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Homework #10. Due: Wednesday, March 28, 2001.
IE 230
Textbook: D.C. Montgomery and G.C. Runger, Applied Statistics and Probability for
Engineers, John Wiley & Sons, New York, 1999. Chapter 6, Sections 6.1–6.2. Pages
12–13 of the concise notes.
1. (Montgomery and Runger, 1999, page 210)
(a) Problem 6–1. Show that the following function satisfies the properties of a joint
probability mass function.
x
y
f XY (x , y )
iiiiiiiiiiiiiiiii
1.5
1.5
2.5
3
2
3
4
5
1/8
1/4
1/2
1/8
(a) Problem 6–2(a). From Exercise 6–1, determine P(X < 2.5, Y < 3).
(b) Problem 6–2(b). From Exercise 6–1, determine P( X < 2.5 ).
(c) Problem 6–3. From Exercise 6–1, determine E( X ) and E( Y ).
(d) Problem 6–4(a). From Exercise 6–1, determine the marginal pmf f X .
(e) Problem 6–4(b). The conditional distribution of Y given that X = 1.5.
(f) Problem 6–4(c). The conditional distribution of X given that Y = 2.
(g) Problem 6–4(d). Determine E(Y | X = 1.5).
2. (A multinomial problem.) Background. In standard two-parent genetics, each parent
contributes one gene to each off-spring. Genes are dominant or recessive. In eye color,
for example, brown-eye genes dominate green-eye genes. That is, to have green eyes
the off-spring must inherit green-eye genes from both parents. If either gene is browneye, then the off-spring’s eyes will be brown. Each parent contributes one of its two
genes with equal probability, regardless of it being dominant or recessive.
Consider two parents with eye genes ( B , G ) and ( G , G ), where B denotes the brown-eye
gene and G denotes the green-eye gene.
(a) What is the probability of a particular off-spring being ( B , B )? ( G , G )? ( B , G )?
(( B , G ) is equivalent to ( G , B ).)
(b) What is the probability of a particular off-spring having brown eyes? Green eyes?
(c) Suppose that the parents have 4 off-spring? Determine the probability that two are
( B , B ), one is ( G , G ) and one is ( B , G ).
(d) Suppose that the parents have 4 off-spring? Determine the probability that three
have brown eyes and one has green eyes.
(Notice the point being made about the relationship between the multinomial distribution
with three outcomes and the binomial distribution (with two outcomes).)
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Homework #10. Due: Wednesday, March 28, 2001.
IE 230
3. Computing with very small and very large numbers. (Do not submit an MSExcel spread
sheet.) First read the handout (on the web page) about computing probabilities using the
gamma function and logarithms.
(a) Determine the largest value of x so that f act (x ) does not overflow. Write the
command, the argument value, and the function value.
(b) Write the MSExcel cell command to evaluate 4! using the gammaln function.
(c) Using "exp", "ln", and "gammaln" functions, write the MSExcel cell command to
evaluate the gamma pdf f X ( 80.2 ) for r = 44.3 and λ = 0.4. Write its value.
(d) Write MSExcel code and value for the probability in 2(c). (You might want to check
your answer with the "gammadist(80.2,44.3,2.5,0)".)
______________________________
Quiz Monday, March 26, 2001. The topic is that of textbook sections 5.6 and 5.8–5.11. You
should have memorized the corresponding material from pages 10 and 11 of the concise
notes, as well as the names and ranges of the named distributions on page 12.
Exam #3, Wednesday, April 11, 2001. 7:00–8:00pm, PHYS 114. Closed book and notes.
No calculator. Cumulative through textbook Section 6.4, which is through Homework
#11. Emphasis will be on material since Exam #2, which covered through textbook
Section 5.5.
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Schmeiser