probabilistic machine learning for civil engineers xxv
[ Union operation for events, i.e., “or”
\ Inters ec t ion operation for events, i.e., “and”
X A random vari abl e
X A vector of random variables
f
(
x
)
⌘ f
X
(
x
)
Probability density function of a random variable X
X ⇠ f
(
x
)
X
is dis t ri b ut e d as described by its marginal probability density
function f(x)
X ⇠ f
(
x
)
X
is distri b ut e d as described by its joint probability density
function f(x)
d
! Converges i n dis t ri b ut i on
x, x
i
Realization of a random vari ab l e x : X ⇠ f(x)
F
(
x
)
⌘ F
X
(
x
)
Cumulat i ve distri b ut i on (or mass) function of a random variable
X
(
x
) Cumulative d i st r ib u ti on function of a standard Normal random
variable with mean eq ual to 0 and variance equal to 1
p
(
x
)
⌘ p
X
(
x
)
Probability mass function of a random variable X
X ?? Y The random variables X and Y are statistic all y independent
X ?? Y |z
The random variables
X
and
Y
are conditi onal l y independent
given z
X|y The random variable X is conditi on al ly dependent on y
E
[
X
]
Expectation operation for a random variable X
var
[
X
]
Variance operation for a random variable X
cov
(
X, Y
)
Covariance operation for a pair of random variables X, Y
X
Coeﬃcient of variation of a random variable X
µ
X
The mean of a random variable X
2
X
The variance of a random variable X
⇢ The correlation coeﬃcient
µ
X
The mean values for a vector of rand om variables X
⌃
X
A covariance matrix for a vector of random variables X
R
X
A correlation matrix for a vector of random variables X
D
X
A standard deviation matrix for a vector of random variables X
N
(
x
;
µ,
2
)
The prob abi l i ty density function of a univariate Normal random
variable X, parameterized by its me an and variance
N
(
x
;
µ
X
, ⌃
X
) The j oint probability density function of a multivariate Nor-
mal random variable
X
, parameteri z ed by its mean vect or and
covariance matrix
ln N
(
x
;
, ⇣
) The p rob ab il i ty density function of a log-normal random vari-
able
X
, parameteri z ed by its mean and standard deviation
deﬁned in the log space
B
(
x
;
↵,
) The p rob ab il i ty density function of a Beta random variable
X
,
parameterized by ↵ and
U
(
x
;0
,
1) The u ni f orm probability density function for a r an dom variable
X deﬁned for the interval (0, 1)
(
x
)
The dirac-delta function