Consider a parameter 0 with density 7(0)=B'0*, where B>0 and y > 2. (a) Find the prior mean of 0. (b) Given the observations Y, = y,Y, =y2,.,Y, = y,, from a uniform (0, 0] distribution, find the posterior mean of 0. (c) Show that the estimator of 0, taking the form y +n max (z, B) (B >0,7 > 2), y+n-1 where z= max(y,y2., y,), is admissible under squared error loss

Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.

The quality characteristic of a product Y consists of three components X1, X2, and X3. The distributions of these components are normal with means 100, 2, 1 and variances 1, 1, 1 respectively. The specifications of the product are 90 ± 8. If Y= X1 + -7X2 + -9X3. Products that are below the lower specification limit are scrapped. Find the proportion of scrapped products = Ф(z).

A variable X has a probability function f(x, 0) = 0e 0x; x≥0 y >0. The parameter 0 is estimated using two different estimators of simple random samples of size two: 01 2x+4x2 and 2=4x+5x2 3 3 Select the best estimator for \theta by comparing the Mean Squared Errors