Totals Betting Market Returns for the NFL Based on Humidity: 2010-2014 Humidity Over Under Push Win % LLR-Fair Bet High Over Win% 80 or more 101 83 2 54.89% 1.7637 75 or more 168 136 4 55.26% 3.3747 (*) 70 or more 238 206 5 53.60% 2.3083 Low Under Win% 60 or less 124 137 1 52.49% 0.6478 55 or less 74 95 1 56.21% 2.6162 50 or less 47 52 1 52.53% 0.2526 Strategy Win% 75 or more & 55 or less 263 210 5 55.60% 5.9512 (**) Humidity LLR-No Profits High 80 or more 0.4662 75 or more 1.0155 70 or more 0.2667 Low 60 or less 0.0013 55 or less 0.9889 50 or less 0.0008 75 or more & 55 or less 1.9747 The log likelihood test statistics have a

chi-square distribution with one degree of freedom.

where df is the degrees of freedom, [d.sup.2] = x'[(X'X).sup.-1] x where X is the design matrix of the linear regression model, m is the number of independent random samples (factors), [PHI](z) is the cumulative distribution function, [phi](z) is the probability density function, F[[chi square].sub.df] is the cumulative distribution function of a

chi-square distribution with df degrees of freedom.

In addition, Cochran's theorem has shown that V [??] (N - 1) v/[[sigma].sup.2] has a

chi-square distribution with df = N - 1 degrees of freedom.

where [chi square]([upsilon]) is a

chi-square distribution with [upsilon] degrees of freedom.

KEY WORDS / Genotype and Environment Contribution / GxE Interaction / Noncentral

Chi-Square Distribution / Modified F Test /

It is well known that sample variances tend to have a

chi-square distribution (Overall & Woodward, 1974).

Checking the

chi-square distribution table, we find that with k-1 = 5 degrees of freedom, the critical [chi square] = 11.07.

is approximately (for sufficiently large [n.sub.j], j = 1, ..., k)

chi-square distribution with [kappa] = sm (k - (m + 1)/2) degrees of freedom (the number of random variables [Z.sub.ij], [[??].sub.ij] is sm(2k - m) and the number of estimated parameters is sm(k - (m -1) / 2)).

,[Z.sub.n] are iid N(0,1) then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] has a [[sigma].sup.2]

chi-square distribution with n degrees of freedom.

Also, the empirical distribution function of D, found under the null composite hypothesis, follows the theoretical

chi-square distribution function, if not as near.

Geared toward graduate students and professionals in statistics, engineering, social sciences and medical science but applicable to other fields as well, this text starts with the statistical decision principle and proceeds to normal distribution,

chi-square distribution and properties, discrete distributions, and large sample theory.

where [chi.sup.2] (n) denotes the

chi-square distribution with n degrees of freedom and [X.sup.i.sub.t] is a square error weighted by [[OMEGA].sub.t].