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joint normal distribution∗
gel†
2013-03-21 19:24:27
A finite set of random variables X1 , . . . , Xn are said to have a joint normal
distribution or multivariate normal distribution if all real linear combinations
λ1 X1 + λ2 X2 + · · · + λn Xn
are normal. This implies, in particular, that the individual random variables Xi
are each normally distributed. However, the converse is not not true and sets of
normally distributed random variables need not, in general, be jointly normal.
If X = (X1 , X2 , . . . , Xn ) is joint normal, then its probability distribution is
uniquely determined by the means µ ∈ Rn and the n × n positive semidefinite
covariance matrix Σ,
µi = E[Xi ],
Σij = Cov(Xi , Xj ) = E[Xi Xj ] − E[Xi ]E[Xj ].
Then, the joint normal distribution is commonly denoted as N(µ, Σ). Conversely, this distribution exists for any such µ and Σ.
Figure 1: Density of joint normal variables X, Y with Var(X) = 2, Var(Y ) = 1
and Cov(X, Y ) = −1.
The joint normal distribution has the following properties:
1. If X has the N(µ, Σ) distribution for nonsigular Σ then it has the multidimensional Gaussian probability density function
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1
fX (x) = p
exp − (x − µ)T Σ−1 (x − µ) .
2
(2π)n det (Σ)
2. If X has the N(µ, Σ) distribution and λ ∈ Rn then
λ · X = λ1 X1 + · · · + λn Xn ∼ N(λ · µ, λT Σλ).
∗ hJointNormalDistributioni
created: h2013-03-21i by: hgeli version: h37204i Privacy
setting: h1i hDefinitioni h62H05i h60E05i
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3. Sets of linear combinations of joint normals are themselves joint normal.
In particular, if X ∼ N(µ, Σ) and A is an m × n matrix, then AX has
the joint normal distribution N(Aµ, AΣAT ).
4. The characteristic function is given by
1 T
ϕX (a) ≡ E [exp(ia · X)] = exp ia · µ − a Σa ,
2
for X ∼ N(µ, Σ) and any a ∈ Cn .
5. A pair X, Y of jointly normal random variables are independent if and
only if they have zero covariance.
6. Let X be a random vector whose distribution is jointly normal. Suppose
the coordinates of X are partitioned into two groups, forming random
vectors X1 and X2 , then the conditional distribution of X1 given X2 = c
is jointly normal.
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