Download ada05

Functions of random variables
• Sometimes what we can measure is not what
we are interested in!
• Example: mass of binary-star system:
V 2a V 3P
M

G
2G
• We want M but can only measure V and P.
Y
• Must conserve probability:
f (Y )dy  f (X)dx
f(Y)
Y  g(X)
g
dY 
dx
X
f(X)
X
Non-linear transformations
• e.g.Flux distributions vs. wavelength, frequency:
f ( )d  f ( )d

c

d  
c

2
d  f ( ) 
• Fluxes and magnitudes:
M  2.5 log X
 2.5 log X  a (M)
c
f ( )
f(M)
M=-2.5 log X
– Gaussian distribution: X ~ G(X0,2)
– Nonlinear transformation induces a bias:
2
– PROBLEM: evaluate a, (M) in terms of X0 , .
f(X)
X
Y
f(Y)
Y=g(X)
Nonlinear transformations
bias the mean
f(X)
X
• To find <Y>, use Taylor expansion around X=<X>:
1
2
Y  g(X)  g( X )  g' ( X ) X  X   g"( X ) X  X   ...
2
• Hence
1
g(X )  g( X )  g' ( X )X  X   g"( X ) X  X 2  ...
2
0
1
 g( X )  g' ( X ) X  X   g"( X ) 2 (X)  ...
2
This is the bias.
Y
f(Y)
Y=g(X)
Variance of a
transformed
variable
f(X)
X
• Get variance of Y from first principles:
 2 (Y )  (Y  Y )2

1
 g( X )  g' ( X ) X  X   g"( X ) X  X 2  ...

2
g( X ) 
2
1

 g"( X ) 2 (X)  ...

2
0
 g' ( X ) X  X   ...  g' ( X )  2 (X )
2
2
What is a statistic?
• Anything you measure or compute from the
data.
• Any function of the data.
• Because the data “jiggle”, every satistic also
“jiggles”.
• Example: the mean value of a sample of N data
points is a statistic:
1 N
X
Xi

N
i1
• It has a definite value for a particular dataset,
but it also “jiggles” with the ensemble of
datasets to trace out its own PDF.
• NB: X  X
Sample mean and variance - 1
1 N
• Sample mean: X   Xi
N i1
• The distribution of sample means has a mean:
1
1
X 
Xi 

N i
N
1
 Xi  N  Xi
i
i
• ...and a variance:
 1
 1 2 
 1
2
 X      Xi   2   Xi   2   Xi 
N i
 N
 i
 N i
2
2
if the Xi are independent
Sample mean and variance - 2
• If the Xi are all drawn from a single parent
distribution with mean <X> and variance 2,
then:
1 N
X   X  X , i.e. X is an unbiased estimator of X .
N i1
• And:
2
2


N

X

Xi 
1


2
2
i
 X   2   Xi  

2
N  i

N
N
  X  
 Xi 
N
, i.e. X " jiggles" much less
than a single data value Xi does.
Other unbiased statistics
• Sample median (half points above, half below)
• (Xmax + Xmin) / 2
• Any single point Xi chosen at random from
sequence
• Weighted average:
wi Xi

w
i
i
i
Inverse variance weighting is best!
• Let’s evaluate the variance of the weighted
average for some weighting function wi:


 wi X i    w i X i   wi2  2 X i 

 i
 i
2  i
 

.
 
2
2




  wi 
 i

 w i 
 w i 
 i

 i

2
• The variance of the weighted average is
minimised when:
1
1
wi 
 2.
Var(Xi )  i
• Let’s verify this -- it’s important!
Choosing the best weighting
function
• To minimise the variance of the weighted
average, set:


2 2
2 2
 wi  i 
2  wi  i
2
  i
 2wk  k
i
0



2  
2 
3
wk 

 wi  
  wi   wi 
   i
  i

 i


2 2
wi  i 


2
1
2

i


 wk  2 .
2 wk  k 




k
 wi 
 wi  
 i

 i
 
(Note :
2 2
2
w


w
for
w

1/

 i i  i
i
i )
Using optimal weights
• Good principles for constructing statistics:
– Unbiased -> no systematic error
– Minimum variance -> smallest possible statistical error
• Optimally (inverse-variance) weighted average:
Xˆ 
 wi Xi
i
 wi
i
• Is unbiased, since:

2
X
/

 i i
i
2
1/

 i
i
Xˆ  X
• And has minimum variance:
 Xˆ  
2
1
2
1
/

 i
i