a) A restaurant manager believed that each entre order will independently be vegetarian with uncertain probability V~Beta(4,36) before she observed 26 vegetarian orders among the last 140 orders. Consider her updated belief about V after her observation. Find the manager's updated distribution, expected value, and variance.b) Let X1 and X2 be indicator variables for whether the first two of the 140 orders placed in the previous problem are vegetarian. Please specify any independence between them before those orders are placed. Are they identically distributed? Are they exchangeable?c) In the previous problem, if the manager believed orders arrive in a Poisson manner at uncertain rate R ~ Gamma(50,60) orders per minute, before she observed 100 orders in the last hour, consider her updated belief about R after her observation. Find the manager's distribution, expected value, and variance.d) Given that X~Gamma(a, b), let Y = cX + d, where c > 0. Find the possible values, probability density function, expected value, and variance for Y as a function of a, b, c, and d.e) Which variables must be independent of A given C and S according to the belief network below? (see the image)f) Which variables must be independent of A given D and S according to the belief network below? (see the image)

e) Which variables must be independent of A given C and S according to the belief network below? f) Which variables must be independent of A given D and 8 according to the belief network below