Question

The article ”The statistics of Phytoxic Air Pollutants” suggests the log-normal distribution as a model for SO2 concentration above a certain forest. Suppose the parameter values are µ = 1.9 and σ = 0.9.

(a) What are the mean value and standard deviation of concentration?

(b) What is the probability that concentration is at most 10? Between 5 and 10? 

Answer 1

Solution:

$\mu=1.9,\sigma=0.9$

(a) Mean and standard deviation of log-Normal distribution:

Mean$=e^{\mu+\sigma^2/2}=e^{1.9+0.9^2/2}=e^{2.305}=10.024$

Variance$=({e^{\sigma}}^2-1)e^{2\mu+\sigma^2}=({e^{0.9}}^2-1)e^{2(1.9)+0.9^2}=125.394$

S.D$=\sqrt{variance}=\sqrt{125.394}=11.197$

(b) $X\sim N(10.024,11.197)$

$P(X\le10)=P(z\le \dfrac{10-10.024}{11.197})=P(z\le0.002)=0.5008$

$P(5\le X\le10)=P(\dfrac{5-10.024}{11.197}\le z\le \dfrac{10-10.024}{11.197})$

$=P(-0.448\le z\le0.002)=P(z\le 0.002)-P(z\le -0.45) \\=P(z\le 0.002-[1-P(z\le 0.45)] \\=0.5008-1+0.67364 \\=0.17444$