Continuous random variable

Log-normal distribution

A log-normally distributed random variable π‘Œ ∼LN(πœ‡,𝜎2) has π‘Œ =e𝑋 for 𝑋 ∼N(πœ‡,𝜎2), prob i.e. lnβ‘π‘Œ has a normal distribution. Applying Distribution of a differentiable injective random function, we have the Probability density function

π‘“π‘Œ(𝑦)=1√2πœ‹eβˆ’(ln⁑𝑦)2/21𝑦

Properties

  1. Moments: 𝔼⁑[π‘Œπ‘›] =𝔼⁑[e𝑛π‘₯] =𝑀𝑋(𝑛) =eπ‘›πœ‡+12𝑛2𝜎2
  2. Expectation: π‘š =𝔼⁑[π‘Œ] =eπœ‡+12𝜎2
  3. Variance: Var⁑[π‘Œ] =π‘š2(e𝜎2 βˆ’1)

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