Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

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. ...