expectation of product of random variables inequality

13 jun expectation of product of random variables inequality

2. In this chapter, we look at the same themes for expectation and variance. 3. Note that this is a vector space, since The expected value of Y is E(Y) = X i ( x i)p X(x i): Especially interesting is the power function ( X) = Xk. The multivariate … Expected values obey a simple, very helpful rule called Linearity of Expectation. Expected … Based on these, we establish several strong laws of large numbers for general random variables and obtain the growth rate of the partial sums. Proofs 10 List of Assumptions, Propositions and Theorems ... Let X and Y be real random variables, and let r and s be real positive constants (r >0, s >0). In probability theory, the expected value of a random variable X {\displaystyle X}, denoted E ⁡ {\displaystyle \operatorname {E} } or E ⁡ {\displaystyle \operatorname {E} }, is a generalization of the weighted average, and is intuitively the arithmetic mean of a large number of independent realizations of X {\displaystyle X}. Distribution, expectation and inequalities. In other words, if X and Y are random variables that take different values with probability zero, then the expectation of X will equal the expectation of Y. . In particular, for a random variable . A well defined expectation implies that there is one number, or rather, one constant that defines the expected value. Moment inequality. This is a consequence of the Chebychev’s useful inequality: Theorem (Chebychev’s inequality) If X is a random variable, then PrrpX ErXsq2 ¥ s⁄ VarrXs : (2) 34/41 106, No. The study manual expands on this results on pg 140. viii. This product formula holds for any expectation of a function \(X\) times a function of \(Y\) Proof. Then 1 p ˇlog2 ˙ … Expectation, also called mean, of a random variable is often referred to as the location or center of the random variable or its distribution. Lecture #19: method of indicators, tail sum formula for expectation, Boole's and Markov's inequalities, expectation of g(X). For example the random variable X with P(X= +1) = 1=2; P(X= 1) = 1=2 and the random variable … 1.Introduction Let (Ω, ,P) be a probability space and let (X,Y) be a bivariate random vector defined on it. We develop an inequality for the expectation of a product of nrandom variables gener-alizing the recent work of Dedecker and Doukhan (2003) and the earlier results of Rio (1993). The expected value of the sum of several random variables is equal to the sum of their expectations, e.g., E[X+Y] = E[X]+ E[Y] . The random variable is just a sum of independent random variables , so the expectation of the product is the product of expectations: This last expectation is easy, since we know the law of exactly: It is 1 with probability and 0 with probability : so our initial probability becomes Theorem 1 (Expectation) Let X and Y be random variables with finite expectations. 1. If g(x) ≥ h(x) for all x ∈ R, then E[g(X)] ≥ E[h(X)]. 2. E(aX +bY +c) = aE(X)+bE(Y)+c for any a,b,c ∈ R. 1 Let’s use these definitions and rules to calculate the expectations of the following random variables if they exist. Example 1 1. Bernoulli random variable. 2. We say that the random variable x is (a version of) E(Xk) is known as the kth moment of X. Independence concept. Moments of sums of random variables 7 6. Moments and behavior of tail areas 3 5. E(X + Y) = E(X) + E(Y) if X and Y are random m × n matrices. the average of independent random variables. Bounds on the Expectation of the Maximum of Samples from a Gaussian Gautam Kamath In this document, we will provide bounds on the expected maximum of nsamples from a Gaussian distri-bution. Under certain assumptions the expectation of a product of functions of a random variable is greater (smaller) than the product of expectations. Imagine observing many thousands of independent random values from the random variable of interest. In a first application, a strong law of large … the Cauchy-Schwarz inequality, Theorem 1.2, allows us to bound the ex-pectation of a product of random variables (obviously, unnecessary if they are independent, but non-trivial in general) in terms of the respective second moments. Say we have two Gaussian random vectors p ( x 1) = N ( 0, Σ 1), p ( x 2) = N ( 0, Σ 2), is there a well known result for the expectation of their product E [ x 1 x 2 T] without assuming independence? Yes, there is a well-known result. Based on your edit, we can focus first on individual entries of the array E [ x 1 x 2 T]. 6.2. Now suppose that X is independent of Y, and let g(Y) be any bounded (measurable) function of Y. On the other hand, the expected value of the product of two random variables is not necessarily the product of the expected values. Proof: This is true by definition of the matrix expected value and the ordinary additive property. Let Gbe a sub-s-algebra of F, and let X 2L1 be a random variable. Applying Jensen’s inequality to g(x) = 1/x gives E 1 X > 1 E(X), when X is a non-constant, positive-valued random variable, and that cer-tainly agrees with the calculation in Example 1.1. The conditional expectation In Linear Theory, the orthogonal property and the conditional ex-pectation in the wide sense play a key role. Consider, for example, a random variable X with standard normal distribution N(0,1). Let T ::=R 1 +R 2. 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 < ∞. 4 ... Law of Iterated Expectation with Inequality Conditional. Moment inequalities 1 3. 2. product of two Sub-Gaussian random variables is Sub-Exponential. to a s-algebra, and 2) we view the conditional expectation itself as a random variable. We will detail four fundamental inequalities, go through their proofs, and demon-strate uses. Our basic vector space V consists of all real-valued random variables defined on (Ω, F, P) (that is, defined for the experiment). It is an entrypoint to more 6 The first part is the additive property —the expected value of a sum is the sum of the expected values. … Before we illustrate the concept in discrete time, here is the definition. Active yesterday. Let Y = max 1 i nX i, where X i˘N(0;˙2) are i.i.d. Lecture #20: expectation of g(X,Y), expectation of the product of random variables, variance and standard deviation. Abstract. In this paper, we obtain the equivalent relations between Kolmogorov maximal inequality and Hájek–Rényi maximal inequality both in moment and capacity types in sublinear expectation spaces. For k = 1 we get the expectation of X. Existence of Expectation of Product of Two Random Variables. Out of the framework of Linear Theory, a significant role plays the independence concept and conditional expectation. 1.4 Probability is a Special Case of Expectation Probability is expectation of indicator functions. vii. E(aX +bY +c) = aE(X)+bE(Y)+c for any a,b,c ∈ R. 1 2 <1is equivalent to X belonging to the class of Sub-Gaussian random variables It is easy to show that: kX2k 1 = (kXk 2)2; kXYk 1 kXk 2 kYk 2 Using Orlicz norms allows to straightforwardly implies the following facts: 1. squared Sub-Gaussian random variable is Sub-Exponential. Theorem 1. The classical form of Jensen's inequality involves several numbers and weights. 436 CHAPTER 14 Appendix B: Inequalities Involving Random Variables Remark 14.3 In fact the Chebyshev inequality is far from being sharp. 1. For any random variables R 1 and R 2, E[R 1 +R 2] = E[R 1]+E[R 2]. 7.1. Featured on Meta The future of Community Promotion, Open Source, and Hot Network Questions Ads Towards that goal let us begin by trying to understand the tail behaviour of a random variable. 1. unless X is a constant random variable. Suppose that E(X2)<∞and E(Y2)<∞.Hoeffding proved that Cov(X,Y)= R2 In this chapter, we explore a number of elementary techniques for 5 obtaining both deviation and concentration inequalities. Calculating probabilities for continuous and discrete random variables. The expectation of a random variable is the long-term average of the random variable. To compute it: (from wikipedia again...) If one considers the joint probability density function of ''X'' and ''Y'', say ''j(x,y)'', then the expectation of ''XY'' is Proof. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): We develop an inequality for the expectation of a product of n random variables gener-alizing the recent work of Dedecker and Doukhan (2003) and the earlier results of Rio (1993). See article general inequality in probabilistic setting. 1. § 1.2. Chebychev’s Inequality The random variable X will rarely deviate from the expectation value if the variance is small. Browse other questions tagged pr.probability probability-distributions inequalities expectation cauchy-schwarz-inequality or ask your own question. 0. Using Markov’s inequality, we can now say: Pr[S ≥ µ+δn] = Pr[Z ≥ eλ(µ+δn)] ≤ E[Z]/eλ(µ+δn). Independence. integrable random variables whose product UV is also integrable, then E(UV) ˘EUEV. Note that E ( X i j + Y i j) = E ( X i j) + E ( Y i j) . Hello, I am trying to find an upper bound on the expectation value of the product of two random variables. If X and Y are random variables with matching corresponding We derive sharp probability bounds on the tails of a product of symmetric nonnegative random variables using only information about their first two moments. random variables. 2. Now we invoke Markov’s inequality with , getting. The expectation of the product of X and Y is the product of the individual expectations: \(E(XY ) = E(X)E(Y )\). Axiomatically, two random sets Aand Markov-type inequalities 2 4. Theorem 1 (Expectation) Let X and Y be random variables with finite expectations. the expectation. However, this holds when the random variables are independent: Theorem 5 For any two independent random variables, X1 and X2, E[X1 X2] = E[X1] E[X2]: Corollary 2 If random variables X1;X2;:::;Xk are mutually independent, then E " Yk i=1 to make heavy use of the fact that for independent random variables, the expected value of the product is the product of the expectations. We now start developing the analogous notions of expected value, variance, standard deviation, and so forth with this new class of random variables. Since any constant, and in particular EX, is trivially a function of Y, Definition 1 implies that EX must be the conditional expectation EW (X jY). If we calculate the probability of the normal using a table of the normal law or using the computer, we obtain Let us suppose we have a random variable X and a random variable Y = ( X) for some function . Using the same approach here, we obtain some inequalities for the expectation, E(X ), and cumulative distribution function F(x) of a random variable having the probability density function f : [a, b] --4 lf8 which is assumed to be absolutely continuous and whose derivative f' E L,, (a, b). Its simplest form says that the expected value of a sum of random variables is the sum of the expected values of the variables. Then EX g(Y) ˘EXEg(Y) ˘E(EX)g(Y). Questions requiring the use of this inequality should be expected. Consider the random variable Z = eλS, where λ is a quantity we will optimize for later. In general, the expected value of the product of two random variables need not be equal to the product of their expectations. 1 Markov Inequality The most elementary tail bound is Markov’s inequality, which asserts that for a positive random variable X 0, with nite mean, P(X t) E[X] t = O 1 t : Let ( Ω, P, F) be a probability space, and let E denote the expected value operator. Suppose that are m independent sequences of independent random variables. If Xis a random variable recall that the expected value of X, E[X] is the average value of X Expected value of X : E[X] = X P(X= ) The expected value measures only the average of Xand two random variables with the same mean can have very di erent behavior. The inequality can be stated quite generally using either the language of measure theory or (equivalently) probability. If the mfg of random variable exists in an interval containing the point t = 0 then d n dt n M X t = M x n 0 = E [X n] the n-th moment of X. Chebyshev Inequalities for Products of Random Variables. Thus, Ω is the set of outcomes, F is the σ -algebra of events, and P is the probability measure on the sample space (Ω, F) . We say that X is acontinuous random variable if there exists a continuous probability density function p(x) such that for any interval I on the real line, we have P(X 2I) = R I p(x)dx. is a given integer. Inequality for Expected Value of Product. Theorem 16.1 : For independent random variables X ; Y, we have E( XY )=E( X )E( Y ) . More intuitively: pose Z = (X,Y), Z is a random variable, apply Jensen inequality to Z. If g(x) ≥ h(x) for all x ∈ R, then E[g(X)] ≥ E[h(X)]. In a variety of settings, it is of interest to obtain bounds on the tails of a random 3 variable, or two-sided inequalities that guarantee that a random variable is close to its 4 mean or median. Ask Question Asked yesterday. Expected value, variance, and Chebyshev inequality. E[XjAi]Pr(Ai): Expected value of a product In general, the expected value of the product of two random variables need not be equal to the product of their expectations. Definition 10.1. For example, if they tend to be “large” at the same time, and “small” at The existence and uniqueness of conditional expectation of a positive infinite mean random variable — Durrett 4.1.3. In the probabilistic setting, the inequality can be further generalized to its full strength. 7. Lecture #18: mean vs. mode vs. median, expectation of the sum of random variables, applications. We develop an inequality for the expectation of a product of n random variables generalizing the recent work of J. Dedecker and P. Doukhan [Stochastic Processes Appl. Consider the random variables f: Ω → { 0, 1, 2 } and g: Ω → [ 0, 1], where Ω ⊆ R n. E [ f ( ⋅) g ( ⋅)] := ∫ Ω f ( ω) g ( ω) P ( d ω). Introduction. The expected value is also known as the expectation, mathematical expectation, mean, average, or first moment. see that, rst we look at the expectation of a product of independent r.v.’s (which is a quantity that frequently shows up in variance calculations, as we have seen). Theorem 1.5.

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