A quality control plan for an assembly line involve sampling

A quality control plan for an assembly line involve sampling n=10 finished item, per day and counting Y the number of defectives. If p denotes the probability of observing a defective then Y has a T assuming that a large number of items are produced by the line. But p var.es from day to day and is assumed have a uniform distribution on the interval from 0 to 1/4. Find the expected value of Y.

Solution

Y | p ~ Binomial(n,p)

E(Y | p)=n*p=10*p

E[Y]=E[E(Y | P)]=E[10*p]=10*E[p]

Now, from p~Unif(0, 1/4)

E[p]=(0+1/4) / 2 =1/8

So, E[Y]=10*E[p]=10/8=1.25