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Random Vectors

Module by: Nick Kingsbury. E-mail the author

Summary: This module introduces random vectors.

Random Vectors are simply groups of random variables, arranged as vectors. E.g.:

X=( X1 … Xn )T X X1 … Xn
(1)
where X1 X1 , …… Xn Xn are nn separate random variables.

In general, all of the previous results can be applied to random vectors as well as to random scalars, but vectors allow some interesting new results too.

Figure 1: pdfs of (a) a 2-D normal distribution and (b) a Rayleigh distribution, corresponding to the magnitude of the 2-D random vectors.
Figure 1 (figure1.png)

Example - Arrows on a target

Suppose that arrows are shot at a target and land at random distances from the target centre. The horizontal and vertical components of these distances are formed into a 2-D random error vector. If each component of this error vector is an independent variable with zero-mean Gaussian pdf of variance σ2 σ 2 , calculate the pdf's of the radial magnitude and the phase angle of the error vector.

Let the error vector be

X=( X1 X2 )T X X1 X2
(2)
X1 X1 and X2 X2 each have a zero-mean Gaussian pdf given by
f⁢x=12⁢π⁢σ2⁢e−x22⁢σ2 f x 1 2 σ 2 x 2 2 σ 2
(3)
Since X1 X1 and X2 X2 are independent, the 2-D pdf of XX is
fX ⁢ x1 x2 =f⁢ x1 ⁢f⁢ x2 =12⁢π⁢σ2⁢e− x1 2+ x2 22⁢σ2 fX x1 x2 f x1 f x2 1 2 σ 2 x1 2 x2 2 2 σ 2
(4)
In polar coordinates x1 =r⁢cosθ x1 r θ and x2 =r⁢sinθ x2 r θ To calculate the radial pdf, we substitute r= x1 2+ x2 2 r x1 2 x2 2 in the above 2-D pdf to get:
Prr<R<r+δ⁢r=∫rr+δ⁢r∫−ππ fX ⁢ x1 x2 ⁢RdθdR r R r δ r R r r δ r θ fX x1 x2 R
(5)
where ∫rr+δ⁢r∫−ππ fX ⁢ x1 x2 ⁢RdθdR≃δ⁢r⁢∫−ππ12⁢π⁢σ2⁢e−r22⁢σ2⁢rdθ=1σ2⁢r⁢e−r22⁢σ2⁢δ⁢r R r r δ r θ fX x1 x2 R δ r θ 1 2 σ 2 r 2 2 σ 2 r 1 σ 2 r r 2 2 σ 2 δ r Hence the radial pdf of the error vector is:
fR ⁢r=limit   δ⁢r → 0 Prr<R<r+δ⁢rδ⁢r=1σ2⁢r⁢e−r22⁢σ2 fR r δ r 0 r R r δ r δ r 1 σ 2 r r 2 2 σ 2
(6)
This is a Rayleigh distribution with variance = 2⁢σ2 2 σ 2 (these are two components of XX, each with variance σ2 σ 2 ).

The 2-D pdf of XX depends only on rr and not on θθ, so the angular pdf of the error vector is constant over any 2⁢π 2 interval and is therefore fΘ ⁢θ=12⁢π fΘ θ 1 2 so that ∫−ππ fΘ ⁢θdθ=1 θ fΘ θ 1

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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