The seasonal catch of a commercial fishing vessel in a certain fishery in the southern hemisphere can be represented by \(Q=c(\mathbf{z}) V\), where \(\mathbf{z}\) is a vector of characteristics of the vessel relating to tonnage, length, number of crew members, holding tank size, etc., \(c(\mathbf{z})\) represents maximum fishing capacity of the boat, \(Q\) represents the tons of fish caught, and \(V \sim \theta v^{\theta-1} \mathrm{I}_{(0,1)}(V)\) represents the proportion of fishing capacity realized.

(a) Derive the density function of seasonal catch.

(b) If \(\theta=10\) and \(c(\mathbf{z})=5,000\), what is the expected value of seasonal catch?