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In my research I'm faced with determining a probability distribution from a function built as follows:

Perform three measurements X, Y, Z that have normally distributed errors.

Impose a constraint and variable change that allows me to reduce the dimensionality to 2.

My question is: Can I assume the resulting function is a chi-square with 2 dof and therefore write my pdf as

[tex]exp(- \chi^2 / 2)[/tex]

The long version with specifics:

I am measuring the energies and opening angle of two photons with a common point of origin and I wish to determine the probability density of true energies and angles from this single measurement. For simplicity I assuming Gaussian errors on the measurements. The opening angle is transformed a bit to make the calculations easier and I start with an initial chi-square of (subscript "m" is my measured value and "z" is my transformed angle measurement):

[tex] \chi^2 = \frac {(E_1 - E_{1m})^2} {\sigma_{E1}^2} + \frac {(E_2 - E_{2m})^2} {\sigma_{E2}^2} + \frac {(z - z_{m})^2} {\sigma_{z}^2} [/tex]

The photons are produced by a common particle and therefore I can impose the constraint that the invariant mass of these photons is a specific value "M" (pis four-momentum).

[tex] C = (\mathbf p_{\gamma 1} + \mathbf p_{\gamma 2})^2 - M^2 = 0 [/tex]

This allows me to reduce the variables from 3 to 2, but in a fairly non-linear way. My final chi-square is a function of energy of the original common particle and the cosine of the center of momentum decay angle of the photons:

[tex] \chi^2 = f(E, \cos{\theta^*}) [/tex]

As one would expect the transformations are quite non-linear, but in practice frequently are "close" to linear for the actual values being considered. I don't want to further burden this post with the ugly transformation details but would happily provide if it is needed.

So the long version of the question is: Can I assume the above expression is still a chi-square? Is the dof 2? Does the fact that E and cos(th*) are not independent play a role in determining the proper dof?

Many, many thanks to anyone that can help. I am especially interested in sources I can reference so I know I'm standing on strong theoretical grounds.

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# Probability Density of a Constrained Chi-Square

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