Question

Proper blending of fine and coarse powders prior to copper sintering is essential for
uniformity in the finished product. One way to check the homogeneity of the blend is to
select many small samples of the blended powders and measure the promotion of the total
weight contributed by the fine powders in each. These measurement should be relatively
stable if a homogeneous blend has been obtained.
(a) Suppose that the proportion of total weight contributed by the fine powders has a
Beta distribution with α = β = 3. Find the mean and variance of the proportion of
weight contributed by the fine powders.
(b) Repeat part (a) if α = β = 2.
(c) Repeat part (a) if α = β = 1.
(d) Which of the cases (a), (b), and (c) yields the most homogeneous blending?

Answer 1

a) The mean and variance for the proportion is given by:

  

therefore mean =1/2=0.5  and variance = 1/28

b) mean = 0.5 and variance = 1/20

c) mean= 0.5 and variance = 1/12

d) The one with least variance will be more stable hence for most homogenous blending a) with alpha=beta=3 is the best