Description
Exercise 5.1. There are five distinct paths that lead from the base of a mountain to the
top, and each of these five paths can also lead from the top of the mountain to the bottom.
A “round trip” up and down the mountain is a pair of path numbers (p1, p2), where p1 is
the number of the path taken from the bottom to the top, and p2 is the number of the path
taken from the top back down to the bottom.
The question is: how many round trips are there? Here are four answers, all correct
under certain circumstances.
1. 20 = 5 × 4
2. 25 = 52
3.
5
2
= 10
4.
6
2
= 15.
For each of these four cases, explain why the given number is the correct answer under
specific assumptions about acceptable round trips, state those assumptions (in words), and
give the complete list of pairs that constitute acceptable round trips.
Exercise 5.2. Consider a family with two children who have different ages. Assume that,
in any birth of one baby, the probability is 1
2
that the baby is a boy, and 1
2
that the baby is
a girl. Let A be the event that the older of the two children is a girl, and let B be the event
that at least one of the two children is a girl.
1. What is the conditional probability that both children are girls, given that the older
child is a girl? Explain how you derived your answer.
2. What is the conditional probability that both children are girls, given that at least one
of the children is a girl? Explain how you derived your answer.
Exercise 5.3. Assume that you have a farm whose main crop is carrots. Over the very
long term, 14% of the carrots harvested are unacceptable because they are too sour. (They
should be sweet.) Consider a sample of 11 carrots, harvested from your farm at random.
In each answer to the following questions, please give the formula you used as well as the
numerical value. (Suggestion: use Matlab.)
2
(a) What is the probability, shown to at least four decimal figures, that all of the carrots
in the sample will be sweet?
(b) What is the probability that 1 or more carrots in the sample are sour?
(c) What is the probability that at most 1 carrot in the sample is sour?
Exercise 5.4. Write a program that generates samples consisting of n components, where
each component is either 1 (“success”) or 0 (“failure”). Suppose that “success” means that
a coin toss has produced heads, where the coin being used is a biased coin in which the
probability of tossing heads is p such that p > 1
2
.
In each sample, the n components should be defined using a random number generator
and the value of p, as described next. To obtain the value of any one component x knowing
p, use the Matlab uniform random number generator rand, which produces random numbers
uniformly distributed in (0, 1):
z = rand
if z <= p,
x = 1
else
x = 0
end
Assume that p = 0.75. Your program should produce and print 8 samples, each containing
12 components. For each sample, count (i) the number of successes and (ii) the ratio of the
number of successes to n. Comment on how closely the computed fraction of successes
corresponds to p.
Exercise 5.5. Let e denote the m-vector whose components are all equal to 1, and let b
denote a specific m-vector, so that
e =
1
1
.
.
.
1
and b =
b1
b2
.
.
.
bm
.
Suppose that we want to find one number, denoted by β, that minimizes various norms
of the residual vector r whose i-th component is bi − β:
r = b − βe =
b1 − β
b2 − β
.
.
.
bm − β
,
3
where
krk1 =
Xm
i=1
|ri
|, krk
2
2 =
Xm
i=1
r
2
i
, and krk∞ = max
i
|ri
|.
(Note that the middle formula gives the squared two-norm.)
(a) Let β2 be the scalar that produces the smallest value of the squared two-norm krk
2
2 =
kb − βek
2
2
. Give a mathematical expression for β2 in terms of the elements of b. (Hint:
use the normal equations.)
(b) Let β∞ be the scalar that produces the smallest value of the infinity norm of the
residual, krk∞ = maxi
|ri
|. Express β∞ in terms of the elements of b and explain why
the infinity norm is minimized for this value.
(c) [Optional.] For the one-norm, the scalar β1 that minimizes kb − βek1 is the median of
b, i.e., the value separating the values {bi} into a higher half and a lower half. (Consider
a set of ordered real scalars {yi}, i = 1, . . . , with yi ≤ yi+1. If there are 2k + 1 values,
the median is yk+1; if there are 2k values, the median is 1
2
(yk + yk+1). The Matlab
command median(y) returns the median when y is a vector.) This result (c) about
the one-norm is not easy to prove, but please give a proof if you feel so inclined. In
any case, you should use this result in part (d).
(d) Let m = 7. For each of the following vectors b, calculate β1, β2, and β∞ (suggestion:
use Matlab), based on the results of (a)–(c).
(i) b = (−150, 25, 0, −70, 70, 150, −25)T
;
(ii) b = (1, 500, 1, 250, 1, 300, 1)T
;
(iii) b = (1, −2, −10, 4, 7, −5, 100000)T
.
For each of these three cases, print the vector b, the three values β1, β2, and β∞ for
that b, and the three residual vectors and their norms:
• for case (i), print r1 = b − β1e and kr1k1;
• for case (ii), print r2 = b − β2e and kr2k2; and
• for case (iii), print r∞ = b − β∞e and kr∞k∞.
In each case, comment on how well (or how badly) the single value β “represents” the
nature of the original vector.
(e) Which norm(s) appear to be least sensitive to “outliers” (a small number of elements
of b that differ widely in magnitude from the other elements)? Explain your answer.
