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p. 41
Random Variables
• A Motivating Example
Experiment: Sample
k
students without replacement from the
population of all
n
students (labeled as 1, 2, …,
n
, respectively)
in our class.
= {all combinations} = {{
i
1
, …,
i
k
}: 1
≤
i
1
<
<
i
k
≤
n
}
A probability measure
P
can be defined on
, e.g, when there is
an equally likely chance of being chosen for each students,
For an outcome
∈
, the experimenter may be more interested
in some quantitative attributes of
, rather than the
itself, e.g.,
The average weight of the
k
sampled students
The maximum of their midterm scores
The number of male students in the sample
Q
: What mathematical structure would be useful to characterize
the
random
quantitative attributes of
’s?
p. 42
• Definition: A
random variable
X
is a function which maps the
sample space
to the real numbers
R
, i.e.,
X
:
→
R
.
The
P
defined on
would be transformed into a
new
probability measure defined on
R
through the mapping
X
the outcome of
X
is random, but the map
X
is deterministic
Example (Coin Tossing): Toss a fair coin 3 times, and let
X
1
= the total number of heads
X
2
= the number of heads on the first toss
X
3
= the number of heads minus the number of tails
={
hhh
,
hht
,
hth
,
thh
,
htt
,
tht
,
tth
,
ttt
}
Ω
R
R
R
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This note was uploaded on 02/15/2012 for the course MATH 2810 taught by Professor Shaoweicheng during the Fall '11 term at National Tsing Hua University, China.
 Fall '11
 ShaoWeiCheng
 Probability

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