Mathematics, 03.03.2020 18:17 nefertitihorne12
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 fX(x) = c x 3 , x > 1, where c is a constant. (a) Find the value of c so that fX is a probability density function. (b) The term PM10 refers to particles 10 µ m or less in diameter. What proportion of contaminating particles are PM10 ? (c) The term PM2.5 refers to particles 2.5 µ m or less in diameter. What proportion of contaminating particles are PM2.5 ? (d) What proportion of the PM10 particles are PM2.5 ?
Answers: 2
![\displaystyle\int^{\infty}_{1} \frac{c}{x^3} = 1\\\\\Rightarrow -\frac{c}{2}\bigg[\frac{1}{x^2}\bigg]^{\infty}_{1} = 1\\\\\Rightarrow \frac{c}{2} = 1\\\\\Rightarrow c = 2](/tpl/images/0532/2992/4d7c0.png)
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