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Example.
In
n
trials, let
X
be the number of successes.
A discrete random variable is one that can only take countably many values. For a discrete random
variable, we deﬁne the probability mass function or the density by
p
(
x
) =
P
(
X
=
x
). Here
P
(
X
=
x
) is an
abbreviation for
P
(
{
ω
∈
S
:
X
(
ω
) =
x
}
). This type of abbreviation is standard. Note
∑
i
p
(
x
i
) = 1 since
X
must equal something.
Let
X
be the number showing if we roll a die. The expected number to show up on a roll of a die
should be 1
·
P
(
X
= 1) + 2
·
P
(
X
= 2) +
· · ·
+ 6
·
P
(
X
= 6) = 3
.
5. More generally, we deﬁne
E
X
=
X
{
x
:
p
(
x
)
>
0
}
xp
(
x
)
to be the expected value or expectation or mean of
X
.
Example.
If we toss a coin and
X
is 1 if we have heads and 0 if we have tails, what is the expectation of
X
?
Answer.
p
X
(
x
) =
±
1
2
x
= 1
1
2
x
= 0
0
all other values of
x
.
Hence
E
X
= (1)(
1
2
) + (0)(
1
2
) =
1
2
.
Example.
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 Fall '06
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