Plot
the cdf. (By hand is fine, but label and scale both axes.)
(d) Does X have a discrete uniform distribution? If not, define another random
variable Y that does (for this experiment).
(e) Compute the mean, E(X). Show this value on your plots of parts (b) and (c).
(f) Compute the variance, V(X). What are the units of V(X)?
(g) Compute the standard deviation of X. What are the units?
(h) Define the event E = {2, 4, 6}. On any one replication of the experiment, how
many outcomes within E can occur?
(i) In replications that E occurs, what are the possible values of X?
(j) In replications that X = $ 9, did E occur?
(k) Is "X = $ 9" a random variable, an event, or a distribution?
2. Let A be an event. Define the random variable Y to be equal to 1 if A occurs and equal
to zero of A' occurs. Y is then an indicator random variable because Y = 1 indicates that A
occurred.
(a) Show that E(Y) = P(A).
(b) What event maximizes E(Y)?
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Homework #5. Due: Wednesday, September 29, 1999.
IE 230
(c) What event minimizes E(Y)?
(d) Recall: A random variable is a function that associates a real number with
every outcome in the sample space. State this function explicitly for the indicator
Y when the event is E from 1(h) above. (Notice that this part requires no
probabilities.)
3. (From Problem 4-30.) Let X denote the thickness of wood paneling (in inches) that a
customer orders. The cdf is
 0
0.2

F( x )  
 0 .9
 1
if    x  1 / 8"
if 1 / 8"  x  1 / 4"
if 1 / 4"  x  3 / 8"
if 3 / 8"  x  
(a) Plot the cdf. (Label and scale both axes.)
(b) Write the pmf. Plot the pmf. (Label and scale both axes.)
(c) Determine P(X  1/8").
(d) Determine P(X < -4").
(e) Determine P(X = .2").
(f) Compute E(X).
4. (From Problem 4-41.) Thickness measurements X of a coating process are made to the
nearest hundred of a millimeter. The thickness measurements are uniformly distributed
with values 0.15, 0.16, 0.17, 0.18, and 0.19 mm.
(a) Compute the mean thickness. (State the units.)
(b) Compute the standard deviation of the thickness. (State the units.)
(c) Plot the pmf.
(d) Plot the cdf.
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Homework #5. Due: Wednesday, September 29, 1999.
IE 230
(e) Let Y denote the thickness in inches. Then Y = aX + b, where a and b are
constants. What are the values of a and b?
(f) Compute E(Y).
(g) For general values of a and b, what is the relationship between E(X) and E(Y)?
Quiz Monday, September 27, 1999. The topic is primarily that of Homework #4:
Textbook
Sections 3.6-3.7. You should have memorized the corresponding material from the
pages 1-4 of the concise notes. Closed book and notes.
Exam 1, 7-8:00pm, Wednesday, September 29, 1999. Physics 114. All material through
Homework #4. Bring your student I.D. Closed book and notes.
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