Continuous random variable

Log-normal distribution

A log-normally distributed random variable 𝑌 ∼LN(𝜇,𝜎2) has 𝑌 =e𝑋 for 𝑋 ∼N(𝜇,𝜎2), #m/def/prob i.e. ln⁡𝑌 has a normal distribution. Applying Distribution of a differentiable injective random function, we have the Probability density function

Properties

1. Moments: 𝔼⁡[𝑌𝑛] =𝔼⁡[e𝑛𝑥] =𝑀𝑋(𝑛) =e𝑛𝜇+12𝑛2𝜎2
2. Expectation: 𝑚 =𝔼⁡[𝑌] =e𝜇+12𝜎2
3. Variance: Var⁡[𝑌] =𝑚2(e𝜎2 −1)

#state/tidy | #lang/en | #SemBr