Mathematics Formula Sheet Page 9
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10Expectation algebra
For independent random variables X and Y
2
2
XY =
±
=
+
E(
)
E(
X
)
E(
Y
)
,
Var(
aX
bY
)
a
Var(
X
)
b
Var(
Y
)
Sampling distributions
For a random sample
X
,
X
,
,
X
of n independent observations from a distribution having
K
1
2
n
2
μ
σ
mean
and variance
2
σ
μ
=
X is an unbiased estimator of
, with
Var(
X
)
n
2
Σ
−
(
X
X
)
2
2
2
σ
i
=
S is an unbiased estimator of
, where
S
−
n
1
μ
σ
2
For a random sample of n observations from
N(
,
)
μ
−
X
~
N(
, 0
) 1
σ
/ n
μ
−
X
~
t
−
(
n
) 1
S
/
n
If X is the observed number of successes in n independent Bernoulli trials in each of which the
X
Y =
probability of success is p , and
, then
n
−
p
1 (
p
)
Y =
=
E(
)
p
and
Var(
Y
)
n
μ
σ
2
For a random sample of
n observations from
N(
,
)
and, independently, a random sample
x
x
x
2
μ
σ
of
n observations from
N(
,
)
y
y
y
μ
μ
−
−
−
(
X
Y
)
(
)
x
y
~
N(
, 0
) 1
2
σ
2
σ
y
x
+
n
n
x
y
9
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