Chebyshev's Inequality : To find upper and lower bounds of probabilitiesP [|X-E(X)|> k σ] and l P [|X-E(X) |≤ kσ].

 

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                                                         Unit...1.3

CHEBYCHEVE’S INEQUALITY                                                                               

                                                                      

                      One of nine children, Chebyshev was born in the central Russian village of Akatovo near Borovsk, to Agrafena Ivanova Pozniakova and Lev Pavlovich Chebyshev. His father had fought as an officer against Napoleon Bonaparte’s invading army. Chebyshev was originally home schooled by his mother and his cousin, Avdotia Kvintillianova Soukhareva. He learned French early in life, which later helped him communicate with other mathematicians. A stunted leg prevented him from playing with other children, leading him to concentrate on his studies instead. Chebyshev studied at the college level at Moscow University, where he earned his bachelor’s degree in 1841. At Moscow University, Chebyshev was a graduate student of Nikolai Brashman. After Chebyshev became a professor of mathematics in Moscow himself, his two most illustrious graduate students were Andrei Andrevevich Markov (the elder) and Aleksandr Lyapunov. Later he moved to St. Petersburg, where he founded one of the most important schools of mathematics in Russia, and there is today a research institute in mathematics called the Chebyshev Laboratory. His name can be alternatively transliterated as Chebychev, Chebysheff, Chebyshov, Tchebychev or Tchebycheff or Tchebyschev or Tchebyscheff (the latter two pairs are French and German transcriptions). Chebyshev is considered to be a founding father of Russian mathematics. According to the Mathematics Genealogy Project, Chebyshev has 7,483 mathematical "descendants" as of 2010.

1.3.1  INTRODUCTION:

               The theorem was first stated without proof by Bienaymé in 1853 and later proved by Chebyshev in 1867. His student Andrev Markov provided another proof in his 1884 Ph.D. thesis.

                          If we know probability distribution of random variable X then we can compute its mean and variance. Conversely if E(X) and V(X) are known then it is not possible to construct probability distribution of random variable X. In other words, hence it is not possible to evaluate, P [|X-E(X) |≤ k]. But E(X) and V(X) can be used to find bounds for such probabilities.

                      Chebychev’s inequality is used to find such bounds. It is used to find upper bound of P [|X-E(X)|> k σ] and lower bound of a P [|X-E(X) |≤ kσ].

1.3.2 Statement of Chebychev’s Inequality:

                         If X is any random variable with mean µ and variance σ2 respectively, then for any positive constant k,

  

Proof:

Case-I: Let X is a continuous random variable with p. d. f. f (x), -∞ < x <∞, with E(X) = µ and V(X) = σ2.

By definition,

                          σ2 =  E[X - E(X)] 2 = E [X - µ] 2

                           

Case-II: Let X is a discrete random variable with p. m. f. p(x) and with E(X) = µ and                    V(X) = σ2. 

             By definition, σ2= E[X - E(X)] 2 = E [X - µ] 2

                                                     

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