18.05
Lecture
8
February
22,
2005
§
3.1
 Random
Variables
and
Distributions
Transforms
the
outcome
of
an
experiment
into
a
number.
De±nitions:
Probability
Space:
(S,
A
,
P
)
S
 sample
space,
A
 events,
P
 probability
Random
variable
is
a
function
on
S
with
values
in
real
numbers,
X:S
R
↔
Examples:
Toss
a
coin
10
times,
Sample
Space
=
{
HTH.
..HT,
....
}
,
all
con±gurations
of
H
T.
Random
Variable
X
=
number
of
heads,
X:
S
R
↔
X:
S
↔ {
0
,
1
,
...,
10
}
for
this
example.
There
are
fewer
outcomes
than
in
S,
you
need
to
give
the
distribution
of
the
random
variable
in
order
to
get
the
entire
picture.
Probabilities
are
therefore
given.
De±nition:
The
distribution
of
a
random
variable
X:S
↔
R
,
is
de±ned
by:
A
√
R
,
P
(
A
)
=
P
(
X
⊂
A
)
=
P
(
s
⊂
S
:
X
(
s
)
⊂
A
)
The
random
variable
maps
outcomes
and
probabilities
to
real
numbers.
This
simpli±es
the
problem,
as
you
only
need
to
de±ne
the
mapped
R
,
P
,
not
the
original
S,
P
.
The
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 Summer '06
 DrStag
 Probability, Probability theory, #, 1 k, 0 K, 10 21 10 2 2 k

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