Chi-squared distribution ?2N

1
Chi-squared distribution ?2N

• N number of degrees of freedom
• Computed using incomplete gamma function
• Moments of ?2 distribution

2
Constructing ?2 from Gaussians - 1

• Let G(0,1) be a normally-distributed random
  variable with zero mean and unit variance.
• For one degree of freedom
• This means that

-a
a
i.e. The ?2 distribution with 1 degree of freedom
is the same as the distribution of the square of
a single normally distributed quantity.
G(0,1)
a2
?21
3
Constructing ?2 from Gaussians - 2

• For two degrees of freedom
• More generally
• Example Target practice!
• If X1 and X2 are normally distributed
• i.e. R2 is distributed as chi-squared with 2 d.o.f

4
Data points with no error bars

• If the individual ?i are not known, how do we
  estimate ??for the parent distribution?
• Sample mean
• Variance of parent distribution
• By analogy, define sample variance
• Is this an unbiased estimator, i.e. is lts2gt?2?

5
Estimating ?2 1

• Express sample variance as
• Use algebra of random variables to determine
• Expand

(Dont worry, all will be revealed...)
6
Aside what is Cov(Xi,X)?
7
Estimating ?2 2

• We now have
• For s2 to be an unbiased estimator for ?2, need
  A1/(N-1)

8
Degrees of freedom 1
ltXgt

• If all observations Xi have similar errors ?
• If we dont know ltXgt use X instead
• In this case we have N-1 degrees of freedom.
  Recall that
• (since lt?2NgtN)

9
Degrees of freedom 2

• Suppose we have just one data point. In this case
  N1 and so
• Generalising, if we fit N data points with a
  function A involving M parameters ?1... ?M
• The number of degrees of freedom is N-M.

10
Example bias on CCD frames

• Suppose you want to know whether the
  zero-exposure (bias) signal of a CCD is uniform
  over the whole image.
• First step determine s2(X) over a few
  sub-regions of the frame.
• Second step determine X over the whole frame.
• Third step Compute
• In this case we have fitted a function with one
  parameter (i.e. the constant X), so M1 and we
  expect lt ?2 gt N - 1
• Use ?2N - 1 distribution to determine probability
  that ?2gt ?2obs