Cheat Sheet For Probability Theory Page 6
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66
Ingmar Land, October 8, 2005
Some authors use the term “normal distribution” equivalently with “Gaussian dis-
tribution”. So, use the term “normal distribution” with caution.
The integral of a normal pdf cannot be solved in closed form and is therefore often
expressed using the Q-function
∞
1
x 2
−
√
Q(z) :=
d x
e
2
2π
z
Notice the limits of the integral. The integral of any Gaussian pdf can also be
expressed using the Q-function, of course.
• A vector-valued RV X with mean µ
and covariance matrix Σ
,
XX
X
Σ
Σ
X
µ
X
X
X
X
X
1
X =
=
Σ
=
,
µ
1
,
1
1
1
2
,
XX
Σ
Σ
X
µ
X
X
X
X
X
X
2
2
2
1
2
2
is called Gaussian if its components have a jointly Gaussian pdf
1
T
1
(x) =
· exp −
(X − µ
)
Σ
(X − µ
) .
p
√
X
XX
2
2
X
X
|Σ
|
2π
1/2
XX
(There is nothing to understand, this is just a definition.) The marginal distri-
butions and the conditional distributions of a Gaussian vector are again Gaussian
distributions. (But this can be proved.)
The corresponding symbolic notation is
X ∼ N (µ
, Σ
)
XX
X
This can be generalized to longer vectors, of course.
• Gaussian RVs are completely described by mean and covariance. Therefore, if they
are uncorrelated, they are also independent. This holds only for Gaussian RVs. (Cf.
above.)
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