pondering Chebyshev’s inequality

Chebyshev’s inequality states that the probability that a random variable falls within k standard deviations of the mean of a probability distribution is at least

k

Checking this out in R for an arbitrary gamma distribution yields:

ci_console

We can compare the two areas by first plotting the area under the gamma curve within k standard deviations of the mean:

ci_area_01

and then superimposing the area specified by Chebyshev’s inequality:

ci_area_02a

Here we see that the area under the gamma curve within k standard deviations of the mean is greater than the limit specified by Chebyshev’s inequality.

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