3.16 Effect of scaling and offset on average and standard deviation. Suppose x is an n-vector and a and ß are scalars. (a) Show that avg(ax +31) = a avg(x) + ß. (b) Show that std(ax + 1) = |a|std(x).
3.16 Effect of scaling and offset on average and standard deviation. Suppose x is an n-vector and a and ß are scalars. (a) Show that avg(ax +31) = a avg(x) + ß. (b) Show that std(ax + 1) = |a|std(x).
MATLAB: An Introduction with Applications
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Transcribed Image Text:3.16 Effect of scaling and offset on average and standard deviation. Suppose x is an n-vector
and a and ß are scalars.
(a) Show that avg(ax +31) = a avg(x) + B.
(b) Show that std(ax +31) = |a|std(x).
Transcribed Image Text:Standard deviation of sum. We can derive a formula for the standard deviation
of a sum from (3.6):
Vstd(a)² + 2p std(a) std(b) + std(b)².
(3.9)
To derive this from (3.6) we let ã and b denote the de-meaned versions of a and b.
Then a + b is the de-meaned version of a + b, and std (a + b)² = ||ã + ỗ||²/n. Now
using (3.6) and p = cos / (a, b), we get
n std (a + b)²
|| +ỗ||2
std (a + b)
=
=
=
=
=
||ã||²+2p||ã|||||| + ||õ|| ²
n std(a)² + 2pn std(a) std(b) + n std(b)².
Dividing by n and taking the squareroot yields the formula above.
If p
1, the standard deviation of the sum of vectors is the sum of their
standard deviations, i.e.,
std(a + b) = std(a) + std(b).
As
Р decreases, the standard deviation of the sum decreases. When p = 0, i.e., a
and b are uncorrelated, the standard deviation of the sum a + b is
std (a + b) = √/std(a)² + std(b)²,
which is smaller than std(a) + std(b) (unless one of them is zero). When p= -1,
the standard deviation of the sum is as small as it can be,
std (a + b) = | std(a) — std(b)|.
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