NTHU MATH 2820, 2008, Lecture Notes
Ch1~6, p.9
Chapters 2 and 3
Definition 2.1
(random variable, TBp. 33)
•
random variable
¾
conditional distribution
¾
independent random variables
¾
function of random variables
• distribution of transformed r.v.
• extrema and order statistics
¾
random variables
¾
distribution
•discrete and continuous
•univariate and multivariate
•cdf, pmf, pdf
Outline
A
random variable
is a function from
Ω
to the real numbers.
Ω
R
R
R
L
L
•
•
•
•
•
•
•
made by ShaoWei Cheng (NTHU, Taiwan)
Ch1~6, p.10
Example 2.1
(cont. Ex. 1.1)
Question 2.1
Why statisticians need random variables? Why they map to real line?
(1)
X
1
= the total number of heads
(2)
X
2
= the number of heads on the
f
rst toss
(3)
X
3
= the number of heads minus the number of tails
Ω
=
{
hhh,hht,hth,thh,htt, tht,tth,ttt
}
1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8
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Ch1~6, p.11
• distribution
Question 2.2
A random variable have a sample space on
real line
. Does it bring
some special ways to describe its probability measure?
• joint pdf
• joint cdf
• joint mgf/cf
• joint pmf
• joint cdf
• joint mgf/cf
multi
variate
r.v.’s
•pdf
•cdf
•mgf/cf
•pmf
•cdf
•mgf/cf
uni
variate
r.v.
continuous
discrete
mgf (moment generating function) and cf (characteristic function) will be
defined in Chapter 4
pmf: probability mass function, pdf: probability density function,
cdf: cumulative distribution function
made by ShaoWei Cheng (NTHU, Taiwan)
Ch1~6, p.12
Definition 2.2
(discrete and continuous random variables, TBp. 35 and 47)
Definition 2.3
(cumulative distribution function, TBp. 36)
A
discrete random variable
cantakeonon
lya
f
nite or at
most a countably in
f
nite number of values. A
continuous ran
dom variable
can take on a continuum of values.
Afunct
ion
F
is called the
cumulative distribution function
(cdf)
of a random variable
X
if
F
(
x
)=
P
(
X
≤
x
)
,x
∈
R
.
R
Q
: Does data following a
continuous distribution
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 Spring '11
 lisa
 Probability, Probability theory, Cumulative distribution function, Xn, ShaoWei Cheng

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