# Essays on Estimation of Joint, Marginal and Posterior Probabilities, and Bayes Theorem Assignment

The paper "Estimation of Joint, Marginal and Posterior Probabilities, and Bayes’ Theorem" is a great example of an assignment on management. The use of probability in risk management is crucial since farmers are able to predict the chances of an event that might affect their crops positively or negatively (Ross, 2014). There are various types of probabilities which include joint probability; shows chances of two events happening at the same time, marginal probability; elaborates the chances of subsets of randomly selected variables i. e. likely and unlikely occurrence, posterior probability; shows chances of an event happen because another event happened (Pitman, 2009). Joint probability; P (Z1S1) =0.4X0.7= 0.28 P (Z1S2) =0.4X0.6= 0.24 P (Z1S3) =0.4X0.2= 0.08 P (Z2S1) =0.4X0.2= 0.08 P (Z2S2) =0.4X0.2= 0.08 P (Z2S3) =0.4X0.3= 0.12 P (Z3S1) =0.2X0.1= 0.02 P (Z3S2) =0.2X0.2= 0.04 P (Z3S3) =0.2X0.5= 0.10 Marginal probability; Marginal probability (PZ1) =P (S1 and Z1) + P (S2 and Z1) + P (S3 and Z1) P (Z1) = (0.4 X 0.7)+(0.4 X 0.6)+(0.2 X 0.2)=0.56 P (Z2) = (0.4X0.2)+(0.4X0.2)+(0.2X0.3)= 0.22 P (Z3) = (0.4X0.1) + (0.4X0.2) + (0.2X0.5) = 0.22 Posterior probability;   Z1 Z2 Z3 P(S1/Zk) = =0.500 = = 0.429 = = 0.143 P(S2/Zk) = =0.364 = = 0.364 = = 0.545 P(S3/Zk) = =0.091 = = 0.182 = = 0.455 Check =1.000 =1.000 =1.000 Bayes’ Theorem;   A B C D E F G H I 1 State Prior Probability P(Zk/Si)   Joint Probabilities(P(Si and Zk)) 2 Sn P(Sn) Z1 Z2 Z3   Z1 Z2 Z3 3 S1 0.4 0.7 0.2 0.1   0.28 0.08 0.04 4 S2 0.4 0.6 0.2 0.2   0.24 0.08 0.08 5 S3 0.2 0.2 0.3 0.5   0.04 0.06 0.1 6 Check 1.0 Marginal Probability P(Zk) 0.56 0.22 0.22 7     Posterior Probability P(S1/Zk) 0.500 0.429 0.143 8       P(S2/Zk) 0.364 0.364 0.545 9       P(S3/Zk) 0.091 0.182 0.455 10           Check 1.0 Probability tree Joint probability; Posterior probability; Question Two The return of the three alternatives Alternative I Investment The option for alternative one is to save the whole amount in a safe money market fund earning 2% (quarterly). Compounded Amount = P (1+i/n) nt P = principal = 500,000 i=rate = 2% n=Times in a year = 3 t = time or years. = 500,000 (1+0.02) 3*3 =\$597,546.28 Interest earned = 597,546.28 -500,000 = \$97,546.3 Alternative II Investment Return of Tomato   Year 1 Year 2 Year 3 Revenue Dry * P=(10,500*0.6)= 6,300 6,300 6,300 Revenue Wet * P=(13,300*0.4)= 5,320 5,320 5,320 Total 11,620 11,620 11,620 Total income = 11,620*3 – 10,000*3 = \$4,860 Return from invested fund in safe money market fund (i=5%): i = 5%

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