Sampling Methods/Monte Carlo method and Log-normal distribution I found a problem from some notes i found online, here is a screenshot:

I am trying to understand this question, it seems this function they define as the LIP() function is basically the quartile/quantile function.
And it seems the question says that the mean does not affect the LIP when we go from a uniform distribution to Log-Normal distribution.
It looks like one has to do some kind of analytical mathematical algebraic or maybe calculus based manipulation.
So the means does not affect the 'shape' of the distribution it seems.
Not sure how or what I should do to the Log-Normal Distribution function.
SO need some help here.
 A: The LIP is not quite equivalent to the quantile function, thanks to that phrase "... fraction of incomes that are below $\alpha$ times the $\beta$ quantile of incomes."  The choice of a Uniform distribution as an example was unfortunate one, as it resulted in an LIP that was indeed equal to the same quantile as the value that resulted from the "below $\alpha$ times" part of the calculation.  
However, if we, for example, choose an Exponential distribution for income with a mean of $1$, and calculate the LIP($1/2$,$1/2$) cutoff, we find that the cutoff is at the $29.3^{rd}$ percentile of the distribution - the $50^{th}$ percentile is at $0.693$, half of that is $0.347$, and $0.347$ happens to be at the $29.3^{rd}$ percentile of the standard exponential.
As @whuber has observed, all that having a nonzero $\mu$ does is rescale the lognormal variate, multiplying it by $\exp\{\mu\}$;  consequently, the resulting LIP remains the same.  This is what you want; the LIP should be unchanged regardless of whether income is measured in dollars or in thousands of dollars, for example.
