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EE 464, Mitra
Spring 2009
Homework 7
Due Friday, March 13, 2009, 9am 464 Box
This problem set investigates tail inequalities, transforms and preliminaries for bivariate random variables.
1. Very Short Problems on Tail Inequalities:
(a) Prove the generalization of the Chebyshev inequality:
P
[

X

c
 ≥
a
]
≤
E
[

X

c

n
]
a
n
Note that
c
is an
arbitrary
real number,
a >
0
and
n
is a positive integer.
(b) Consider a positive random variable
X
with mean
a
, show that
P
£
X
≥
√
a
/
≤
√
a
(c) Consider the various bounds we have considered so far. If we simultaneously consider the
moments in the following set,
'
E
[
X
]
,
E
£
X
2
/
,
E
£
X
3
/
,
E
£
X
4
/“
, can we come up with a
tighter bound?
2. Compare the Markov, Chebyshev and Chernoﬀ bounds for
P
[
X
≥
a
]
for a uniform random variable
on
[0
,b
]
and Bernoulli random variable with parameter
p
. Plot and compare the bounds for diﬀerent
values of
a,b
and
p
.
3. For a Laplacian random variable with parameter
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