Chi-squared distribution  2 N

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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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Presentation Transcript
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 =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

• 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
• 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