(Solved): Particles are a major component of air pollution in many areas. It is of interest to study the siz ...

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 $f(x)=\{\begin{array}{cc}\frac{c}{x^3}& x\ge 1\\ 0& x<1\end{array}$ 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 ${\mathrm{PM}}_{10}$ refers to particles $10\mu \mathrm{m}$ or less in diameter. What proportion of the contaminating particles are ${\mathrm{PM}}_{10}$ ? f. The term ${\mathrm{PM}}_{2.5}$ refers to particles $2.5\mu \mathrm{m}$ or less in diameter. What proportion of the contaminating particles are ${\mathrm{PM}}_{2.5}$ ? g. What proportion of the ${\mathrm{PM}}_{10}$ particles are ${\mathrm{PM}}_{2.5}$ ?