F.3 BIENAYMÉ, CHEBYSHEV, AND MARKOV INEQUALITIES
Theorem F.5 (Bienaymé's inequality). For random variable X and nonrandom c:
(F.16)
for every and .
Proof. Define the translated random variable with probability density function (pdf) fY(y). Since Y and fY(y) are nonnegative:
(F.17)
where the lower limit has been increased from zero. Using the smallest value of y in the integrand and factoring it from the integral gives
(F.18)
Rearranging the last expression completes the proof.
Theorem F.6 (Chebyshev's inequality). For random variable X with mean and variance :
(F.19)
for every .
Proof. ...
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