Log-Normal Distribution

Log-normal Distribution

If a continuous random variable Y = log X follows normal distribution with meanμ and varianceσ2 i.e. if Y = log_e X ~ N (μ,σ²), then X = e^y is called a log-normal distribution. The log-normal distribution is used in solving the problems of biology, economics, reliability theory etc. For example, the hourly median power of radio signals received and transmitted between two places follows the log-normal distribution.

Definition:

If log_e X ~ N (μ,σ²),then the positive continuous random variable X is said to follow log-normal distribution, log N(μ,σ²), with probability density function (pdf)

$$f(x) \ = \ \frac{1}{x \sigma \ \sqrt{ 2 \pi}} \ {e^-}^{\frac{1}{2} \ \left ( \frac{log x - \ \mu}{\sigma} \ \right )^2} \ \ \ ; \ x \ > \ 0.$$

Hereμ and σ² are two parameters of the log-normal distribution.

Ifμ = 0 and σ² = 1 i.e. if log X ~ N (0, 1) then the random variable X is said to follow standard log-normal distribution, Log N (0, 1), and its probability density function (pdf) is given by

$$f(x) \ = \ \frac{1}{x \ \sqrt{2 \pi}} \ {e^-}^{\frac{1}{2} \ \left ( \ log x \ \right )^2} \ \ \ ; \ x \ > \ 0.$$

Moments of Log-normal Distribution :

Let X ~ log N(μ,σ²). The r^th moment about origin of log-normal distribution is given by

$$\mu_r^{'} \ = \ E \ (X^r) \ = \ \int_0^{\infty} \ x^r \ f(x) \ dx$$

$$= \ \int_0^{\infty} \ x^r \\frac{1}{x \sigma \ \sqrt{ 2 \pi}} \ {e^-}^{\frac{1}{2} \ \left ( \frac{log x - \ \mu}{\sigma} \ \right )^2}$$

$$put \ z \ = \ \frac{log x - \mu}{\sigma} \ then \ x \ = \ {e^\mu}^{+ \sigma z} \ so \ that \ dx \ = \ {e^\mu}^{+ \ \sigma z} \ . \ \sigma \ dz$$

$$\therefore \ \mu_r^{'} \ = \ \int_{- \infty}^{\infty} \ {e^(}^{\mu \ + \ \sigma z)r} \ \frac{1}{ \ {e^\mu}^{+ \ \sigma z} \ \sigma \ \sqrt{2\pi}} \ {e^-}^{\frac{1}{2} \ z^2} \ {e^\mu}^{+ \sigma z} \ . \ \sigma \ dz$$

$$= \ \frac{{e^\mu}^{r}}{\sqrt{2\pi}} \ \int_{- \ \infty}^{\infty} \ {e^-}^{\frac{1}{2} \ \left ( z^2 \ - \ 2z \ \sigma r \ \right )} \ dz$$

$$= \ \frac{{e^\mu}^{r} \ {e^\frac{1}{2}}^{\sigma^2 \ r^2}}{\sqrt{2\pi}} \ \int_{-\infty}^{\infty} \ {e^-}^{\frac{1}{2} \ \left ( \ z - \sigma r \ \right )^2} \ dz$$

$$\therefore \ \mu_r^{'} \ = \ {e^\mu}^{r \ + \ \frac{1}{2} \ \sigma^2 \ r^2 }$$

$$When \ r \ = \ 1, \ \ \ \mu_1^{'} \ = \ {e^\mu}^{+ \ \frac{1}{2} \ \sigma^2} \ = \ Mean$$

$$When \ r \ = \ 2, \ \ \ \mu_2^{'} \ = \ {e^2}^{\mu \ + \ 2 \ \sigma^2}$$

$$When \ r \ = \ 3, \ \ \ \mu_3^{'} \ = \ {e^3}^{\mu \ + \ \frac{9}{2} \ \sigma^2}$$

$$When \ r \ = \ 4, \ \ \ \mu_4^{'} \ = \ {e^4}^{\mu \ + \ 8 \ \sigma^2}$$

$$ Now, \ the \ Variance \ = \ \mu_2 \ = \ \mu_2^{'} \ - \ (\mu_1^{'})^2$$

$$= \ {e^2}^{\mu \ + \ 2 \ \sigma^2} \ - \ \left ( {e^\mu}^{+ \ \frac{1}{2} \ \sigma^2} \ \right )^2$$

$$= \ {e^2}^{\mu \ + \ 2 \ \sigma^2} \ - \ {e^2}^{\mu \ + \ \sigma^2}$$

$$= \ {e^2}^{\mu \ + \sigma^2} \ \left ( \ {e^\sigma}{^2} \ - \ 1 \ \right )$$

$$\mu_3 \ = \ \mu_3^{'} \ - \ 3 \mu_2^{'} \ \mu_1^{'} \ + \ 2 \ \left ( \ \mu_1^{'} \ \right )^3$$

$$= \ {e^3}^{\mu \ + \ \frac{9}{2} \ \sigma^2} \ - \ 3 \\ {e^2}^{\mu \ + \ 2 \ \sigma^2}\ {e^\mu}^{+ \ \frac{1}{2} \ \sigma^2} \ + \ 2 \ \left (\ {e^\mu}^{+ \ \frac{1}{2} \ \sigma^2} \ \right )$$

$$= \ {e^3}^{\mu \ + \ \frac{3}{2} \ \sigma^2} \ \left ( \ {e^\sigma}^{2} \ - 1 \ \right )^2 \ \left ( {e^\sigma}^{2} \ + \ 2 \ \right )$$

$$\mu_4 \ = \ \mu_4^{'} \ - \ 4 \mu_3^{'} \ \mu_1^{'} \ + \ 6 \ \mu_2^{'} \ (\mu_1^{'} )^2 \ - \ 3 \ (\mu_1^{'})^4$$

$$= \ {e^4}^{\mu \ + \ 8 \ \sigma^2} \ - \ 4 {e^3}^{\mu \ + \ \frac{9}{2} \ \sigma^2} \ .\ {e^\mu}^{+ \ \frac{1}{2} \ \sigma^2} \ + \ 6\ {e^2}^{\mu \ + \ 2 \ \sigma^2} \ \left (\ {e^\mu}^{+ \ \frac{1}{2} \ \sigma^2} \ \right )^2 \ - \ 3 \ \left (\ {e^\mu}^{+ \ \frac{1}{2} \ \sigma^2} \ \right )^4$$

$$= \ {e^4}^{\mu \ + \ 2 \sigma^2} \ \left ( \ {e^\sigma}^{2} \ - \ 1 \ \right )^2 \ \left ( \ {e^4}^{\sigma^2} \ + \ 2 \ {e^3}^{\sigma^2} \ + \ 3 \ {e^2}^{\sigma^2} \ - \ 3 \ \right )$$

The moment coefficient of skewness is given by

$$\gamma_1 \ = \ ({e^\sigma}^{2} \ - \ 1)^{\frac{1}{2}} \ ( {e^\sigma}^{2} \ + \ 2 )$$

The moment coefficient of kurtosis is given by

$$\gamma_2 \ = \ ({e^\sigma}^{2} \ - \ 1) \ ({e^3}{\sigma^2} \ + \ 3 \ {e^2}^{\sigma^2} \ + \ 6 \ {e^\sigma}^{2} \ + \ 6 \ )$$

Since,γ₁ > 0, the log-normal distribution is positively skewed. Similarly, sinceγ₂ > 0, the log-normal distribution is leptokurtic.

Bibliography

Sukubhattu N.P. (2013). Probability & Inference - II. Asmita Books Publishers & Distributors (P) Ltd., Kathmandu.

Larson H.J. Introduction to Probability Theory and Statistical Inference. WileyInternational, New York.