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# Chi-squared distribution  2 N - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about ' Chi-squared distribution  2 N' - carter-vaughn

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Presentation Transcript

• 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:

• 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

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

• 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

• Express sample variance as:

• Use algebra of random variables to determine:

• Expand:

(Don’t worry,

all will be

revealed...)

<X>

Xi

<Xi>

Aside: what is Cov(Xi,X)?

Estimating 2 – 2

• We now have

• For s2 to be an unbiased estimator for 2, need A=1/(N-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)

• 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.

• 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