expectation of product of random variables inequality

We develop an inequality for the expectation of a product of random variables generalizing the recent work of Dedecker and Doukhan (2003) and the earlier results of Rio (1993). Markov's inequality. We start with an example. Recall that the expected value of a random variable \(X\) is defined by $$ E[X] = \sum_{x} {xp(x)} $$ where \(X\) is a discrete random variable with probability mass function \(p(x)\), and by $$ E[X] = \int_{-\infty}^{\infty . probability - Expectation over a max operation - Cross Validated The Cauchy-Schwarz Inequality implies the absolute value of the expectation of the product cannot exceed | σ 1 σ 2 |. Some inequalities for the expectation of a product of functions of a ... Expected Value of Random Variables — Explained Simply A random variable X: S → R is called continuous if the probability Q it induces is such that there is some f: R → [ 0, ∞) for which. X and Y, such that the final expression would involve the E (X), E (Y) and Cov (X,Y). Then, E[XY] = P!2 X(!)Y(!)P(! Suppose that E(X2)<∞and E(Y2)<∞.Hoeffding proved . variance of sum of correlated random variables Remarks The expectation is a one-number summary of a distribution. She is interested to see if the synthia cells can survive in cold conditions. In particular, if Z = X + Y, then. Example 1.3 (Lq Lp for q > p >1). 5 4 = E ( max ( X, c)) > max ( E ( X), c) = 1. Rio-type inequality for the expectation of products of random variables Properties of Expectation. Choose τ ∈ (0, 1) so that f(x) = τxalogx and g(x) = xa have equal integrals with respect to μ. In this paper we investigate whether there are anal-ogous notions for random variables with values in a local field (that is, After defining an inner product on the set of random variables using the expectation of their product, If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the lesser values (that is, the variables tend to show similar behavior), the covariance is positive. Applying Holder's inequality to the product of random variables¨ jXjp 1 with conjugate variables p0, q p >1 . Expectation of product of independent random variables. To motivate the de nition of the inner product given above, rst consider the case when the probability space is nite. Since Y 0, by Markov's inequality Pr(jX E(X)j ˙X) = Pr(Y 2E(Y)) 1 2: Intuitively, the probability of a random variable being k standard deviations from the mean is 1=k2. Upper bound on expectation value of the product of two random variables PDF Rio-type inequality for the expectation of products of random variables Unlike expectation, variance is not linear. Before we illustrate the concept in discrete time, here is the definition. 3. • Expectation is a linear operator on L1(P), This means that E(aX +bY) = aEX +bEY. Expectation and conditional expectation of real-valued random vari-ables (or, more generally, Banach space-valued random variables) and the corresponding notion of martingale are fundamental objects of probability theory.

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