Random
walks
Def:
Let
{
X
i
;
i
≥
1
}
be
a
sequence
of
IID
rv’s,
and
let
S
n
=
X
1
+
X
2
+
· · ·
+
X
n
for
n
≥
1
.
The
integertime
stochas
tic
process
{
S
n
;
n
≥
1
}
is
called
a
random
walk,
or,
specif
ically,
the
random
walk
based
on
{
X
i
;
i
≥
1
}
.
Our
focus
will
be
on
thresholdcrossing
problems.
For
example,
if
X
is
binary
with
p
X
(1)
=
1
,
p
X
(
−
1)
=
q
= 1
−
p
,
then
°
±
∞
k
p
Pr
{
S
n
≥
k
}
=
²
if
p
≤
1
/
2
.
1
p
n
=1
−
2
6.262:
Discrete
Stochastic
Processes
4/27/11
L21:
Hypothesis
testing
and
Random
Walks
Outline:
•
Random
walks
•
Detection,
decisions,
&
Hypothesis
testing
•
Threshold
tests
and
the
error
curve
•
Thresholds
for
random
walks
and
Cherno
f
1
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View Full DocumentDetection,
decisions,
&
Hypothesis
testing
The
model
here
contains
a
discrete,
usually
binary,
rv
H
called
the
hypothesis
rv.
The
sample
values
of
H
,
say
0
and
1,
are
called
the
alternative
hypotheses
and
have
marginal
probabilities,
called
a
priori
probabilities
p
0
=
Pr
{
H
= 0
}
and
p
1
= Pr
{
H
= 1
}
.
Among
arbitrarily
many
other
rv’s,
there
is
a
sequence
Y
°
(
m
)
= (
Y
1
,Y
2
,...
m
)
of
rv’s
called
the
observation.
We
usually
assume
that
Y
1
2
,
are
IID
conditional
on
H
=
0
and
IID
conditional
on
H
= 1
.
Thus,
if
the
Y
n
are
continuous,
m
f
Y
°
(
m
)
(
°
y
±
) =
f
(
y
n
±

)
.
H

°
Y

H
n
=1

3
Assume
that,
on
the
basis
of
observing
a
sample
value
°
y
of
Y
°
,
we
must
make
a
decision
about
H
,
i.e.,
choose
H
= 0
or
H
= 1
,
i.e.,
detect
whether
not
H
is
1.
Decisions
in
probability
theory,
as
in
real
life,
are
not
necessarily
correct,
so
need
a
criterion
for
making
a
choice.
might
maximize
the
probability
of
choosing
correctly,
for
example,
or,
given
a
cost
for
the
wrong
choice,
might
minimize
the
expected
cost.
Note
that
the
probability
experiment
here
includes
not
only
the
experiment
of
gathering
data
(i.e.,
measuring
the
sample
value
°
y
of
Y
°
)
but
also
the
sample
value
of
the
hypothesis.
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 Fall '11
 staff
 Probability, Probability theory, Stochastic process

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