Suppose that a system has allowed energy levels n?, with n = 0,1,2,3,4,... There

Question

Suppose that a system has allowed energy levels n?, with n = 0,1,2,3,4,... There are three distinguishable particles, with total energy U = 4?.
(a) Tabulate all possible distributions of the three particles among the energy levels, satisfying U = 4?.
(b) Evaluate wi for each of above distributions, and also ? = ?iwi.
(c) Calculate the average occupation numbers Nn =?k Nnkwk/? for the three particles in the energy states. Here Nn is the average occupation number of the energy level with energy n?. You should find Nn for all n?4(and find that Nn>4 = 0)

Explanation / Answer

Construct a Chi-Square Goodness of Fit Test 1. Fit the distribution based on data (method of moments or maximum likelihood). 2. Per the fit, calculate the expected # in a multi-cell partition of the distribution. 3. Tabulate the actual observed in these cells. 4. Form the sum of squares of the differences between actual and fitted. Divide by the degrees of freedom (function of the data set and the # of parameters estimated for the postulated distribution). This is the Ch-Square variable. See if significant at the 5% level (so called selected Type I error). I would also be interested in the Power of the Test (1- Type II error). Possibly more powerful modification: Calculate a statistic of the distribution (like the ratio of adjacent energy levels) for which you can compute an expected value to compare to your observed statistic. Perform the same Chi-Square Test, but adjust the degrees of freedom as you are no longer consuming degrees of freedom to estimate parameters of the distribution. Good luck.

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