SA421 – Simulation Modeling
Asst. Prof. Nelson Uhan
●
A random variable is continuous if it can take on a continuum of values
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Let X be a continuous random variable
●
The cumulative distribution function (cdf)
F
X of
X is
F
X ( a
) =
Pr
{
X
≤ a
}
●
The probability density function (pdf) p
X of X is
Fall 2014
●
Another way the pdf and cdf of a continuous random variable are related:
●
Let
Y
0
, . . . ,
Y n
− 1 be n independent and identically distributed (iid) random variables with cdf
F
Y
●
Let y 0
, . . . , y n
− 1 be observations of
Y 0
, . . . ,
Y n
− 1
●
In words, F
Y ( a
) =
Pr
{
Y
≤ a
} ≈
●
The empirical cdf is
○
Note that
F e ( a
) is a random variable for any fixed value of a
●
The observed empirical cdf is
1

Example 1.
Let n
=
4. Suppose the observations of
Y
0
F e
.
,
Y
1
,
Y
2
,
Y
3 are y
0 =
3, y
1 =
1, y
2 =
8, y
3 =
4. Plot the
F e ( y
)
1
2 4 6 8 10 y
●
Let y ( 0 )
, y ( 1 )
, . . . , y ( n
− 1 ) be the observations y
0
, . . . , y n
− 1 sorted from smallest to largest
⇒
F e ( y ( i
) ) = for i
=
0, 1, . . . , n
−
1.
●
Let Y
0
, . . . , Y n
− 1 be n iid continuous random variables
●
Let y
0
, . . . , y n
− 1 be observations of
Y
0
, . . . ,
Y n
− 1
●
Let
X be the proposed continuous random variable with cdf
F
X
●
The Kolmogorov-Smirnov (K-S) goodness-of-fit test compares the empirical cdf of the cdf of the proposed random variable
X
Y j
’s with the
●
Question: Do the
Y j
’s share the same distribution as
X
?
●
Null hypothesis
H 0
: for any
Y j
,
●
The test statistic is
●
The observed test statistic is
2

●
The p
-value is Pr
{
D
≥ d
}
○
○
○
○
√ nD follows a Kolmogorov distribution
Important caveat:
√ nD does not follow a Kolmogorov distribution if the proposed distribution of
X depends on estimates based on the observations y
0
, . . . , y n
− 1 e.g. if you propose
X as an exponential random variable, but guess the mean based on y 0
, . . . , y n
− 1
There are ways around this, some quick-and-dirty, some more rigorous
●
How do we compute d
? Do we really need to consider all values of x
?
ˆ e ( y
)
1 y y
0 y
1 y
2
●
So, we can compute the observed test statistic as y
3
3