In the opposite case, when the greater values of one . Weight this by P ( X = x), to get the product g ( x) P ( X = x). The expectation of Bernoulli random variable implies that since an indicator function of a random variable is a Bernoulli random variable, its expectation equals the probability. PDF Random Variables: Distribution and Expectation The volatility is the square root of the variance. Tchebychev's inequality asserts that if a random variable X has expected value „ †Tchebychev's inequality then, for each †>0, PfjX ¡„j>†g•var.X/=†2 The inequality becomes obvious if we define a new random variable Z that takes the value 1 when jX ¡„j>†, and 0 otherwise: clearly Z •jX¡„j2=†2, from which it follows . To motivate the de nition of the inner product given above, rst consider the case when the probability space is nite. In general, the expected value of the product of two random variables need not be equal to the product of their expectations. Definition 13.3 (expectation): The expectation of a discrete random variable X is defined as E(X)= åa2A Theorem 1 (Chebyshev's Inequality). A while back we went over the idea of Variance and showed that it can been seen simply as the difference between squaring a Random Variable before computing its expectation and squaring its value after the expectation has been calculated. Abstract: We develop an inequality for the expection of a product of n random variables generalizing the recent work of Dedecker and Doukhan (2003) and the earlier results in Rio (1993). to a s-algebra, and 2) we view the conditional expectation itself as a random variable. Imagine observing many thousands of independent random values from the random variable of interest. Image by author. Lecture #19: method of indicators, tail sum formula for expectation, Boole's and Markov's inequalities, expectation of g(X). PDF Chapter 3: Expectation and Variance - Auckland STA 711: Probability & Measure Theory Robert L. Wolpert 5 Expectation Inequalities and Lp Spaces Fix a probability space (Ω,F,P) and, for any real number p > 0 (not necessarily an integer) and let \Lp" or \Lp(Ω,F,P)", pronounced \ell pee", denote the vector space of real-valued (or sometimes complex-valued) random variables X for which E|X|p < ∞.