State Rain Sun Fog Snow Mean Variance CovarianceProb 0.2 0.4 0.3 0.1Return_A 0.23 -0.3 0.35 0.45 0.076Return_B 0.1 0.01 0.12 0.25 0.085

RetA*P 0.046 -0.12 0.105 0.045RetB*p 0.02 0.004 0.036 0.025

p*dRA 0.004743 0.05655 0.022523 0.013988 0.097804p*dRB 4.5E-05 0.00225 0.000368 0.002722 0.005385

p*dRA*dRB 0.00039 0.00635 -0.000426 0.000186 0.0065

Correlation

0.283236

State Rain Sun Fog Snow Expectations CovarianceVariance Correlationprobability 0.2 0.15 0.5 0.15R1 0.1 0.3 0.2 0.4R2 -0.2 0.1 0.3 0.25

p*R1 0.02 0.045 0.1 0.06 0.225p*R2 -0.04 0.015 0.15 0.0375 0.1625p*dR1*dR2 0.009063 -0.000703 -0.001719 0.002297 0.008938p*dR1^2 0.003125 0.000844 0.000312 0.004594 0.008875p*dR2^2 0.026281 0.000586 0.009453 0.001148 0.037469

0.490114

Correlation

R1 R2 Covariancecorrelation0.8224252 0.063295 -0.04888 -0.4800040.8359938 0.1878960.8542775 0.0592670.0103438 0.650810.9994831 0.629070.7954981 0.0341290.5040988 0.1599180.0054377 0.316733

0.14275 0.574867