14 / 68
Mean and Variance of Random Variable
X
•
Discrete Probability Distribution:
Mean
, µ
X
=
x
∈
S
x
·
f
(
x
)
Variance
, σ
2
X
=
x
∈
S
x
2
·
f
(
x
)
−
[
µ
X
]
2
where
f
(
x
)
is the value of its probability distribution at
x
.
•
Continuous Probability Distribution:
Mean
, µ
X
=
∞
−∞
x
·
f
(
x
)
dx
Variance
, σ
2
X
=
∞
−∞
x
2
·
f
(
x
)
dx
−
[
µ
X
]
2
where
f
(
x
)
is the value of its probability at
x
.
Standardizing Normally Distributed Random Variables
X
and
¯
X
•
If
X
∼
N
(
µ, σ
2
)
, then
Z
=
X
−
µ
σ
•
If
¯
X
∼
N
(
µ,
σ
2
n
)
, then
Z
=
¯
X
−
µ
σ
√
n
where
Z
∼
N
(0
,
1)
.
4
Statistical Tables and Formulae 2.0