Lognormal Distribution

• The lognormal distribution applies in cases where a natural log transformation results in a normal distribution.
• Lognormal Distribution: The continuous random variable $X$ has a lognormal distribution if the random variable $Y=ln(X)$ has a normal distribution with mean $\mu$ and standard deviation $\sigma$.
• The resulting density function of $X$ is
  $$f(x)=\left\lbrace \begin{array}{lr} \frac{1}{\sqrt{2\pi}\s... ...-\mu)/\sigma]^2}, & x\geq0 \\ 0,& x<0\\ \end{array}\right.$$

• The normal distribution has 2-tails. The lognormal distribution has a single tail.
• The normal distribution extends to $-\infty$ and $\infty$.
• The lognormal only extends to $\infty$ but is 0 for $x<0$.

• Theorem 6.5:
  

Figure 8: Lognormal Distributions.

• Example 6.22: Suppose it is assumed that the concentration of a certain pollutant produced by chemical plants, in parts per million, has a lognormal distribution with parameters $\mu = 3.2$ and $\sigma = 1$.
• What is the probability that the concentration exceeds 8 parts per million? (Table A.3)

• Solution:

  Let $X$ be the pollutant concentration
  Since $ln(X)$ has a normal distribution with $\mu = 3.2$ and $\sigma = 1$

  $$P(X > 8)=1-P(x \leq 8)=1-\Phi\left[ \frac{ln(8)-3.2}{1}\right]$$
  $$=1-\Phi\left[\frac{2.08-3.2}{1}\right] =1-\Phi(-1.12)$$
  $$=1-0.1314=0.8686$$

  Here,we use the $\Phi$ notation to denote the cumulative distribution function of the standard normal distribution.

• Example 6.23: The life, in thousands of miles, of a certain type of electronic control for locomotives has an approximate lognormal distribution with $\mu= 5.149$ and $\sigma = 0.737$.
• Find the $5^{th}$ percentile of the life of such locomotive?
• Solution:
  $$P(Z < z_1)=0.05 \Rightarrow z_1=-1.645$$

  $ln(x)$ has a normal distribution with $\mu= 5.149$ and $\sigma = 0.737$

  $$\frac{ln(x)-5.149}{0.737}=-1.645$$
  $$\Rightarrow ln(x)=0.737*(-1.645)+5.149=3.937$$
  $$\Rightarrow x=51.265$$

  5% of the locomotives will have lifetime less than 51.265 thousand miles