# Safe values of $\mu$ and $\sigma$ when randomly sampling from a Log-Normal Distribution?

I believe I'm obtaining overflow errors when randomly sampling from a log-normal distribution with the command:

RandomVariate[LogNormalDistribution[μ, σ], 1]


Specifically, I can obtain an accurate looking distribution with values of $\mu \approx 1$ and $\sigma \approx 1$, but I get significant outliers for values of $\mu > 10$ and $\sigma > 10$. Why would this occur? Isn't it true that the ratio $\dfrac{\mu}{\sigma}$ should govern the probability of obtaining values $\gg \mu$?

What values of $\mu$ and $\sigma$ will give accurate values properly reflecting a random sample from LogNormalDistribution?

• What definition of "outlier" are you using? A simple "visual test" usually does not work here. Try to generate your data, compute the IQR (interquartile range) and then you can effectively say if the observation is an outler... – Rod May 13 '13 at 6:04
• Have you compared a histogram of your random variates with the PDF of your distribution? – J. M. will be back soon May 13 '13 at 6:07

If $X\sim N\left(\mu ,\sigma ^2\right)$ and $Y=e^X$, then $Y\sim \text{Lognormal}(\mu ,\sigma )$. So, by selecting LogNormalDistribution[10, 10], you are effectively generating values from a $N(10, 100$) distribution (which is a very large variance), and then raising them to $e^X$ ... which will generate deliciously large variates.

To see this:

Here are 6 values generated from a Lognormal, with a given random seed:

SeedRandom;
RandomVariate[LogNormalDistribution[10, 10], 6]


{0.886492, 8.54449*10^7, 0.0194899, 1.42431*10^6, 1572.01, 8.07229*10^8}

... and here are the same 6 values generated from the associated Normal, given the same random seed, and raised to $e^X$:

SeedRandom;
Exp[RandomVariate[NormalDistribution[10, 10], 6]]


{0.886492, 8.54449*10^7, 0.0194899, 1.42431*10^6, 1572.01, 8.07229*10^8}

In summary: there is nothing wrong with the values being generated ... they are 'safe' /// you are just getting what you asked for.

• But why would $\mu = 1000$ and $\sigma = 2000$ give me values $>10^{1269}$ when $\mu = 1$ and $\sigma = 2$ barely breaks $500$ for its largest value for $\approx 10^3$ samples? This scaling doesn't seem reasonable? – B.E. May 13 '13 at 6:50
• Ahh - I think you are doing a plot of the pdf with something like {x, 0, 1000}, and it seems to disappear to 0 for x > 500?? That is a bit deceptive ... the tails are very very very long ... Plot from {x, 1000, 2000} and you will see it is still positive ... forever and ever and ever ... :) – wolfies May 13 '13 at 6:57
• @B.E.: You increase mu by a factor of 1000, so the output increases by a factor of e^1000. What is so surprising about that? – Aditya Jun 19 '13 at 3:44