Expectation Definition and 689 Threads

Expectation damages are damages recoverable from a breach of contract by the non-breaching party. An award of expectation damages protects the injured party's interest in realising the value of the expectancy that was created by the promise of the other party. Thus, the impact of the breach on the promisee is to be effectively "undone" with the award of expectation damages.The purpose of expectation damages is to put the non-breaching party in the position it would have occupied had the contract been fulfilled. Expectation damages can be contrasted to reliance damages and restitution damages, which are remedies that address other types of interests of parties involved in enforceable promises.The default for expectation damages are monetary damages which are subject to limitations or exceptions (see below)
Expectation damages are measured by the diminution in value, coupled with consequential and incidental damages.

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  1. V

    The expectation of 'z' and 'x+iy'

    can anyone give me any ideas on how to evaluate this: <z>=<\Phi1|z|\Phi2> (for say hydrogen wavefunctions). Similarly <x+iy>=<\Phi1|x+iy|\Phi2> FYI, I'm trying to understand how radiation is polarised (an external B field polarises radiation, so we must consider the dipole...
  2. P

    What Is E{x|x+y+z=1} for Independent Standard Normal Variables?

    Given x,y and z are standard normal distributions with mean 0 and standard deviation 1. x,y and z are also statistically independent. Find E{x|x+y+z=1}.
  3. G

    Expectation of Negative Binomial Distribution

    I am re-writing up some lecture notes and one of the proofs that E[X] for the negative binomial is r/p where r is the number of trials...The problem is there are a number of books that say r(1-p)/p is the correct expectation whilst others agree with 1/p Which one is correct...for what its...
  4. R

    Expectation for # Exchanges (Quicksort Algorithm)

    Hi I am reading Hoare's original paper where he derives the complexity of quicksort. I am trying to figure how he derives the expectation for the number of exchanges (sorry if this is a very CS-specific question): \frac{(N-r-1)(r-1)}{N} \frac{N}{6}+\frac{5}{6N} I can't see...
  5. P

    Find Expectation Value for Particle Moving in N Steps of Length L

    Homework Statement A particle moves in a sequence of steps of length L. The polar angle \theta for each step is taken from the (normalized) probability density p(\theta). The azimuthal angle is uniformly distributed. Suppose the particle makes N steps. My question is how do I find the...
  6. N

    How Do Quantum Operators Affect Position and Momentum Expectations?

    I'm trying to evaluate the expectation of position and momentum of \exp\left(\xi (\hat{a}^2 - \hat{a}^\dag^2)/2\right) e^{-|\alpha|^2} \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} |n\rangle} where \hat{a},\hat{a}^\dag are the anihilation/creation operators respectively. Recall \hat{x}...
  7. B

    Need help with expectation value

    Homework Statement I have a random, uniformly distributed vector with Cartesian components x,y,z. I should calculate the expectation value of the products of the components, e.g. <x\cdot x>, <x\cdot y>, ..., <z\cdot z>. Homework Equations In spherical coordinates the x,y,z components...
  8. B

    How Do You Calculate E(Y) and Var(Y) for Independent Random Variables?

    Homework Statement Say X has a density f(x) = 3x^(-4) if x > 1, and 0 otherwise. Now say X1,...,X16 are independent with density f. Let Y = (X1X2...X16)^(1/16). Find E(Y) and Var(Y). Homework Equations Var(Z) = E(Z^2) - [E(Z)]^2 E(Z) = Integral from -inf to +inf of z*f(z)dz The...
  9. C

    Conditional expectation and variance

    Let X, Y be independent exponential random variables with means 1 and 2 respectively. Let Z = 1, if X < Y Z = 0, otherwise Find E(X|Z) and V(X|Z). We should first find E(X|Z=z) E(X|Z=z) = integral (from 0 to inf) of xf(x|z). However, how do we find f(x|z) ?
  10. A

    How Is Conditional Expectation Derived in Normal Distributions?

    Help me in conditional expectation Hi all.. I read one article couple days ago, yet, there is some equations that I could not understand. let assume that y = u + v where u is normally distributed with mean = 0 and variance = s -> u ~ N (0, s) and v is normally distributed with mean =...
  11. J

    Is the formula for conditional expectation valid for multiple random variables?

    [SOLVED] Conditional Expectation I'm trying to understand the following proof I saw in a book. It says that: E[Xg(Y)|Y] = g(Y)E[X|Y] where X and Y are discrete random variables and g(Y) is a function of the random variable Y. Now they give the following proof: E[Xg(Y)|Y] = \sum_{x}x g(Y)...
  12. T

    In an experiment, do we measure the eigenvalue or expectation value?

    In an experiment, do we measure the eigenvalue or the expectation value ? If both can be measured, how can we distinguish one from another ?
  13. J

    What is the relationship between two random variables X and Y?

    I have a random variable Y that represents the size of a population. I know that the expectation E(Y) = a. Now suppose, I have another random variable X that represents the number of people in that population that have a certain disease. The expectation is that on average half the population...
  14. 3

    Expectation value of P^2 for particle in 2d box

    [SOLVED] expectation value of P^2 for particle in 2d box I am having difficulty in finding the right way to find this value. my book only give the 1d momentum operator as: ih(bar)*d/dx(partials). i see its much like finding the normalization constant. which i have done using a double integral...
  15. R

    Expectation Values of Spin Operators

    [SOLVED] Expectation Values of Spin Operators Homework Statement b) Find the expectation values of S_{x}, S_{y}, and S_{z} Homework Equations From part a) X = A \begin{pmatrix}3i \\ 4 \end{pmatrix} Which was found to be: A = \frac{1}{5} S_{x} = \begin{pmatrix}0 & 1 \\ 1 & 0...
  16. M

    Expectation of Normal Distribution

    Let Y = a + bZ + cZ2 where Z (0,1) is a standard normal random variable. (i) Compute E[Y], E[Z], E[YZ], E[Y^2] and E[Z^2]. HINT: You will need to determine E[Z^r], r = 1, 2, 3, 4. When r = 1, 2 you should use known results. Integration by parts will help when r = 3, 4. I am struggling with the...
  17. P

    Expectation of maximum of a multinormal random vector

    Hello, This time my question is not about Catalan numbers but something much more interesting (to me at least;)) I was wondering how the maximum of a multinormal random vector is distributed, for example let X \approx N(\mu_1,\sigma_1^2) Y \approx N(\mu_2,\sigma_2^2) be normally...
  18. H

    Angular Momentum Expectation Values help for noobie

    For a particle in the state Y(l=3, m=+2), how do I find <Lx^2> + <Ly^2> ? I'm lost. THanks!
  19. M

    Finite expectation value <-> finite sum over Probabilties

    Homework Statement If X is a real valued random variable with E[|X|] finite. <-> \sum(P(|X|>n)) finite , with the sum over all natural numbers from 1 to infinity. Homework Equations As a tip I am given that for all integer valued X>0 E(X) = \sum(P(X)>k , where the sum goes over all k =1 to...
  20. C

    Calculating Expectation Value of Particle in Square Potential Well

    I'm confused re a particle of energy E < V inside a square potential of width 'a' centered at x = 0 with depth V. They give the wavefunction for outside the well as \Psi(x) = Ae^{k|x|} for |x| > a/2 and k^2 = -\frac{2ME}{\hbar^2} => k = i\frac{\sqrt{2ME}}{\hbar} ? I need the probability that...
  21. T

    Expectation value using ladder operators

    I wonder if someone could examine my argument for the following problem. Homework Statement Using the relation \widehat{x}^{2} = \frac{\hbar}{2m\omega}(\widehat{A}^{2} + (\widehat{A}^{+})^{2} + \widehat{A}^{+}\widehat{A} + \widehat{A}\widehat{A}^{+} ) and properties of the ladder operators...
  22. N

    Expectation of an Hermitian operator is real.

    Homework Statement WTS \langle \hat{A} \rangle = \langle \hat{A} \rangle^\ast The Attempt at a Solution \langle \hat{A} \rangle^\ast = \left(\int \phi_l^\ast \hat{A} \phi_m dx\right)^\ast=\left(\int (\hat{A}\phi_l)^\ast \phi_m dx\right)^\ast= \int \phi_m^\ast \hat{A}\phi_l dx. So...
  23. T

    Expectation value for a superposition

    Homework Statement u(x) = \sqrt{\frac{8}{5}}\left(\frac{3}{4}u_{1}(x)-\frac{1}{4}u_{3}(x)\right) Determine the time-dependent expectation value of position of this wave function (the particle is in an infinite potential well between x = 0 and x = a). The Attempt at a Solution I...
  24. C

    How to Calculate Expectation Value Using Trace?

    Homework Statement How do I get the expectation value of operator \sigma using density matrix \rho in a trace: Tr\left(\sigma\rho\right) I have \sigma and \rho in matrix form but how do I get a number out of the trace?
  25. W

    What is the proof for the expectation value of a quantum system?

    I am trying to show that \frac{d}{dt}<x^2>=\frac{1}{m}(<xp>+<px>)....(1) With the wavefunction \Psi being both normalized to unity and square integrable Here is what I tried... <xp> = \int_{-\infty}^{\infty}{\Psi}^*xp{\Psi}dx <px> =...
  26. T

    QM Harm. Osc. expectation values

    QM Harmonic Oscillator, expectation values Hello. I am working on a problem involving the 1-dimensional quantum harmonic oscillator with energy eigenstates |n>. The idea of the exercise is to use ladder operators to obtain the results. I feel I am getting a reasonably good hang of this, but my...
  27. C

    Expectation Value Homework: Integrating Gaussian Distribution

    Homework Statement Can somebody help me integrate \int{x\cdot p(x)} where p(x) is the Gaussian distribution (from here http://hyperphysics.phy-astr.gsu.edu/hbase/math/gaufcn.html) The Attempt at a Solution I can't really get anywhere. It's true that \int{e^{x^2}} has no analytical...
  28. R

    How Does the Expectation of a Normal Variable Derivation Work?

    Hi More of a general integration question, but I just saw the following proof for the derivation of the expectation of a normal variable: E[X] = \frac{1}{\sqrt{2\pi\sigma^2}}\int_{-\infty}^{\infty}{x exp\left( -\frac{1}{2\sigma^2}(x-\mu)^2 \right) dx} Set z=(x-mu): E[X] =...
  29. R

    Physical Meaning of QM Expectation Values and other ?s

    I am just starting an introduction to quantum mechanics this semester, and it's hard for me to do some of my homework and follow some of the lectures because I can't grasp the actual 'physical' meaning of some of the concepts. What do they mean by the expectation values? For example...
  30. E

    Find Expectation Value of x for \psi(x,t)

    I am trying to find <x> for \psi(x,t) = A exp\left(-|x|/L - i*E*t/\hbar\right) I found the normalization factor of 1/L and I took \int_{-\infty}^{\infty}\left( x * exp(|x|/L) \right) in two integrals however I got as a final result: L * -\infty * exp(-\infty/ L) - L *...
  31. P

    Expectation value for Hydrogen radius

    Homework Statement Find the expectation value for a hydrogen atom's radius if n=25 and l=0. Homework Equations expectation value = <f|o|f> where f=wavefunction and o=operator The Attempt at a Solution So I know that to find an expectation value, you integrate over all relevant...
  32. quantumdude

    Expectation Value of Momentum in H-Atom

    Here's a silly question. I'm sure I should know the answer, but alas my most recent QM course was 9 years ago. I sat down to calculate the expectation value of momentum in the H-atom today, because some kid on another forum wanted to know how fast an electron in an atom is. I was going to...
  33. D

    Calculating Conditional Expectation for Continuous and Discrete Random Vectors

    Hi, Let x,z continuous random vectors and n discrete random vector: n=[n1,n2,...]. I'm trying to find for instance, E_z|n3{ E_n|z(x)} = ?. Thanks...
  34. M

    What Are the Expectation Values of Observables A and B in State |b2>?

    Homework Statement We have an observable A, that has eigen vectors l a1 > and l a2 > , with eigenvalues a1 and a2 respectively. A second observable B has eigenvectors l b1 > and l b2 > with eigenvalues b1 and b2 respectively. The eigenstates of B can be written in terms of the eigenstates...
  35. J

    Generating function expectation

    A probability distribution,f(x) ,can be represented as a generating function,G(n) , as \sum_{x} f(x) n^x . The expectation of f(x) can be got from G'(1) . A bivariate generating function, G(m,n) of the joint distribution f(x,y) can be represented as \sum_{x} \sum_{y} f(x,y) n^x m^y ...
  36. B

    How to Calculate E(XY) for Random Variables X and Y?

    I have two random variables X and Y, and I need to calculate E(XY). The expectation of X, E(X) = aZ, and the expectation of Y, E(Y) = bZ, where a and b are known constants and Z is a random variable. So the question is how would I calculate E(XY)? I was thinking that I could do the...
  37. N

    Standard deviation of expectation values

    Very basic question which has confused me: if the variance of an expectation value <A> is: uncertainty of A=<(A-<A>)^2>^0.5 how is this equal to: (<A^2>-<A>^2)^0.5 ??
  38. T

    Expectation Inequality for Positive Random Variables

    Homework Statement Prove that E(X) > a.P(X>a) Homework Equations E(X) is expectation, a is a positive constant and X is the random variable. (Note, > should be 'greater than or equal to' but I'm not too sure how to do it) The Attempt at a Solution Well I can show it easy enough...
  39. E

    Computing Expectation values as functions of time.

    Homework Statement 6) A particle in the infinite square well has the initial wave function Ψ(x,0)= Ax when 0<=x<=a/2 Ψ(x,0)= A(a-x) when a/2<=x<=a a) Sketch Ψ(x,0), and determine the constant A. b) Find Ψ(x,t) c) Compute <x> and <p> as functions of time. Do they oscillate? With what...
  40. V

    Expectation value of the square of the observable

    Homework Statement I know how to compute the expectation value of an observable. But how does one compute the expectation value of an observable's square? Homework Equations \langle Q \rangle = \int_{-\infty}^{\infty} \Psi^* \hat{Q} \Psi \; dx \langle Q^2 \rangle = \int_{-\infty}^{\infty}...
  41. V

    Expectation value of an operator (not its corresponding observable value)

    1. Problem statement This isn't a homework question itself, but is related to one. More specifically, I'm computing the time-derivative of \langle x \rangle using the correspondence principle. One side simplifies to \left\langle \frac{\hat{p}}{m} \right\rangle, but what is the physical meaning...
  42. T

    Calculating Expectation Values for x, x^2 in 1D Box

    Homework Statement Calculate the expectation values of x, x^2 for a particle in a one dimensional box in state \Psi_n Homework Equations \Psi_n = \sqrt{\frac{2}{a}}sin(\frac{n\pi x}{a}) The Attempt at a Solution i formed the integral \int_{-\infty}^{+\infty}\Psi ^2 x dx as the...
  43. G01

    Does a Particle in an Infinite Potential Well Obey the Uncertainty Principle?

    Homework Statement I need to show that a particle in an infinite potential well in the nth energy level, obeys the uncertainty principle and also show which state comes closest to the limit of the uncertainty principle. This means i have to calculate <x>, <x^2>, <p> and <p^2>Homework...
  44. G

    Computing Expectation Values: What Makes Sense?

    How much sense does it make to compute expectation value of an observable in a limited interval? i.e. \int_a^b \psi^* \hat Q \psi dx. rather than \int_{-\infty}^{\infty} \psi \hat Q \psi dx Apparently, it shouldn't make any sense for it gives weird results when you compute e.v. of momentum for...
  45. M

    3-Dimension Expectation Values (QM)

    Hello, I have a problem that wants me to find the expectation value of <r> <r^2> for the ground state of hydrogen (part a.). My friend and I already completed the exercise but I'm concerned about how we found the expectation value. Since the ground state of hydrogen is only dependent on r do...
  46. L

    Harmonic oscillator expectation values

    I'm given that there is a harmonic oscillator in a state that is a superposition of the ground and first excited stationary states given by \psi = \frac{1}{\sqrt{2}}\abs{\psi_0(x,t) + \psi_1(x,t)}, where \psi_0 = \psi_0(x)e^{\frac{-iE_0t}{\hbar}} and \psi_1 = \psi_1(x)e^{\frac{-iE_1t}{\hbar}}...
  47. S

    How Do You Find <r^2> for Uncertainty in Position?

    When trying to work out the uncerainty in position of the expectation value I have read that you have to find <r^2> as well as <r>^2. I have worked out the value of 3a/2 for <r> but what do I have to do to find <r^2>. Do I just sqare the whole function before I integrate? Also as I am...
  48. D

    Expectation value of 1s state of hydrogen driving me absolutely nuts

    My question says: "Evaluate the expectation value <1/r> of the 1s state of hydrogen. How does this result compare to the result found using the Bohr theory?" Firstly, I have been told that <1/r> does not mean <1/r> but rather that it means 1/<r>. Having made it this far I now do the 1/<r>...
  49. L

    Find Expectation Value of Wavefunction in 1-D Box

    OK, here is the problem: An electron is in a 1-D box of length L. Its wavefunction is a linear combination of the ground and first-excited stationary states (and here it is): \phi(x,t) = \sqrt\frac{2}{L}[sin (\frac {\pi x} {L})e^{-i \omega_1 t} + sin\frac {2 \pi x}{L} e^{-i \omega_2 t}]...
  50. H

    Hermiticity and expectation value

    is there a better way to check for hermicity than doing expecation values? for example, what if you had xp (operators) - px (operators), or pxp (operators again); how can you tell if these combos are hermetian or not, without going through the clumsy integration (that doesn't give a solid...
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