Answer 9.60 **9.45**Suppose that \( Y_1, Y_2, \ldots, Y_n \) is a random sample from a probability density function in the (one-parameter) exponential family so that\[f(y \mid \theta) = \begin{cases} a(\theta)b(y)e^{-[c(\theta)d(y)]}, & a \leq y \leq b, \\0, & \text{elsewhere},\end{cases}\]where \( a \) and \( b \) do not depend on \(\theta\). Show that \(\sum_{i=1}^n d(Y_i)\) is sufficient for \(\theta\). **Exercise 9.60**Let \( Y_1, Y_2, \ldots, Y_n \) denote a random sample from the probability density function\[f(y \mid \theta) =\begin{cases} \theta y^{\theta - 1}, & 0 < y < 1, \, \theta > 0, \\ 0, & \text{elsewhere}.\end{cases}\]**a.** Show that this density function is in the (one-parameter) exponential family and that \( \sum_{i=1}^{n} -\ln(Y_i) \) is sufficient for \( \theta \). (See Exercise 9.45.)**b.** If \( W_i = -\ln(Y_i) \), show that \( W_i \) has an exponential distribution with mean \( 1 / \theta \).**c.** Use methods similar to those in Example 9.10 to show that \( 2\theta \sum_{i=1}^{n} W_i \) has a \(\chi^2\) distribution with \( 2n \) degrees of freedom.**d.** Show that\[E \left( \frac{1}{2\theta \sum_{i=1}^{n} W_i} \right) = \frac{1}{2(n-1)}.\][Hint: Recall Exercise 4.112.]**e.** What is the MVUE for \( \theta \)?

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(10) = {01²-¹0 0, elsewhere. a Show that this density function is in the (one-parameter) exponential family and that -In(Y) is sufficient for 9. (See Exercise 9.45.) b If W₁ = -In(Y), show that W, has an exponential distribution with mean 1/0. c Use methods similar to those in Example 9.10 to show that 20 with 2n df. W, has a x² distribution