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(This article was first published on <strong><a href="https://xianblog.wordpress.com/2016/05/30/the-random-variable-that-was-always-less-than-its-mean/"> R – Xi'an's Og</a></strong>, and kindly contributed to <a rel="nofollow" href="http://www.r-bloggers.com/">R-bloggers)</a> </div>
<p><strong>A</strong>lthough this is far from a paradox when realising why the phenomenon occurred, it took me a few lines to understand why the empirical average of a log-normal sample is apparently a biased estimator of its mean. And why the biased plug-in estimator does not appear to present a bias. The picture below compares two estimators of the mean of a log-normal LN(0,σ²) distribution when σ² increases: blue stands for the empirical mean, while gold corresponds to the plug-in estimator exp(σ²/2) when σ² is estimated from the log-sample. (The sample is of size 10⁶.)</p>
<p><a target="_blank" href="https://xianblog.files.wordpress.com/2016/05/ragnor.jpg"><img height="450" alt="" width="450" class="wp-image-30730" src="https://xianblog.files.wordpress.com/2016/05/ragnor.jpg?w=450&h=450"></a>The question came on <a target="_blank" href="http://stats.stackexchange.com/a/214743/7224">X validated</a> and my first reaction was to doubt the implementation which outcome was so counter-intuitive. But then I thought about the representation of a log-normal variate as exp(σξ) when ξ is a standard Normal variate. When σ grows large enough, it is near impossible for σξ to be larger than σ². More precisely,</p>
<p>P(X>E[X])=P(σξ>σ²/2)=1-Φ(σ/2)</p>
<p>which can be arbitrarily small.</p>
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