Question

Consider a random variableXhaving the lognormal distribution with parametersμ andσ². Determine the p.d.f. ofX.

Short answer

The p.d.f of X is \[f\left( x \right) = \frac{1}{{x\sigma \sqrt {2\pi } }}{e^{ - \frac{1}{{2{\sigma ^2}}}{{\left( {\log x - \mu } \right)}^2}}}; - \infty < x < \infty \]

Step-by-step solution

Given information

X is a lognormal random variable.

Determine the probability density function

The p.d.f of X is

\[f\left( x \right) = \frac{1}{{x\sigma \sqrt {2\pi } }}{e^{ - \frac{1}{{2{\sigma ^2}}}{{\left( {\log x - \mu } \right)}^2}}}; - \infty < x < \infty \]

Hence, the p.d.f of X is \[f\left( x \right) = \frac{1}{{x\sigma \sqrt {2\pi } }}{e^{ - \frac{1}{{2{\sigma ^2}}}{{\left( {\log x - \mu } \right)}^2}}}; - \infty < x < \infty \]