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(This article was first published on <strong><a href="https://xianblog.wordpress.com/2018/01/03/correlation-for-maximal-coupling/"> R – Xi'an's Og</a></strong>, and kindly contributed to <a rel="nofollow" href="https://www.r-bloggers.com/">R-bloggers)</a> </div>
<p><a rel="nofollow" href="https://xianblog.files.wordpress.com/2017/12/maxima.jpg" target="_blank"><img data-orig-file="https://xianblog.files.wordpress.com/2017/12/maxima.jpg" data-orig-size="480,480" data-medium-file="https://xianblog.files.wordpress.com/2017/12/maxima.jpg?w=300" data-attachment-id="35098" data-image-meta="{" data-image-description="" src="https://xianblog.files.wordpress.com/2017/12/maxima.jpg?w=450&h=450&fit=450%2C450&resize=450%2C450" data-permalink="https://xianblog.wordpress.com/2018/01/03/correlation-for-maximal-coupling/maxima/" data-large-file="https://xianblog.files.wordpress.com/2017/12/maxima.jpg?w=450&amp;h=450&amp;fit=450%2C450&amp;resize=450%2C450" alt="" data-image-title="maxima" sizes="(max-width: 450px) 100vw, 450px" width="450" data-comments-opened="1" class="wp-image-35098" data-recalc-dims="1" height="450"></a><strong>A</strong>n interesting (if vaguely formulated) <a rel="nofollow" href="https://stats.stackexchange.com/q/320272/7224" target="_blank">question on X validated</a>: given two Gaussian variates that are maximally coupled, what is the correlation between these variates? The answer depends on the parameters of both Gaussian, with a correlation of one when both Gaussians are identical. <a rel="nofollow" href="https://stats.stackexchange.com/a/320305/7224" target="_blank">Answering the question</a> by simulation (as I could not figure out the analytical formula on Boxing Day…) led me back to <a rel="nofollow" href="https://statisfaction.wordpress.com/2017/09/06/sampling-from-a-maximal-coupling/" target="_blank">Pierre Jacob’s entry</a> on the topic on <a rel="nofollow" href="https://statisfaction.wordpress.com/2017/09/06/sampling-from-a-maximal-coupling/" target="_blank">Statisfaction</a>, where simulating the maximal coupling stems from the decompositions</p>
<p>p(x)=p(x)∧q(x)+{p(x)-p(x)∧q(x)}  and  q(x)=p(x)∧q(x)+{q(x)-p(x)∧q(x)}</p>
<p>and incidentally to the R function image.plot (from the R library fields) for including the side legend.</p>
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