Respuesta :
Answer:
a) c = 2
b) 0.99
c) 0.84
d) 0.8485
Step-by-step explanation:
We are given the following in the question:
[tex]f(x) = \dfrac{c}{x^3}, x > 1[/tex]
a) Value of c
Property of probability density function
[tex]\displaystyle\int^{\infty}_{-\infty} f(x) = 1[/tex]
Putting values, we get,
[tex]\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[/tex]
Thus, the value of c is 2.
[tex]f(x) = \dfrac{2}{x^3}, x > 1[/tex]
b) proportion of contaminating particles are PM10
We have to evaluate
[tex]P( x \leq 10) =\displaystyle\int ^{10}_{1}\frac{2}{x^3}dx\\\\=\bigg(\frac{2}{-2x^2}\bigg)^{10}_{1}\\\\=-(\frac{1}{100}-1)\\\\=0.99[/tex]
c) proportion of contaminating particles are PM2.5
[tex]P( x \leq 2.5) =\displaystyle\int ^{2.5}_{1}\frac{2}{x^3}dx\\\\=\bigg(\frac{2}{-2x^2}\bigg)^{2.5}_{1}\\\\=-(\frac{1}{6.25}-1)\\\\=0.84[/tex]
d) proportion of the PM10 particles are PM2.5
[tex]P(PM ~2.5|PM~10) = \dfrac{0.84}{0.99} = 0.8485[/tex]
