### Probability density function | Mathematics homework help

Particles are a major component of air pollution in many areas. It is of interest to study the sizes of contaminating particles. Let X represent the diameter, in micrometers, of a randomly chosen particle. Assume that in a certain area, the probability density function of X is inversely proportional to the volume of the particle; that is, assume that

where c is a constant.

a. Find the value of c so that f (x) is a probability density function.

b. Find the mean particle diameter.

c. Find the cumulative distribution function of the particle diameter.

d. Find the median particle diameter.

e. The term PM10 refers to particles 10 μm or less in diameter. What proportion of the contaminating particles are PM10?

f. The term PM2.5 refers to particles 2.5 μm or less in diameter. What proportion of the contaminating particles are PM2.5?

g. What proportion of the PM10 particles are PM2.5?