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Unformatted text preview: Exercises ' .. ._._._ 21 NeymanPearson approach to decision making (hypothesis testing) when prior probabilities are
absent. The latter, in Section 134, could be omitted. Chapter 14 formally treats independence.
It is possible to omit the initial discussion of independent events and start with Section 14.4
on independent random variables. One could then reconstruct the story of independent events by
replacingthem by their indicator function random variables. Chapter 15 on conditional expectation
is essential, although Section 15.6 on Kalman ﬁltering can be easily omitted. All of Chapter 15
should be covered with the exception of Section 15.10.. If time permits, elements of Chapter
16 on Bayesian inference are well worth presenting as important applications of conditional
expectation. I believe that characteristic functions, as discussed in Chapter 17, provide an essential
alternative deseription of probability that enables us to solve many problems that would otherwise
be intractable. Chapter 18 on probability bounds has material of value in practice, but it can be
omitted with the eitception of Section 18.2 on Chebychev bounds. Chapters 19 and 20 really
belong in a second course on random processes. They were included here to accommodate those who think otherwise. EXERCISES E0.1 Evaluate the following difference of two sums: 100 _ mo
6 = glazed Z haze—m.
k=0  m=l E0.2 Prove by induction on n that
I ll
. . 2 1
’ 2k = (2n+1)(n+ 1)n.
i=0 6 E0..3 Evaluate the following inﬁnite sums: a. I 22 _ Chapter 0 README: Learning and Teaching Probabiﬁjtjiﬁeasorﬂig .I I l 5 '5‘ 1
bz (72—3113xm +13"
6" a . 2 1_
. “3222171”
' n=lm=0 E05 Evaluate the dot product it ~ y for
' " ' x7:(—1 o 1),y7=_(1'1 ~1‘).. . Are x and y orthbgo’nal? . .
E0.6 For each of the following equations, determine whether the calculation 7 4 —1
y=AxforA= 0 2 1 —1 —2 3 can be carried out, and if so, carry it out: a.
I x = (1 1 6)
b..
1
x = l
6
c” .
l
x __ l
_ 0
_ 6 '
E007 For the matrix A of the preceding problem, evaluate the malﬁx
B = 11111.. E0..8 Evaluate the Quadratic foim q: "7fo for 0 312
x=2,A\.=012.‘
71 114 E03 Complete the square in the follOwing expression:
' ‘ 3 1 2
x7 1 1 3 x+(1 0 2)x+5..
2 3 4 _ 120,10 If‘x > 0, how does
ex
'1 + x + éxz compare to l? Exercises ' 'E0..11 Can you ﬁnd x > 0 such that
1+x,+.i:2 > e"? £0.12 Without using a calculator, arrange a, b, and c in nondecreasing order: 33 65 b 20 80 22 Ia=e e, =e e , c=d5whered=e .. E0..13 By using the power series for (3“, show that, for positive 1:,
er  I > i i
x x 2 £10.14 If the function ¢(x) has derivative
d<I>(x) 1 _ = —e ’
dx V21:
then evaluate the derivativelf" of
ﬁx) a “Ks/J?) _ 120.15 Evaluate dg/dx. given that _ . x2
gcx) = f [logos +y)12dy_»
I
E0.16 Show that the deﬁnite integral
[Ar (2:, )0 dx dy of the function if (x; y) = ex”
over the region '  A: {(x,y):0$x 5y $1}
is
$9 —1)2‘. E017 Using integration by parts, show that b
j JezeJr dx = (a2 + 2a ,+ 2):?“  (19+ 21; + 2)e"'b.,
a. £0.18 Using integration byparts, show that if
(X)
1k =f xke""dx,
0 . then
I}; = Hk_.1, for k 2 1.,
Conclude from this that I}; is k! (k factorial). 23 24 Chapter 0 README: Learning and Teaching Probabilistic Reason_i!1_g __._..___.___.____mmmm £0.19 Without carrying out the indicated calculations, provide reasons why the suggested an
swers are incorrect: a‘. HUIUJN Jnt‘Uht HIPNU) 1
4
3
2 Loweu: for (cos(x))4dic = %(sin(x))5 ' 100 100 2 292 “m < 0 n=0 m=0 20 16 17 18
,AJB= 29 35 35 34 41. 39 38 38 WHOINN
NWNWLR 1
l
,13=1
2
l ...
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 Spring '05
 HAAS
 Probability theory, independent random variables, function random variables, Teaching Probabilistic Reason_i

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