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Log-normal distribution and its uses.

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Answer

  • X ~ LogNormal(μ,  σ2)\mathrm{LogNormal}({\mu}, \;{\sigma}^{2}) if log⁡(X)\operatorname{log}(X) ~ N(μ,  σ2)N({\mu}, \;{\sigma}^{2}).
  • Right-skewed, always positive.
  • Mean: exp⁡(μ  +  σ2/2)\operatorname{exp}({\mu}\; + \;{\sigma}^{2} / 2); Median: exp⁡(μ)\operatorname{exp}({\mu}).
  • Uses: incomes, prices, response times, file sizes — quantities that are strictly positive with right skew.
  • Products of many small independent factors (like the multiplicative CLT) tend log-normal.
  • In DS: use log-transform to make skewed features more Gaussian-like for linear models.
Check yourself — multiple choice
  • Symmetric bell curve
  • log⁡(X)\operatorname{log}(X) ~ N; right-skewed, positive; incomes / prices / response times; mean  =  exp⁡(μ  +  σ2/2)\mathrm{mean}\; = \;\operatorname{exp}({\mu}\; + \;{\sigma}^{2} / 2)
  • Never used
  • Discrete

Log-normal: log⁡(X)\operatorname{log}(X) is normal; right-skewed positive; incomes, prices, response times.

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