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Chi-squared distribution 2 N

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Chi-squared distribution 2N

- 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

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

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

(Don’t worry,

all will be

revealed...)

Estimating 2 – 2

- We now have
- For s2 to be an unbiased estimator for 2, need A=1/(N-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

- 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

- 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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