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 chosenparticle. Assume that in a certain area, the probability densityfunction of X is inversely proportional to the volume of theparticle; that is, assume that
f(x) = c /(x^3) x ≥ 1
f(x) = 0 x < 1 where c is a constant.
a. Find the value of c so that f (x) is a probability densityfunction.
b. Find the mean particle diameter.
c. Find the cumulative distribution function of the particlediameter.
d. The term PM10 refers to particles 10 μm or less in diameter.What proportion of the contaminating particles are PM10?
e. The term PM2.5 refers to particles 2.5 μm or less indiameter. What proportion of the contaminating particles arePM2.5?
f. What proportion of the PM10 particles are PM2.5?
g. Consider the pdf of the particle diameter. The median of theparticle diameter must be
(i) larger than
(ii) smaller than
(iii) approximately equal to the mean, 2 micrometers. Choose anappropriate answer and explain why.
h. Compute the first quartile of the particle diameter.
