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§
2 Review of some probability results
§
2.1 Statistics and sampling distributions
2.1.1
Samples are subject to variation. Repetition of the same observation process may give rise to
diﬀerent samples, even though they are all drawn from the same
population/distribution:
Example
§
2.1.1
•
if you toss a coin 30 times everyday, you may not get the same number of heads everyday;
•
if you select 30 persons randomly in the streets everyday and ask for their annual incomes,
you may not get the same 30 answers everyday;
•
if you generate 30 random exp(1) numbers from your PC everyday, you may not get the
same 30 numbers everyday.
In each of the above examples, the sample (of size 30) is subject to variation because of the
random nature of our observation process.
[ In the rest of the course, a sample is taken to be either a collection of random variables or their realisations,
depending on the context. ]
2.1.2
Deﬁnitions.
A
statistic
is a function of the sample, and is also subject to variation.
In other words, a statistic is itself a random variable and so has a distribution. We call its
distribution the
sampling distribution
.
2.1.3
Many statistics represent properties of a sample which have a population counterpart.
Examples of population properties (of cdf
F
) vs sample properties (calculated from
X
1
,...,X
n
):
Properties
Population
Sample
Mean
μ
1
=
E
F
[
X
] =
R
xdF
(
x
)
¯
X
=
n

1
∑
n
i
=1
X
i
r
th moment
μ
r
=
E
F
[
X
r
] =
R
x
r
dF
(
x
)
n

1
∑
n
i
=1
X
r
i
r
th central moment
E
F
{
(
X

μ
1
)
r
}
=
R
(
x

μ
1
)
r
dF
(
x
)
n

1
∑
n
i
=1
(
X
i

¯
X
)
r
Median
F

1
(1
/
2)
X
((
n
+1)
/
2)
if
n
odd,
{
X
(
n/
2)
+
X
(
n/
2+1)
}
/
2 if
n
even
p
quantile
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 Spring '10
 SMSLee
 Statistics, Probability

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