Solution Found!
Particles are a major component of air pollution in many
Chapter 2, Problem 24E(choose chapter or problem)
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
\(f(x)=\left\{\begin{array}{cc} \frac{c}{x^{3}} & x \geq 1 \\ 0 & x<1 \end{array}\right.\)
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 \(\mathrm {PM_{2.5}}\)?
Questions & Answers
(1 Reviews)
QUESTION:
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
\(f(x)=\left\{\begin{array}{cc} \frac{c}{x^{3}} & x \geq 1 \\ 0 & x<1 \end{array}\right.\)
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 \(\mathrm {PM_{2.5}}\)?
ANSWER:Step 1 of 8
Given,
Let X represent the diameter.
We assume that in a certain area.
Then the probability density function is
\(f(x)=\left\{\begin{array}{cc} \frac{c}{x^{3}} & x \geq 1 \\ 0 & x<1 \end{array}\right.\)
Reviews
Review this written solution for 18919) viewed: 5701 isbn: 9780073401331 | Statistics For Engineers And Scientists - 4 Edition - Chapter 2.4 - Problem 24e
Thank you for your recent purchase on StudySoup. We invite you to provide a review below, and help us create a better product.
No thanks, I don't want to help other students