Since BM has continuous sample paths, and variance equal to t (in your formulation), it would seem that |W(\tau_n)| = n, almost by definition. There may be a tacit assumption that n is an integer, but that doesn't appeaqr necessary because of path continuity.
You need to evaluate the following: \sum 12!/x_1!...x_8! where the sum goes over all possible combinations so that each x_i is at least 1, and \sum x_i is at least 8. Then multiply the result by (1/8)^{12}. For example, there are 8 ways that exactly one x_i contains 2, with the other 7...
Most likely someone employed an approximation to the multinomial distribution. it is possible to calculate such a probablility exactly using a computer program, but it is somewhat involved. It is essentially a problem in combinatorics.
If you suspect an outlier, you are better off with the median, independent of other distributional assumptions. One outlier can severely distort the correlation coefficient. This is one reason you should always graph your data in some form or fashion. However, you cannot compute a reliable...
Re: How to look for the critical value of Pearson Correlatio
If you can compute t-distribution critical points, you can use t = r \frac{\sqrt{n-2}}{\sqrt{1-r^2}} which has a t-distribution with n-2 degrees of freedom.
To generate random samples from the bivariate normal distribution, do the following. Generate Z_1, Z_2 as standard normal variates. Then let X = \mu_1 + \sigma_1 Z_1, Y = \mu_2 + \sigma_2 ( Z_1 \rho + Z_2 \sqrt{1-\rho^2} }).
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