expectation value inequalities

13 jun expectation value inequalities

Chebychev's inequality for random variables limits the probability that a random variable differs from its expected value by any multiple of its SE. Under the conditions of the previous theorem, for any >0, (1 n Xn i=1 Xi> exp n 2 2(˙2 + =3) Bernstein’s inequality points out an interesting phenomenon: if ˙2 < , then the upper bound behaves like e n instead of the e n 2 guaranteed by Hoe ding’s inequality. See Example 7. where F(x) is the distribution function of X. Notice that if M is a bound in absolute value for f, then −M and M are lower and upper bounds for f, and conversely that if L and U are lower and upper bounds, then max(|L|,|U|) is a bound for f in absolute value. Now, this is nothing more than a fairly simple double inequality to solve so let’s do that. So it is a function of y. The Schwarz inequality applies to quaternions, not quaternion operators. We look at f of x1 up to f of xn. All mistakes are mine. Since our absolute … The expectation of a random variable plays an important role in a variety of contexts. From this bound, it is shown that the difference of expectation values also obeys AWEC- and ANEC-type integral conditions. 3. Hence, taking expectation over equation ( 1) and using linearity of expectation, I obtain. Expectation Values. To relate a quantum mechanical calculation to something you can observe in the laboratory, the "expectation value" of the measurable parameter is calculated. For the position x, the expectation value is defined as This integral can be interpreted as the average value of x that we would expect to obtain from a large number... Hence we have the coincidence experiments e 13, e 14, e 23 and e 24, but instead of concentrating on the expectation values they introduce the coincidence probabilities p 13, p 14, p 23 and p 24, together with the probabilities p 2 and p 4.Concretely, p ij means the probability that the coincidence experiment e ij gives the outcome (o In our specific case, if we know that Y = 2, then w = a or w = b, and the expected value of X, given that Y = 2, is 1 2 X(a)+ 1 2 X(b) = 2. For example, in decision theory, an agent making an optimal choice in the context of incomplete information is often assumed to maximize the expected value of their utility function. If we set a= k˙, where ˙is the standard deviation, then the inequality takes the form P(jX )j k˙) We study a class of stochastic bi-criteria optimization problems with one quadratic and one linear objective functions and some linear inequality constraints. able guess is the expected value of the object. Knowledge of the fact that Y = y does not necessarily reveal the “true” w, but certainly rules out all those w for which Y(w) 6= y. Theorem 1 (Expectation) Let X and Y be random variables with finite expectations. asked Apr 9 '16 at 21:03. james42 james42. 2. The normal curve depends on x only through x 2.Because (−x) 2 = x 2, the curve has the same height y at x as it does at −x, so the normal curve is symmetric about x=0. Starting from the inequality (1) from the set, we continued with other inequalities one after one until all the generated states violate one inequality from the set. A bound in absolute value, which is what we will usually refer to as just a bound, is a number M so that |f(x)| ≤ M for all x. Example: Roll a die until we get a 6. The expectation value of a Bell operator is optimized by considering all possible measurement settings for all observables. This is surprising, since it is well known that the expectation value of T μν u μ u ν in the renormalized Casimir vacuum state alone satisfies neither quantum inequalities nor averaged energy conditions. A natural way to proceed is to find a value ` for which P[Ln `] is “small.” More formally, we bound the expectation as follows ELn `P[Ln <`]+nP[Ln `] `+nP[Ln `], (2.6) for an ` … The following properties of the expected value are also very important. Let be an integrable random variable defined on a sample space . Let for all (i.e., is a positive random variable). Then, Intuitively, this is obvious. The expected value of is a weighted average of the values that can take on. But can take on only positive values. A typical version of the Chernoff inequalities, attributed to Herman Chernoff, can be stated as follows: Theorem 3.1. provided the expectation on the right hand side exists. Let q= p/(p− 1). expectation. variable, the value (X − E[X])k may be negative for odd values k. Therefore Markov’s inequality would not apply. Define a new operator A' based on A whose expectation value is always zero. Expected value or Mathematical Expectation or Expectation of a random variable may be defined as the sum of products of the different values taken by the random variable and the corresponding probabilities. And in general, of course, the expected--this covariance matrix I could express with that E notation. There is one more concept that is fundamental to all of probability theory, that of expected value. 2. The expectation operator has inherits its properties from those of summation and integral. A.2 Conditional expectation as a Random Variable Conditional expectations such as E[XjY = 2] or E[XjY = 5] are numbers. A special case of the Holder inequality. An absorbing state is a state Here we present various concentration inequalities of this flavor. Furthermore, the Lipschitz condition on … That is, E(x + y) = E(x) + E(y) for any two random variables x and y. ... arguments in each expectation value have opposite meaning the value of the expectation value is –1. the Schwarz inequality: E(|XY|) ≤ [E(X2)E(Y2)]1/2. Putting these inequalities together, we have E[Y] 0:999 1 … The Holder inequality follows. These negative energy densities lead to many problems. Cite. By the Holder inequality, Note that the proof holds for any finite dimensions as long as … 2. nonnegative values all terms in the sum giving the expectations are nonnegative we have E[X] = X P(X= ) X a P(X= ) a X a P(X= ) = aP(X a) and thus P(X a) E[X] a: To prove Chebyshev we will use Markov inequality and apply it to the random variable Y = (X E[X])2 which is nonnegative and with expected value E[Y] = E (X E[X])2 = var(X): We have then Then 1/p+ 1/q= 1. One advantage of Markov’s inequality is that the computation of the expectation value is su–cient, so it typically easy to apply. Bounding the expectation of Ln is not straightfor-ward as it is the expectation of a maximum. Hint: by definition, E(X) = E(X +) − E(X −) where X + = ( | X | + X) / 2 and X − = ( | X | − X) / 2. Usually we center the expected value to 0 { we use moments of ( X) = X E(X). In particular, the following theorem shows that expectation preserves the inequality and is a linear operator. The exp value (averaged over all X’s) of the conditional exp value (of Y |X) is the plain old exp value (of Y ). Therefore, one can the same experimental situation as that considered by Bell. Inequalities of this type are known as Bell inequalities, or sometimes, Bell-type inequalities. Expectation is just an integral over the underlying probability space, and integrals respect inequalities. CONDITIONAL EXPECTATION: L2¡THEORY Definition 1. Proof. 2.1 Jensen’s Inequality. That is Follow edited May 27 '19 at 5:29. Tags: expectation expected value inequality probability quadratic function upper bound variance. For the quantum case, we replace the classical stochastic var-iables with hermitian operators acting on a Hilbert space. Lecture 3: Chain Rules and Inequalities Last lecture: entropy and mutual information This time { Chain rules { Jensen’s inequality { Log-sum inequality { Concavity of entropy { Convex/concavity of mutual information Dr. Yao Xie, ECE587, Information Theory, Duke University For example, if a continuous random variable takes all real values between 0 and 10, expected value of the random variable is nothing but the most probable value among all the real … So, with this first one we have, − 10 < 2 x − 4 < 10 − 10 < 2 x − 4 < 10. Share. In quantum field theory, there exist states in which the expectation value of the energy density for a quantized field is negative. ; The positive real number λ is equal to the expected value of X and also to its variance As applications, the convergence rates of the law of large numbers and the Marcinkiewicz–Zygmund-type law of large numbers about the random variables in upper expectation spaces are obtained. E ( a b s ( X + Y) | X) ≥ | E ( X + Y | X) | = | X + E ( Y | X) | = | X |. Thus, as with integrals generally, an expected value can exist as a number in \( \R \) (in which case \( X \) is integrable), can exist as \( \infty \) or \( -\infty \), or can fail to exist.In reference to part (a), a random variable with a finite set of values in \( \R \) is a simple function in the terminology of general integration. Define a new operator A' based on A whose expectation value is always zero. Suppose X It can be very useful if such a prediction can be accompanied by a guarantee of its accuracy (within a certain error estimate, for example). Preservation of almost sure inequalities The re nements will mainly be to show that in many cases we can dramatically improve the constant 10. which implies half of the claim. Vague Expectation Value Loss. suggests the following definition of the expected value E(X) of a random variable X. Markov’s Inequality The expectation can be used to bound probabilities, as the following simple, but fundamental, result shows: Theorem (Markov’s Inequality) If X is a nonnegative random variable and t a positive real number, then PrrX ¥ts⁄ ErXs t: 25/41 Second, A 1 B 2 is always either 1 or − 1. Solve |x + 4| – 6 < 9. We require this value to be at least 9 n: 1 q 1 n 2ˇ 9 n, 1 9 n q 1 n 2ˇ, 1 18 n + 81 n2 1 1 n2 ˇ, n2 2 ˇ 18n 81 , 2 2 ˇ logn log(18n 81) , 2 2 ˇ logn log(18n 81) 1 This inequality holds for all n 2835, as desired. The program is designed to use patterns, modeling, and conjectures to build student understanding and competency in mathematics. 3.1 Expectation The mean, expected value, or expectation of a random variable X is writ-ten as E(X) or µ X. By virtue of the equivalence, ... mistake bound ˚is equivalent to a simple statement about the expected value of ˚with respect to the uniform distribution. expectation is the value of this average as the sample size tends to infinity. There is one more concept that is fundamental to all of probability theory, that of expected value. In this letter, we present a new method to study the tail inequalities for sums of random matrices. Extremal Density Matrices for the Expectation Value of a Qudit Hamiltonian Entropy–energy inequalities for qudit states. Share. Concentration Inequalities 219 Theorem 3. bernstein’s inequality. In this definition, π is the ratio of the circumference of a circle to its diameter, 3.14159265…, and e is the base of the natural logarithm, 2.71828… . So suppose x, y are two non-independent random variables, given that I know the distribution of x p(x) and the distribution of y q(y), how can I find an upper bound on E[|x * y … This identity enables us to extend the definition of integrals of non-negative random variables to integrals of any random variables.) Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Expectation of sum of two random variables is the sum of their expectations. Recall that a random variable X for the experiment is simply a measurable function from (Ω,F) into another measurable space (S,S). 'The positive wealth effect of the upper income segments juxtaposed with the negative income effects of the lower income households tells a story of a very uneven recovery and sharpening inequalities. 6. Let (›,F,P) be a probability space and let G be a ¾¡algebra contained in F.For any real random variable X 2 L2(›,F,P), define E(X jG) to be the orthogonal projection of X onto the closed subspace L2(›,G,P). I use the following graph to remember them. Conditional expectation: the expectation of a random variable X, condi- (Both expectations involve non-negative random variables. Engaging with extant literature on transnational communication, digital and gender inequalities vis-à-vis ICTs, we seek to deepen understanding of the power relations and emotional hierarchies in transnational households and their impact on experiences and outcomes of mediated communication. Define the square of the operator in a way designed to link up with the standard deviation. The standard rule of conditionalization can be straightforwardly adapted to this. Now for the final step: First, A 2 B 1 is always either 1 or − 1. E [ g ( X, Y)] ≥ a + b T ( E [ X], E [ Y]) T = g ( E ( X), E [ Y]) which is the Jensen's inequality in two variables. Lets say we have random variables X, Y and Zdistributed with In X all If we observe N random values of X, then the mean of the N values will be approximately equal to E(X) for large N. The expectation is defined differently for continuous and discrete random variables. If Xi denotes the value of the ith toss, then the expected number of rolls is E XN i=1 Xi = E[N]E[X1] = 10(3.5) = 35. Remarks: Yikes, what the heck is this!? A discrete random variable X is said to have a Poisson distribution, with parameter >, if it has a probability mass function given by:: 60 (;) = (=) =!,where k is the number of occurrences (=,,; e is Euler's number (=! 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. y = (2×π) −½ ×e −x 2 /2. ≤2. If we take x = 1 + A 1 B 2, we get E ( | A 2 B 1 ( 1 + A 1 B 2) |) = E ( | 1 + A 1 B 2 |). take the expected value (and use the tower rule, aka. Then the Law of the Unconscious (5). We also provide equivalence conditions for monogamy and polygamy inequalities of quantum entanglement and quantum discord distributed in three-party quantum systems of arbitrary dimension with respect to q-expectation value for q≥1. |x + 4| – 6 < 9 → |x + 4| < 15. Markov’s inequality and Chebyshev’s inequality place this intuition on firm mathematical ground. In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value Well, we've got n possible outputs, x1 to xn. One interpretation of expectation was, if the random variable is repeated large number of times, then the average is close to the expected value with high probability. Despite being more general, Markov’s inequality is actually a little easier to understand than Chebyshev’s and can also be used to simplify the proof of Chebyshev’s. Definition of Expectation The expectation (also called expected value, or mean) of a random variable X, is the mean of the distribution of X, denoted E(X). Adding these two inequalities together and using that EZ1 A+ EZ1 Ac = EZ, which follows from linearity of expectation for simple random variables (Theorem 1.1), we get E(X1 A+ Y1 Ac) " 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. B B. In several important cases, a random variable from a special distribution can be decomposed into a sum of simpler random variables, and then part (a) of the last theorem can be used to compute the expected value. 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 theorem By introducing the expectation level, the bi-criteria problem is … However, how can we tell how good the expected value is to the actual outcome of the event? Define the square of the operator in a way designed to link up with the standard deviation. This definition may seem a bit strange at first, as it seems not to have any connection with Hello, I am trying to find an upper bound on the expectation value of the product of two random variables. −. CONDITIONAL EXPECTATION 1. 2. If g(x) ≥ h(x) for all x ∈ R, then E[g(X)] ≥ E[h(X)]. 1. The black dotted line stands for B = 2, above which a violation occurs. satisfies the inequality "Expectation(f) > x' if and only if EXP(p; f) > x, and similarly for other inequalities. Although quantum field theory introduces negative energies, it also provides constraints in the form of quantum inequalities (QI's). Proof. − 6 < 2 x < 14 − 3 < x < 7 − 6 < 2 x < 14 − 3 < x < 7. For a different example, in statistics, where one seeks estimates for unknown parameters based on available data, the estimate itself is a random variable. It is often useful to bound the probability that a random variable deviates from some other value, usually its mean. 3,647 10 10 gold badges 22 22 silver badges 33 33 bronze badges. Obviously, the quantity can be only or −2, and thus, the absolute value of its expectation value is bounded by 2 This is the classical Bell bound. In such settings, a desirable criterion for a "good" estimator is that it is unbiased; that is, the expected value of the estimate is equal to the true value of the underlying parameter. But let me just stretch it as far as what would be the expected value of any function of x? AB. Expectation (Moments) MIT 14.30 Spring 2006 Herman Bennett 7 Expected Value 7.1 Univariate Model Let X be a RV with pmf/pdf f (x). expectation Exp[ R 1 + R 2] = Exp[R 1] + Exp[R 2]: 2 Tail Bounds It is usually easy to compute the expectation of the quantities of interest, while analyzing algorithms. For the quantum case, we replace the classical stochastic variables with hermitian operators acting on a Hilbert space. However, Markov’s inequality does not depend on any property of the probability distribution of the random variable. A Gentle Introduction to Concentration Inequalities Karthik Sridharan Abstract This notes is ment to be a review of some basic inequalities and bounds on Random variables. Assume p>1. Kolmogorov's exponential inequalities are basic tools for studying the strong limit theorems such as the classical laws of the iterated logarithm for both independent and dependent random variables. This second level of complexity includes the first as part, since the probability of a proposition A is just the expectation value of its indicator function I(A) which takes value 1 if A is true and value 0 otherwise. Abstract Book Stocholm. Formally, the expected value is the Lebesgue integral of, and can be approximated to any degree of accuracy by positive simple random variables whose Lebesgue integral is positive. Integrated 1 is year one of a three-year high school mathematics sequence. As with equations p p simply represents whatever is inside the absolute value bars. In this section we will study a new object E[XjY] that is a random variable. Isolate the absolute value. 6 Chebyshev’s Inequality: Example Chebyshev’s inequality gives a lower bound on how well is X concentrated about its mean. Proof. The expectation and variance of the Poisson distribution P ... variables is to the expected value, various concentration inequalities are in play. 1 An identical set of tools can be defined for B. Suppose X is B100;1=2 and we want a lower bound on expected value) with an associated probability. We will repeat the three themes of the previous chapter, but in a different order. Section 5 Conditional Expectation Section 6 Inequalities Section 7 General Expectations (Advanced) Section 8 Further Proofs (Advanced) In the first two chapters we learned about probability models, random variables, and distributions. 5. We can not use quantum mechanical expectation value measured in experiments to show the violation of Bell's inequality and then further deny the local hidden-variables theory. As usual, our starting point is a random experiment modeled by a probability space (Ω,F,P). expectation on both sides. This is exactly the role that the concentration inequalities play. We can not use quantum mechanical expectation value measured in experiments to show the violation of Bell's inequality and then further deny the local hidden-variables theory. It's the expected value, the average value of x minus m squared. This is pretty neat and almost directly gives us something called the Weak Law of Large Numbers (but we will return to this). Properties of Expected Value. Expected Value and Markov Chains Karen Ge September 16, 2016 Abstract A Markov Chain is a random process that moves from one state to another such that the next state of the process depends only on where the process is at the present state. Section 5 Conditional Expectation Section 6 Inequalities Section 7 General Expectations (Advanced) Section 8 Further Proofs (Advanced) In the first two chapters we learned about probability models, random variables, and distributions. Markov’s Inequality. When S⊆Rn, we assume that S is Lebesgue measurable, and we take S to the σ-algebra of Lebesgue measurable subsets of S. As noted above, here is the measure-theoreti… is the operator for the x component of momentum. We start with an example. Find the variance in the number of fixed points of a permutation chosen uniformly at random from all permutations. The equality is strict unless X is a constant random variable. Different from other work (Ahlswede & Winter, 2002; Tropp, 2012; Hsu, Kakade, & Zhang, 2012 ), our tail results are based on the largest singular value (LSV) and independent of the matrix dimension. Consider the random variable Z = eλS, where λ is a quantity we will optimize for later. 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 < ∞. If we consider E[XjY = y], it is a number that depends on y. value N = n we can easily derive the conditional expectation of T by E(TjN = n) = Xn i=1 E(X i) = nE(X): (5) Using the theorem of total expectation we get E(T) = X n nE(X)p N(n) = E(X) X n np N(n) = E(X)E(N): (6) Jan Bouda (FI MU) Lecture 3 - Expectation, moments and inequalities March 21, 2012 16 / 56 Look at ways to express the expectation value of A. 2. The normal curve has the form . A typical version of the Cherno inequalities, attributed to Herman Cherno , can be stated as follows: Vague Expectation Value Loss Fraassen, Bas 2004-06-04 00:00:00 Vague subjective probability may be modeled by means of a set of probability functions, so that the represented opinion has only a lower and upper bound. Bell’s inequalities can be violated by a classical system as well. Bell’s theorem shows that no theory that satisfies the conditions imposed can reproduce the probabilistic predictions of quantum mechanics under all circumstances. By Giuseppe Marmo, F. … The subject of this post is Doob’s inequalities which bound the distribution of the maximum value of a martingale in terms of its terminal distribution, and is a consequence of the optional sampling theorem.We work with respect to a filtered probability space. (MU 3.21) A fixed point of a permutation π : [1,n] → [1,n] is a value for which π(x) = x. Expectation value of the Bell operator B (t a, t b, t a ′, t b ′) as a function of ℓ, where the parameters specifying the state of the systems at times t a, t b, t a ′, and t b ′ have been fixed to the values given in the figure. Note that this is a vector space, since • For any X ∈ Lp and a ∈ R, Expected value is also called as mean. Lecture 20 - Conditional Expectation, Inequalities, Laws of Large Numbers, Central Limit Theorem This lecture is based on the materials from the Courant Institute’s Theory of Probability taught by Professor Austin in Spring 2016. can be only 2or−2, and thus, the absolute value of its expectation value is bounded by 2 | C | AB AB AB. Therefore, also the Lebesgue integral of must be positive. For X the number of successes in n trials, this definition makes E(X) = np. Markov's Inequality for random variables limits the probability that a nonnegative random variable exceeds any multiple of its expected value. Glorfindel.

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