6.7 Markov and Chebyshev go to the casino
In this section we work through the math from estimating π in section 1.5
There are two important inequalities involving expected values, variances, and error terms. Let X be a finite random variable with distribution so that
E(X) = p1 x1 + p2 x2 + ··· + pn xn
and each xk ≥ 0.
Markov’s Inequality: For a real number a > 0,
P(X > a) ≤ E(X)/a
In Markov’s Inequality, the expression P(X > a) means ‘‘look at all the xk in X and for all those where xk > a, add up the pk to get P(X > a).’’
Question 6.7.1
Show that Markov’s Inequality holds for the distribution at the beginning of this section for the values a = 3 and a = 10.
Andrey Markov, circa 1875. Photo is in the public domain.
