Consider a random sample X1,X2, . . . ,Xn from a

QUESTION:

Consider a random sample \(X_{1}, X_{2^{\prime} \cdots}, X_{n}\) from a distribution with pdf

                    \(f(x \mid \theta)=3 \theta x^{2} e^{-\theta x^{3}}, 0<x<\infty\).

Let  have a prior pdf that is gamma with \(\alpha=4\) and the usual \(\theta=1 / 4\). Find the conditional mean of \(\theta\), given that \(X_{1}=x_{1}, X_{2}=x_{2^{2} \cdots, X} X_{n}=x_{n}\).

Equation Transcription:

Text Transcription:

X_1,X_2,...,X_n

f(x|theta)=3theta x^2 e^-theta x^3, 0<x<infinity

alpha=4

theta=1/4

theta

X_1=x_1,X_2=x_2,...,X_n=x_n