Problem 14E

In a process of sintering (heating) two types of copper powder (see Exercise 5.152), the density function for Y1, the volume proportion of solid copper in a sample, was given by

Reference

In the production of a certain type of copper, two types of copper powder (types A and B) are mixed together and sintered (heated) for a certain length of time. For a fixed volume of sintered copper, the producer measures the proportion Y1 of the volume due to solid copper (some pores will have to be filled with air) and the proportion Y2 of the solid mass due to type A crystals. Assume that appropriate probability densities for Y1 and Y2 are

The proportion of the sample volume due to type A crystals is then Y1Y2. Assuming that Y1 and Y2 are independent, find P(Y1Y2 ≤ .5).

Solution 14E

Step1 of 2:

Let us consider the random variables denotes two types of copper powder, the density function for is given by:

The density function for , the proportion of type A crystal among the solid copper, was given as:

We need to find the probability density function of U.

Step2 of 2:

Let us assume that are independent so,

Where, , .

Then

Now,

Now,

Therefore, the probability density function of U is given by: