Chi squared distribution 2 n
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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

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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 <s2>=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...)


Aside what is cov x i x

X

<X>

Xi

<Xi>

Aside: what is Cov(Xi,X)?


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

<X>

  • If all observations Xi have similar errors :

  • If we don’t know <X> 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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