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p. 51
Continuous Random Variables
• Recall
: For
discrete
random variables, only a
finite
or
countably
infinite
number of possible values with positive probability.
Often, there is interest in random variables that can take (at least
theoretically) on an
uncountable
number of possible values, e.g.,
the weight of a randomly selected person in a population,
the length of time that a randomly selected light bulb works,
the error in experimentally measuring the speed of light.
Example (Uniform Spinner, LNp.214):
= (
,
]
For (
a
,
b
]
⊂
,
P
((
a
,
b
]) =
b
a
/(2
)
Consider the random variables:
X
:
→
R
,
and
X
(
) =
for
∈
Y
:
→
R
,
and
Y
(
) =
tan
(
) for
∈
Then,
X
and
Y
are random variables that takes on an
uncountable number of possible values.
p. 52
• Probability Density Function and Continuous Random Variable
Definition. A function
f
:
R
→
R
is called a probability density
function (pdf) if
1.
f
(
x
)
≥
0, for all
x
∈
(
∞
,
∞
), and
2.
∞
−∞
f
(
x
)
dx
=1
.
Definition: A random variable
X
is called
continuous
if there
exists a pdf
f
such that for any set
B
of real numbers
P
X
({
X
∈
B
}) =
∫
B
f
(
x
)
dx
.
For example,
Notice that:
P
X
({
X
=
x
})=0, for any
x
∈
R
,
But, for
≤
a
<
b
≤
,
P
X
({
X
∈
(
a
,
b
]})=
P
((
a
,
b
]) =
b
a
/(2
) > 0.
Q
: Can we still define a probability mass function for
X
? If
not, what can play a
similar
role like pmf for
X
?
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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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