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Unformatted text preview: (Ø
)
ØS
Ø
8bE [X ]
Ø ˘n
Ø
Pr Ø
− E [X ]Ø ≥ ≤ ≤
for large enough b.
Øn
Ø
n≤2 56 CHAPTER 1. INTRODUCTION AND REVIEW OF PROBABILITY Exercise §
1.27. Let {Xi ; i ≥ 1} be IID rv’s with mean 0 and inﬁnite variance. Assume that
£
E Xi 1+h = β for some given h, 0 < h < 1 and some given β . Let Sn = X1 + · · · + Xn .
a) Show that Pr {Xi  ≥ y } ≤ β y −1−h b : Xi ≥ b
X : −b ≤ Xi ≤ b
i
−b : Xi ≤ −b
hi
£ § R1
1−h
˘
Show that E X 2 ≤ 2β b h Hint: For a nonnegative rv Z , E X 2 = 0 2 z Pr {Z ≥ z } dz
1−
(you can establish this, if you wish, by integration by parts).
n
o
˘
˘
˘
˘
c) Let Sn = X1 + . . . + Xn . Show that Pr Sn 6= Sn ≤ nβ b−1−h
h 1−h
i
©Ø Ø
™
n
d Show that Pr Ø Sn Ø ≥ ≤ ≤ β (12bh)n≤2 + b1+h .
n
−
˘
˘
b) Let {Xi ; i ≥ 1} be truncated variables Xi = e) Optimize your bound with respect to b. How fast does this optimized bound approach
0 with increasing n?
Exercise 1.28. (MS convergence → convergence in probability) Assume that {§ n ; n ≥
Y
£
2 = 0.
1} is a sequence of rv’s and z is a number with the property that limn→1 E (Yn − z )
a) Let ε > 0 be arbitrary and show that for each n ≥ 0,
£
§
E (Yn − z )2
Pr {Yn − z  ≥ ε} ≤
ε2 b) For the § above, let δ > 0 be arbitrary. Show that there is an integer m such that
ε
£
E (Yn − z )2 ≤ ε2 δ for all n ≥ m.
c) Show that this implies convergence in probability. Exercise 1.29. Let X1 , X2 . . . , be a sequence of IID rv’s each with mean 0 and variance
√
√
σ 2 . Let Sn = X1 + · · · + Xn for all n and consider the random variable Sn /σ n − S2n /σ 2n.
Find the limiting distribution function for this rv as n → 1. The point of this exercise is
√
to see clearly that the distribution function of Sn /σ n is converging but that the sequence
of rv’s is not converging.
Exercise 1.30. A town starts a mosquito control program and the rv Zn is the number
of mosquitos at the end of the nth year (n = 0, 1, 2, . . . ). Let Xn be the growth rate of
mosquitos in year n; i.e., Zn = Xn Zn−1 ; n ≥ 1. Assume that {Xn ; n ≥ 1} is a sequence of
IID rv’s with the PMF Pr {X =2} = 1/2; Pr {X =1/2} = 1/4; Pr {X =1/4} = 1/4. Suppose
that Z0 , the initial number of mosquitos, is some known constant and assume for simplicity
and consistency that Zn can take on noninteger values.
a) Find E [Zn ] as a function of n and ﬁnd limn→1 E [Zn ].
b) Let Wn = log2 Xn . Find E [Wn ] and E [log2 (Zn /Z0 )] as a function of n. 1.8. EXERCISES 57 c) There is a constant α such that limn→1 (1/n)[log2 (Zn /Z0 )] = α with probability 1. Find
α and explain how this follows from the strong law of large numbers.
d) Using (c), show that limn→1 Zn = β with probability 1 for some β and evaluate β .
e) Explain carefully how the result in (a) and the result in (d) are possible. What you
should learn from this problem is that the expected value of the log of a product of IID rv’s
is more signiﬁcant that the expected value of the product itself.
Exercise 1.31. Use Figure 1.7 to verify (1.43). Hint: Show that y Pr {Y ≥y } ≥
R
and show that limy→1 z≥y z dFY (z ) = 0 if E [Y ] is ﬁnite. R z ≥y z dFY (z ) Q
Exercise 1.32. Show that m≥n (1 − 1/m) = 0. Hint: Show that
µ
∂
µµ
∂∂
µ
∂
1
1
1
1−
= exp ln 1 −
≤ exp −
.
m
m
m Exercise 1.33. Let A1 , A2 , . . . , denote an arbitrary set o events. This exercise shows that
of
nS
o
nT S
limn Pr
m≥n Am = 0 if and only if Pr
n m≥n Am = 0.
nS
o
nT S
o
a) Show that limn Pr
= 0 implies that Pr
= 0. Hint: First
m≥n Am
n m≥n Am
show that for every positive integer n,
n\ [
o
n[
o
Pr
Am ≤ Pr
Am .
n m≥n m≥n nT S
o
nS
o
b) Show that Pr
Am = 0 implies that limn Pr
Am = 0. Hint: First
n m≥n
m≥n
show that each of the following hold for every positive integer n,
n[
o
n\
o
[
Pr
Am = Pr
Am .
m≥n lim Pr n→1 n[ m≥n k≤n o
n\
Am ≤ Pr k≤n m≥k [ m≥k o
Am . Exercise 1.34. Assume that X § a zeromean rv with ﬁnite second and fourth moments,
£§
£ is
i.e., E X 4 = ∞ < 1 and E X 2 = σ 2 < 1. Let X1 , X2 , . . . , be a sequence of IID rv’s,
each with the distribution of X . Let Sn = X1 + · · · + Xn .
£ 4§
a) Show that E Sn = n∞ + 3n(n − 1)σ 4 .
4 n n−
b) Show that Pr {Sn /n ≥ ≤} ≤ n∞ +3≤4(n4 1)σ .
P
c) Show that n Pr {Sn /n ≥ ≤} < 1. Note that you have shown that Lemma 1.1 in the
proof of the strong law of large numbers holds if X has a fourth moment. d) Show that a ﬁnite fourth moment implies a ﬁnite second moment. Hint: Let Y = X 2
and Y 2 = X 4 and show that if a nonnegative rv has a second moment, it also has a mean.
e) Modify the above parts for the case in which X has a nonzero mean. Chapter 2 POISSON PROCESSES
2.1 Introduction A Poisson process is a simple and widely used stochastic process for modeling the times
at which arrivals enter a system. It is in many ways the continuoustime version of the
Bernoulli process that was described brieﬂy in Subsection 1.3.5.
Recall that...
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 Spring '09
 R.Srikant

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