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Chi-squared distribution  2 N. N = number of degrees of freedom Computed using incomplete gamma function: Moments of  2 distribution:. 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:

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chi squared distribution 2 n
Chi-squared distribution 2N
  • N = number of degrees of freedom
  • Computed using incomplete gamma function:
  • Moments of 2 distribution:
constructing 2 from gaussians 1
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

constructing 2 from gaussians 2
X2

X1

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
data points with no error bars
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 =2?
estimating 2 1
Estimating 2 – 1
  • Express sample variance as:
  • Use algebra of random variables to determine:
  • Expand:

(Don’t worry,

all will be

revealed...)

estimating 2 2
Estimating 2 – 2
  • We now have
  • For s2 to be an unbiased estimator for 2, need A=1/(N-1):
degrees of freedom 1
Degrees of freedom – 1

  • If all observations Xi have similar errors :
  • If we don’t know use X instead:
  • In this case we have N-1 degrees of freedom. Recall that:
  • (since <2N>=N)
degrees of freedom 2
Degrees of freedom – 2
  • Suppose we have just one data point. In this case N=1 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.
example bias on ccd frames
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 M=1 and we expect < 2 > = N - 1
  • Use 2N - 1 distribution to determine probability that 2> 2obs
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