While looking at Tao’s post on the law of large numbers, I claimed to Alap that Markov’s inequality was “obvious”. Asked to back that up, though, I couldn’t remember *why* it was obvious. Take some distribution . For simplicity, assume this distribution is over the non-negative reals. Then, Markov’s inequality states, for

.

Though I find it more intuitive in the form

.

Now, a proof is pretty trivial, basically by writing down the definitions of both sides.

However, I think more intuition can be conveyed with pictures.

Matlab Code:

function markov_inequality fsize = 16; lambda = 5; x = 0:.01:15; p = .5*exp(-(x-3).^2) + exp(-(x-7).^2/4); p=p/sum(p)*100; figure(1) plot(x,p,'k-','LineWidth',2) xlim([min(x) max(x)]) ylim([0 max(x.*p)+.25]) legend('p(x)'); set(gca,'FontSize',fsize) figure(2) plot(x,p.*x,'k-','LineWidth',2); xlim([min(x) max(x)]) ylim([0 max(x.*p)+.25]) legend('x p(x)'); set(gca,'FontSize',fsize) figure(3) plot(x,p.*x,'k-',x,p.*lambda,'r-','LineWidth',2); xlim([min(x) max(x)]) ylim([0 max(x.*p)+.25]) hold on; legend('x p(x)', '\lambda p(x)'); title('\lambda=5'); set(gca,'FontSize',fsize) figure(4) area(x,x.*p,'FaceColor',[.7 .7 .7],'LineWidth',2) hold on; good = find(x>=lambda); area(x(good),p(good)*lambda,'FaceColor',[.8 0 0],'LineWidth',2) hold off legend('E[x]','\lambda P[x \geq \lambda]') xlim([min(x) max(x)]) ylim([0 max(x.*p)+.25]) set(gca,'FontSize',fsize) end

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Excellent.This is first time I have seen an intuitive explanation of Markov’s inequality.

Well done!