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Level
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Code
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Display
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Definition
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1
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B
|
beta
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The
beta-distribution
is
used
for
data
that
is
bounded
on
both
sides
and
may
or
may
not
be
skewed
(e.g.,
occurs
when
probabilities
are
estimated.)
Two
parameters
a
and
b
are
available
to
adjust
the
curve.
The
mean
m
and
variance
s2
relate
as
follows:
m
=
a/
(a
+
b)
and
s2
=
ab/((a
+
b)2
(a
+
b
+
1)).
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1
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E
|
exponential
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Used
for
data
that
describes
extinction.
The
exponential
distribution
is
a
special
form
of
g-distribution
where
a
=
1,
hence,
the
relationship
to
mean
m
and
variance
s2
are
m
=
b
and
s2
=
b2.
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1
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F
|
F
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Used
to
describe
the
quotient
of
two
c2
random
variables.
The
F-distribution
has
two
parameters
n1
and
n2,
which
are
the
numbers
of
degrees
of
freedom
of
the
numerator
and
denominator
variable
respectively.
The
relationship
to
mean
m
and
variance
s2
are:
m
=
n2
/
(n2
-
2)
and
s2
=
(2
n2
(n2
+
n1
-
2))
/
(n1
(n2
-
2)2
(n2
-
4)).
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1
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G
|
(gamma)
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The
gamma-distribution
used
for
data
that
is
skewed
and
bounded
to
the
right,
i.e.
where
the
maximum
of
the
distribution
curve
is
located
near
the
origin.
The
g-distribution
has
a
two
parameters
a
and
b.
The
relationship
to
mean
m
and
variance
s2
is
m
=
a
b
and
s2
=
a
b2.
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1
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LN
|
log-normal
|
The
logarithmic
normal
distribution
is
used
to
transform
skewed
random
variable
X
into
a
normally
distributed
random
variable
U
=
log
X.
The
log-normal
distribution
can
be
specified
with
the
properties
mean
m
and
standard
deviation
s.
Note
however
that
mean
m
and
standard
deviation
s
are
the
parameters
of
the
raw
value
distribution,
not
the
transformed
parameters
of
the
lognormal
distribution
that
are
conventionally
referred
to
by
the
same
letters.
Those
log-normal
parameters
mlog
and
slog
relate
to
the
mean
m
and
standard
deviation
s
of
the
data
value
through
slog2
=
log
(s2/m2
+
1)
and
mlog
=
log
m
-
slog2/2.
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1
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N
|
normal
(Gaussian)
|
This
is
the
well-known
bell-shaped
normal
distribution.
Because
of
the
central
limit
theorem,
the
normal
distribution
is
the
distribution
of
choice
for
an
unbounded
random
variable
that
is
an
outcome
of
a
combination
of
many
stochastic
processes.
Even
for
values
bounded
on
a
single
side
(i.e.
greater
than
0)
the
normal
distribution
may
be
accurate
enough
if
the
mean
is
"far
away"
from
the
bound
of
the
scale
measured
in
terms
of
standard
deviations.
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1
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T
|
T
|
Used
to
describe
the
quotient
of
a
normal
random
variable
and
the
square
root
of
a
c2
random
variable.
The
t-distribution
has
one
parameter
n,
the
number
of
degrees
of
freedom.
The
relationship
to
mean
m
and
variance
s2
are:
m
=
0
and
s2
=
n
/
(n
-
2)
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1
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U
|
uniform
|
The
uniform
distribution
assigns
a
constant
probability
over
the
entire
interval
of
possible
outcomes,
while
all
outcomes
outside
this
interval
are
assumed
to
have
zero
probability.
The
width
of
this
interval
is
2s
sqrt(3).
Thus,
the
uniform
distribution
assigns
the
probability
densities
f(x)
=
sqrt(2
s
sqrt(3))
to
values
m
-
s
sqrt(3)
>=
x
<=
m
+
s
sqrt(3)
and
f(x)
=
0
otherwise.
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1
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X2
|
chi
square
|
Used
to
describe
the
sum
of
squares
of
random
variables
which
occurs
when
a
variance
is
estimated
(rather
than
presumed)
from
the
sample.
The
only
parameter
of
the
c2-distribution
is
n,
so
called
the
number
of
degrees
of
freedom
(which
is
the
number
of
independent
parts
in
the
sum).
The
c2-distribution
is
a
special
type
of
g-distribution
with
parameter
a
=
n
/2
and
b
=
2.
Hence,
m
=
n
and
s2
=
2
n.
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