Does anybody know the mean of log(x+c) if x is log-normally distributed and c is any constant?

Question

X ~ log-N(mu, sigma), so log(x) is normally distributed with mean mu and variance sigma^2

But what´s about y=log(x+c) with the exp(y) following the so called shifted log-normal distribution? Is there any closed-form for the distribution of y? And more important: how can I calculate the mean of y?

Please help -- I have googled but could not find the answer. Hopefully I can get here an answer, if possible also with references. Thank you!!!

Morewood · Accepted Answer

First, may we assume c≥0?  Otherwise, "y" is undefined with positive probability...

The cumulative probability distribution for y would be:

Prob( Y<y ) = Prob ( Log(X+c)<y )
= Prob( X < Exp(y)-c )
...
= Prob ( (Log(X)-μ)/σ < ... ) , but Z=(Log(X)-μ)/σ is standard normal.
= Φ( ... ) ,   where Φ( ) is the standard normal cumulative distribution function.

To find the mean of Y, you want the probability density function, which is just the derivative:

Exp(-(...)²/2) / √(2π) × (d/dy)(...)

The mean of Y is the integral (from -∞ to ∞) of "y" times that:

(Mean of Y) = ∫ y × Exp(-(...)²/2) / √(2π) × (d/dy)(...) dy

which does NOT look nice to me!  (Even with the obvious substitution: z= ... )

elewa · Answer

best way is to apply GDC. you receives the naswer right away. without using calculator, you've gotten to apply tables and word the cost of c resembling 0.seventy 2. it is going to likely be (x -4.2)/a million.03. Now artwork algebra backwards and locate X.

Does anybody know the mean of log(x+c) if x is log-normally distributed and c is any constant?

2 Antworten