6.042/18.062J
Mathematics
for
Computer
Science
April
28,
2005
Srini
Devadas
and
Eric
Lehman
Lecture
Notes
Random
Variables
We’ve
used
probablity
to
model
a
variety
of
experiments,
games,
and
tests.
Through
out,
we
have
tried
to
compute
probabilities
of
events
.
We
asked,
for
example,
what
is
the
probability
of
the
event
that
you
win
the
Monty
Hall
game?
What
is
the
probability
of
the
event
that
it
rains,
given
that
the
weatherman
carried
his
umbrella
today?
What
is
the
probability
of
the
event
that
you
have
a
rare
disease,
given
that
you
tested
positive?
But
one
can
ask
more
general
questions
about
an
experiment.
How
hard
will
it
rain?
How
long
will
this
illness
last?
How
much
will
I
lose
playing
6.042
games
all
day?
These
questions
are
fundamentally
different
and
not
easily
phrased
in
terms
of
events.
The
problem
is
that
an
event
either
does
or
does
not
happen:
you
win
or
lose,
it
rains
or
doesn’t,
you’re
sick
or
not.
But
these
new
questions
are
about
matters
of
degree:
how
much,
how
hard,
how
long?
To
approach
these
questions,
we
need
a
new
mathematical
tool.
1
Random
Variables
Let’s
begin
with
an
example.
Consider
the
experiment
of
tossing
three
independent,
un
biased
coins.
Let
C
be
the
number
of
heads
that
appear.
Let
M
= 1
if
the
three
coins
come
up
all
heads
or
all
tails,
and
let
M
= 0
otherwise.
Now
every
outcome
of
the
three
coin
ﬂips
uniquely
determines
the
values
of
C
and
M
.
For
example,
if
we
ﬂip
heads,
tails,
heads,
then
C
= 2
and
M
= 0
.
If
we
ﬂip
tails,
tails,
tails,
then
C
= 0
and
M
= 1
.
In
effect,
C
counts
the
number
of
heads,
and
M
indicates
whether
all
the
coins
match.
Since
each
outcome
uniquely
determines
C
and
M
,
we
can
regard
them
as
functions
mapping
outcomes
to
numbers.
For
this
experiment,
the
sample
space
is:
S
=
{
HHH,
HHT,
HTH, HTT, THH, THT, TTH, TTT
}
Now
C
is
a
function
that
maps
each
outcome
in
the
sample
space
to
a
number
as
follows:
C
(
HHH
)
=
3
C
(
THH
)
=
2
C
(
HHT
)
=
2
C
(
THT
)
=
1
C
(
HTH
)
=
2
C
(
TTH
)
=
1
C
(
HTT
)
=
1
C
(
TTT
)
=
0
Similarly,
M
is
a
function
mapping
each
outcome
another
way:
M
(
HHH
)
=
1
M
(
THH
)
=
0
M
(
HHT
)
=
0
M
(
THT
)
=
0
M
(
HTH
)
=
0
M
(
TTH
)
=
0
M
(
HTT
)
=
0
M
(
TTT
)
=
1
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2
Random
Variables
The
functions
C
and
M
are
examples
of
random
variables
.
In
general,
a
random
variable
is
a
function
whose
domain
is
the
sample
space.
(The
codomain
can
be
anything,
but
we’ll
usually
use
a
subset
of
the
real
numbers.)
Notice
that
the
name
“random
variable”
is
a
misnomer;
random
variables
are
actually
functions!
1.1
Indicator
Random
Variables
An
indicator
random
variable
(or
simply
an
indicator
or
a
Bernoulli
random
variable
)
is
a
random
variable
that
maps
every
outcome
to
either
0
or
1.
The
random
variable
M
is
an
example.
If
all
three
coins
match,
then
M
= 1
;
otherwise,
M
= 0
.
Indicator
random
variables
are
closely
related
to
events.
In
particular,
an
indicator
partitions
the
sample
space
into
those
outcomes
mapped
to
1
and
those
outcomes
mapped
to
0.
For
example,
the
indicator
M
partitions
the
sample
space
into
two
blocks
as
follows:
HHH
��
TTT
HHT
HTH
HTT
��
THH
THT
TTH
�
M
= 1
M
= 0
In
the
same
way,
an
event
partitions
the
sample
space
into
those
outcomes
in
the
event
and
those
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 Spring '11
 Dr.EricLehman
 Computer Science, Probability theory, HHT, T HT

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