Question

According to Zimmels (1983), the sizes of particles used in sedimentation experiments often have a uniform distribution. In sedimentation involving mixtures of particles of various sizes, the larger particles hinder the movements of the smaller ones. Thus, it is important to study both the mean and the variance of particle sizes. Suppose that spherical particles have diameters that are uniformly distributed between 0.02 and 0.07 centimeters. Find the mean and variance of the volumes of these particles. (Recall that the volume of a sphere is (4/3) πr³. Round your answers to four decimal places.)

E(Y)

= ____________✕10⁻⁵ cm³

V(Y)

= ____________✕10⁻⁹

Answer 1

We are given the distribution of diameter here as:

Therefore the distribution of radius here is obtained here as:

The third and sixth moment of Radius is computed here as:

Now the volume of a sphere with above radius is computed here as:

The mean and variance of volume is computed here as:

Therefore the mean here is 6.2439 x 10^-5 cm³

The second moment of Volume is now computed here as:

The variance of volume is thus computed here as:

Therefore the variance of the volume here is given as: 2.5512 x 10^-9 cm⁶ is the required variance here.