Validity of the Formula for the Uncertainty in the Fit Coefficients

   My derivation of the uncertainty in the fit coefficients assumed that the expected value <δy_jδy_k>=0 for k≠j.

  It is not obvious that this is so but we can demonstrate it in the following manner. One starts by creating an array of random numbers with mean μ=0 and standard deviation σ=0.3. In this case the array contained 30 random numbers. 

  Next we multiply two numbers in the array with each other. Since there are 30 choices for each number we end up with a 30x30 grid of products.

  We can now compare the sum of all the products with the sum of the squared products, the diagonal terms in the 30x30 grid. This process was repeated 100 times to get the mean values and the variations.

  The two averages for the 100 trials are approximately equal which suggests that the expected value for the product for two uncorrelated random numbers is negligible. Only the sum of the squared terms appear to contribute to variations of the a_i and we conclude that the two formulas are approximately equivalent.  The lower values for the variation of simpler sum suggests that it is the better estimate. One would expect these results to hold for a large number of trials.

Supplemental (May 18): Perhaps a simpler way of demonstrating the expected value of the product of two normally distributed random numbers is negligible is to do a large number of products and compute the average.

  The bottom column shows the averages of 200 products repeated 50 times with the average and standard deviation of the 50 averages.

Supplemental (May 19): One can show that the expected value for the product of two independent normally distributed random errors is,