An incoming shipment of 1,000 toys from a toy manufacturer is received by a large department store for a pre-Christmas sale. The store randomly samples 50 toys from the shipment, without replacement, and records whether or not the sample item is defective. The store wants to generate a MVUE estimate of the proportion of defectives in the shipment of toys. The statistical model it uses for the sampling experiment is given by the hypergeometric density with the following parameterization:

\(f(x ; \Theta)=\frac{\left(\begin{array}{c}1000 \Theta \\ x\end{array}ight)\left(\begin{array}{c}1000(1-\Theta) \\ 50-x\end{array}ight)}{\left(\begin{array}{c}1000 \\ 50\end{array}ight)} I_{\{0,1,2, \ldots, 50\}}(x)\), where \(\boldsymbol{\Theta} \in \Omega=\{0, .001, .002, \ldots, 1\}\) represents the proportion of defectives in the shipment.

a. Show that the \(X \sim f(x ; \Theta)\) is a minimal, complete sufficient statistic for \(f(x ; \Theta)\).

b. Define the MVUE for the proportion of defectives in the shipment.

c. Suppose that the outcome of \(X\) was 3. Define a MVUE estimate of the proportion of defectives in the shipment.

d. Define an MVUE for the number of defective toys in the shipment, and provide an MVUE estimate of the this number.