Inégalités probabilistes et indépendance. Inégalité de Markov. 1. Rappelez l'inégalité de Bienaymé-Tchebychev et redémontrez-la à partir de l'inégalité de
Inégalités de Markov et de Bienaymé-Tchebychev loi des grands nombres. Exercice 1. 1/6 en proba (loi faible) ou p.s. (loi forte).
tions of Markov's lectures on the theory of probability. These notes forme Romanovskii V. I.
25/10/1999 Mots Clés: Forme de Dirichlet Inegalite de deviation
15/02/2010 Inégalité de Markov. Elle est aussi appelée de Tchebychev de Bienaymé-Tchebychev (prouvée vers 1869)
16/01/2009 Markov semigroups appear naturally in the study of Markov processes where the probability measure Pt(x
Definition 1 (Markov chain). A Markov chain with parameters (X P
Inégalités de Markov et de Tchebychev. 1. Inégalité de Markov. 2. Inégalité de Tchebychev. 3. Inégalité exponentielle. 4. Inégalité de concentration.
4.3.5 Inégalité de Markov et de Bienaymé Tchebychev . En utilisant le théorème d'extension de Carthéodory on montre qu'il existe une unique proba-.
Keywords: Markov chains; nonreversible chains; rates of convergence I could see how to derive approximations using combinatorial probability ...
Markov's inequality Proposition 15 3 (Markov's inequality) Suppose X is a nonnegative random variable then for any a > 0 we have P (X > a) 6 E X a 198 15 PROBABILITY INEQUALITIES Proof We only give the proof for a continuous random variable the case of a discrete random variable is similar
Use Markov’s inequality to compute upper bounds on Pr[X 2] Pr[X 3] Pr[X 4] Now compute the probabilities directly and compare them to the upper bounds Pr[X 2] = Pr[X 3] = Pr[X 4] = (The point is that sometimes Markov’s gives exact bounds (above) but other times they are loose ) The answers are the in BT book 2
First Moment Method One use of Markov’s inequality is to use the expectation to control the probability distribution of a random variable For example let X be a non-negative random variable; if E[X] < t then Markov’s inequality asserts that Pr[X ‚ t] • E[X]=t < 1 which implies that the event X < t has nonzero probability The next
1 Markov’s Inequality Recall that our general theme is to upper bound tail probabilities i e probabilities of the form Pr(X cE[X]) or Pr(X cE[X]) The rst tool towards that end is Markov’s Inequality Note This is a simple tool but it is usually quite weak It is mainly used to derive stronger tail bounds such as Chebyshev’s Inequality
)Discrete state discrete time Markov chain 1 1 One-step transition probabilities For a Markov chain P(X n+1 = jjX n= i) is called a one-step transition proba-bility We assume that this probability does not depend on n i e P(X n+1 = jjX n= i) = p ij for n= 0;1;::: is the same for all time indices In this case fX tgis called a time
1 –Inégalité de Markov Proposition 10 1 – Inégalité de Markov Soit X une variable aléatoire positive (discrète ou à densité) admettant une espérance Alors pour tout réel a strictement positif on a P(X >a) 6 E(X) a Remarque 10 2 – On a également P(X ¨a) 6 E(X) a Corollaire 10 3 Soit X une variable aléatoire (discrète ou