Inequalities

Markov’s Inequalities

Let X be a non-negative random variable and suppose that E(X) exists. For any t > 0,
markov's inequality


Chebyshev’s inequality

Let µ = E(X) and σ2 = V(X) then,
Chebyshev
where Z = (X – µ) / σ. In particular P(|Z| > 2) ≤ 1/4 and P(|Z| > 3) ≤ 1/9.


Hoeffding’s Inequality

Theorem 1:
Let Y1, . . . , Yn be independent observations such that E(Yi) = 0 and ai ≤ Yi ≤ bi. Let ε > 0. Then, for any t > 0,
hoeffding

Theorem 2:
Let X1, . . . , Xn ~ Bernoulli(p). Then for any ε > 0,
hoeffding2


Cauchy-Schwarz inequality

If X and Y have finite variances then
Cauchy Schwarz


Jensen’s Inequality

If g is convex then
Eg(X) ≥ g(EX)

If g is concave then
Eg(X) ≤ g(EX)

Leave A Comment