Geoscience Reference
In-Depth Information
Marginalization:
p(x)
=
−∞
p(x,y)dy
.
The expectation of a function
f(x)
under the probability distribution
P
for a discrete random vari-
able
x
is
]=
x
p(x)f(x)dx
.
In particular, if
f(x)
=
x
, the expectation is the mean of the random variable
x
.
]=
a
P(a)f (a)
, and for a continuous random variable is
E
P
[
f
E
p
[
f
The variance o
f
x
is
Var
(x)
= E[
(x
− E[
x
]
)
2
]=E[
x
2
2
. The standard deviation of
]−E[
x
]
=
√
Va r
(x)
.
x
is
std(x)
The covariance between two random variables
x,y
is Cov
(x, y)
= E
x,y
[
(x
− E[
x
]
)(y
− E[
y
]
)
]=
E
x,y
[
xy
]−E[
x
]E[
y
]
.
When
x
,
y
are
D
-dimensional vectors,
E[
x
]
is the mean vector with the
i
-th entry being
E[
x
i
]
.Cov
(
x
,
y
)
is the
D
×
D
covariance matrix with the
i, j
-th entry being Cov
(x
i
,y
j
)
.
DISTRIBUTIONS
Uniform
distribution with
K
outcomes (e.g., a fair
K
-sided die):
P(A
=
a
i
)
=
1
/K, i
=
1
,...,K
.
Bernoulli
distribution on binary variable
x
∈{
0
,
1
}
(e.g., a biased coin with head probabil-
μ
x
(
1
μ)
(
1
−
x)
. The mean is
ity
μ
):
P(x
|
μ)
=
−
E[
x
]=
μ
, and the variance is Var
(x)
=
μ(
1
−
μ)
.
Binomial
distribution: the probability of observing
m
heads in
N
trials of a
μ
-biased coin.
N
m
μ
m
(
1
μ)
N
−
m
, with
N
m
N
!
(N
−
m)
!
m
!
P(m
|
N,μ)
=
−
=
.
E[
m
]=
Nμ
,Var
(m)
=
Nμ(
1
−
μ)
.
Multinomial
distribution: for a
K
-sided die with probability vector
μ
=
(μ
1
,...,μ
K
)
, the
probability of observing outcome counts
m
1
,...,m
K
in
N
trials is
P(m
1
,...,m
K
|
μ, N)
=
N
m
1
...m
K
k
=
1
μ
m
k
.
k
Gaussian (Normal)
distributions
univariate:
p(x
√
2
πσ
exp
, with mean
μ
, variance
σ
2
.
(x
−
μ)
2
2
σ
2
1
μ, σ
2
)
|
=
−
exp
−
μ)
, where
x
and
μ
are
D
-
1
(
2
π)
2
1
−
μ)
−
1
(
x
multivariate:
p(
x
|
μ, )
=
−
2
(
x
1
2
|
|
dimensional vectors, and
is a
D
×
D
covariance matrix.
LINEAR ALGEBRA
A scalar is a 1
×
1 matrix, a vector (default column vector) is an
n
×
1 matrix.
