TITLE: Visualizing Markov and Chebyshev inequalities
QUESTION [5 upvotes]: I am helping a class on introductory probability covering Markov and Chebyshev's inequalities.  I would like to give the students a nice visualization for why they are true or at least to show what they mean.  What would be a good way to do this?

REPLY [9 votes]: Fix a non-negative integer random variable $X$.  Plot $\Pr[X\ge x]$ as a function of $x$.  By splitting the area under the curve into horizontal rectangles, we see that the area under curve is $\sum_x \Pr[X\ge x] = \sum_x x\cdot \Pr[X=x] = E[X]$.  For any particular value of $x$, the (shaded) rectangle with width $x$ and height $\Pr[X\ge X]$ fits under the curve.  We conclude that $x\cdot \Pr[X\ge x] \le E[X]$.
The same argument works for continuous random variables; the only difference is that the curve may be smooth.