Functions of random variables

1
Functions of random variables

• Sometimes what we can measure is not what we are
  interested in!
• Example mass of binary-star system
• We want M but can only measure V and P.
• Must conserve probability

2
Non-linear transformations

• e.g.Flux distributions vs. wavelength, frequency
• Fluxes and magnitudes
• Gaussian distribution X G(X0,?2)
• Nonlinear transformation induces a bias
• PROBLEM evaluate a, ?(M) in terms of X0 , ?.

f(M)
M-2.5 log X
f(X)
X
3
Nonlinear transformations bias the mean

• To find ltYgt, use Taylor expansion around XltXgt
• Hence

0
This is the bias.
4
Variance of a transformed variable

• Get variance of Y from first principles

5
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
• 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

6
Sample mean and variance - 1

• Sample mean
• The distribution of sample means has a mean
• ...and a variance

if the Xi are independent
7
Sample mean and variance - 2

• If the Xi are all drawn from a single parent
  distribution with mean ltXgt and variance ?2, then
• And

8
Other unbiased statistics

• Sample median (half points above, half below)
• (Xmax Xmin) / 2
• Any single point Xi chosen at random from
  sequence
• Weighted average

9
Inverse variance weighting is best!

• Lets evaluate the variance of the weighted
  average for some weighting function wi
• The variance of the weighted average is minimised
  when
• Lets verify this -- its important!

10
Choosing the best weighting function

• To minimise the variance of the weighted average,
  set

11
Using optimal weights

• Good principles for constructing statistics
• Unbiased -gt no systematic error
• Minimum variance -gt smallest possible statistical
  error
• Optimally (inverse-variance) weighted average
• Is unbiased, since
• And has minimum variance