Find the
Find the variance of the volume of the particles.
Answer to Problem 57E
The mean volume of the particles is
The variance of the volume of the particles is
Explanation of Solution
The mean volume of the particles is obtained below:
The probability density
It is given that the spherical particles have diameters that are uniformly distributed between 0.01 and 0.05 centimeters.
Let Y be the random variable associated with the diameter.
The random circle has a radius that is uniformly distributed on the interval (0, 1). Thus, the probability density function is given below:
Let V be the random variable associated with the volume and the random variable associated with the radius be R.
The volume of a sphere is given as follows:
Let V be denoted as the volume and R be denoted as the radius.
The required value is calculated below:
Then,
The mean of the area of the particle is
The variance of the area of the particle is calculated below:
The formula for variance is
The required value is obtained as follows:
The required value is obtained as follows:
Then,
Thus, the variance of the area of the particle is
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Chapter 4 Solutions
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