Many thanks in advance
Suppose x is normal variable x~N(a,b)
and y=160*x^2
I need calculate E(y)=∫yf(y)d(y)
f(y) is the density function of y
how can I write it as an integral of x since we know x's distribution, I mean use the density function of x to substitute the original integral...
Hi:
If we want to work out the expectation of:
<0|T(φ1φ2)|0>
ie. <0|<0|T(φ1φ2)|0>|0>
apparently it is acceptable to pull out the <0|T(φ1φ2)|0>:
So <0|<0|T(φ1φ2)|0>|0>=<0|T(φ1φ2)|0><0|I|0>
I do realize this is a really stupid question, but I want to be 100% sure. Is this simply...
Question 1)
I have X and Y independent stoch. variables
What is E[X^2 * Y | X] ?
does it generally hold that if X and Y are independent, then every function of X (eg X^2) is independent of Y?
Does E[X^2 * Y | X] then become E[X^2|X]*E[Y|X] = E[X^2|X]*E[Y] since X^2 is independent of...
Hi had this question on my last "Statistical Inference" exam. And I still have some doubts about it. I determined that the maximum likelihood estimator of an Uniform distribution U(0,k) is equal to the maximum value observed in the sample. That is correct. So say my textbooks. After that the...
Homework Statement
i. Confirming the wavefunction is normalised
ii. Calculating the expectation values: <\hat{x}> , <\hat{x^{2}}> , <\hat{p}> , <\hat{p^{2}}> as a function of \sigma
iii. Interpreting the results in regards to Heisenberg's uncertainty relation.
Homework Equations...
Hello,
I am trying to learn about some basic quantum mechanics.
http://farside.ph.utexas.edu/teaching/qmech/lectures/node35.html this website shows that the time derivative of the momentum expectation d<p>/dt = -<dV/dx>
The part that i am not getting is how the writer goes from the...
Homework Statement
Consider a Negative Binomial random variable Y ~ NB(r, p). Show (from first principles!) that E[Y] is r/p. Why does this imply Y is proper?
Homework Equations
I have no idea how to use latex, so this may be messy:
pmf of Y: [ (k+r-1)! / k!(r-1)! ] * (1-p)^r * p^k...
Homework Statement
Let X1...XN be independent and identically distributed random variables, N is a non-negative integer valued random variable. Let Z = X1 + ... + XN (assume when N=0 Z=0).
1. Find E(Z)
2. Show var(Z) = var(N)E(X1)2 + E(N)var(X1)
Homework Equations
E(Z) = EX (E(X|Z))...
Homework Statement
An email is sent on the network in which the recipients (0,1,2,3,4,5} are in communication.
1 can send to 4 and 2
2 to 1,3,5
3 to 0,2,5
4 to 1, 5
5 to 0,2,4
0 to 3 and 5
If a message is sent to 2,3,4,5 it is forwarded randomly to a neighbour (even if this means a...
Homework Statement
A particle of mass m is in the state
Psi(x,t) = Ae^(-a[(mx^2)+it])
where A and a are positive real constants.
a) Find A
b) For what potential energy function V(x) does Psi satisfy the Shrodinger equation?
c) Calculate the expectation values of x, x^2, p, and...
Homework Statement
I'm told that of n couples, each of whom have at least one child, with couples procreating independently and no limits on family size, births single and independent, and for the ith couple the probability of a boy is p_i and of a girl is q_i with p_i + q_i = 1.
1. Show...
Homework Statement
Let X1,...,Xn denote a random sample from a N(\mu , \sigma) distribution. Let Y = \Sigma \frac{(X_i - \overline{X})^2}{n}
Homework Equations
The Attempt at a Solution
How would I find E(Y)?
Any help would be greately appreciated.
Hello,
Can someone explain to me how the expectation values are calculated in the following picture:
I mean , What did they do after the brackets? What did they multiply with what?
thanks
I am aware of the expectation value \left\langle\ r \right\rangle. But I was wondering what is physically meant by the expectation value: \left\langle\frac{1}{r}\right\rangle
The reason I am asking is because calculating this (reciprocal) expectation value for the 1s state of hydrogen, one...
Homework Statement
This has been driving me CRAZY:
Show that \langle a(t)\rangle = e^{-i\omega t} \langle a(0) \rangle
and
\langle a^{\dagger}(t)\rangle = e^{i\omega t} \langle a^{\dagger}(0) \rangle
Homework Equations
Raising/lowering eigenvalue equations:
a |n...
Let's consider eigenstates |nlm\rangle of hamiltonian of hydrogen atom.
Can anyone prove that
\langle r \rangle = \langle nlm|r|nlm\rangle = \frac{a}{2}(3 n^2-l(l+1)).
Where a - bohr radious.
I've been trying to prove it using some property of Laguerre polynomials (which are
radial part...
Given X follows an exponential distribution \theta
how could i show something like
\operatorname{E}(X|X \geq \tau)=\tau+\frac 1 \theta
?
i have get the idea of using Memorylessness property here,
but how can i combine the probabilty with the expectation?
thanks.
casper
Homework Statement
The ground state (lowest energy) radial wave function for an electron bound to a proton to form a hydrogen atom is given by the 1s (n=1, l=0) wave function:
R10 = (2 / a3/2) exp(-r / a)
where r is the distance of the electron from the proton and a is a constant.
a)...
" The law of total expectation is: E(Y) = E[E(Y|X)].
It can be generalized to the vector case: E(Y) = E[E(Y|X1,X2)].
Further extension:
(i) E(Y|X1) = E[E(Y|X1,X2)|X1]
(ii) E(Y|X1,X2) = E[E(Y|X1,X2,X3)|X1,X2] "
====================
I understand the law of total expectation itself, but...
Homework Statement
Using the fact that ,\left\langle \hat{L}_{x}^{2} \right\rangle = \left\langle \hat{L}_{y}^{2} \right\rangle show that \left\langle \hat{L}_{x}^{2} \right\rangle = 1/2 \hbar^{2}(l(l+1)-m^{2}.
The Attempt at a Solution
L^{2} \left|l,m\right\rangle = \hbar^{2}l(l+1)...
Homework Statement
Express Lx in terms of the commutator of Ly and Lz and, using this result, show that <Lx>=0 for this particle.
The Attempt at a Solution
[Ly,Lz]=i(hbar)Lx
<Lx>=< l,m l Lx l l,m>
then what?
Homework Statement
In my textbook, the formula for the expectation value is written as:
<x> = \int \Psi^{*}\Psi dx
Shouldn't there be an x next to |\Psi|^{2} ?
Thanks.
Homework Equations
The Attempt at a Solution
Homework Statement
Calculate the expectation value of the operator _{}Sz for a spin-half particle known to be in an eigenstate of the operator _{}Sz
Homework Equations
The Attempt at a Solution
I know the eigenvalues for the _{}Sz but how can I find the expectation values...
Hi
I have a question.
Let X1 & X2 be stochastic variables and X1<=X2, then can we say E[X1]<=E[X2] or SD[X1]<=SD[X2]? why or why not?
Looking forward to some reply
Thanks!
Expectation value of operator A is given by following formula in Dirac notation.
<A> = <x|A|x>
where
A : Operator
<A> : Expectation value of A
|x> : State
Somehow I am unable to convince myself that this formula is true.
Would someone please explain it to me?
Thanks
Homework Statement
Evaluate the expectation value of p and p² using the momentum-space wave function
Homework Equations
Momentum-space wave function:
\sqrt{\frac{d}{\hbar\sqrt{\pi}}}e^{\frac{-\left(p'-\hbar k\right)^2d^2}{2\hbar^2}}
The Attempt at a Solution
I can get \langle...
Homework Statement
An electron is in the spin state in the Sz representation
|ψ> = A (1-2i 2)T <- this is a 2 X 1 matrix
If Sx is measured, what values and probabilities do you get?
What is the expectation value of Sx?
Homework Equations
The Attempt at a Solution...
Homework Statement
Consider a particle in an infinite one-dimensional box that has a length L and is centered at the origin. (Use h for Planck's constant, n, and L, as necessary.) Evaluate <x^2> for <x^2> at n=1.
Homework Equations
<x^2>= (2/L)[(L^3/24)-(L^3/4n^2*pi^2)cos(n*pi)]
The...
I'm trying to check that the expectation value <E> is E for the wavefunction
sqrt(2/L) sin(2pix / L)
I know the shortcut way of doing it by saying that the hamiltonian multiplied by the function is just the eigenvalue E multiplied by the function, and since the function is normalized the...
My teacher said that angular momentum doesn't have orientation in space - but how can that be? Isn't cos(theta) = L_z / |L vector| ?
Also (an unrelated question) could somebody give an example of how the integration process goes when you are trying to get an expectation value for something...
Why should symmetries require a field that acquires vacuum expectation value to have the same quantum numbers as the vacuum? Please give me a reference also..if possible...
Homework Statement
The wave function of a state is Psi(x)= N*a(x)exp(i*p0*x/h)where a(x) is a quadratically integrable real valued function Show that the expectation value of the function is p0.
Homework Equations
The Attempt at a Solution
The only thing I'm having a problem...
I want to find <x> and ,<x^2>, <p>, and <p^2> of a particle in an infinite well where:
V(x)=0, \frac{-a}{4}<x<\frac{3a}{4}
Using the usual method, I found the wavefunction to be:
\psi(x)=\sqrt{\frac{2}{a}}sin[\frac{n\pi}{a}(x+\frac{a}{4})]
I also found...
Homework Statement
I am trying to find the variance of p for a wave function \Psi(x,0)=A(a^2-x^2)
I'm confused about how to set up the integral.
it should be something like -i^2h^2\int_{-a}^a A(a^2-x^2) (\frac{\partial\Psi}{\partial x})^2 dx
I'm confused about the partial...
Homework Statement
Using the Feynman-Hellman theorem, determine the expectation values of 1/r and 1/r^2 for the hydrogen atom.
Homework Equations
Hamiltonian: H=-\frac{\hbar^2}{2m}\frac{d^2}{dr^2}+\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}-\frac{e^2}{4\pi\epsilon_0}\frac{1}{r}
energy...
Homework Statement
Suppose \theta ~ ~ gamma(\alpha , \lambda) where alpha is a positive integer. Conditional on \theta, X has a Poission distribution with mean \theta . Find the unconditional distribution of X by finding it's MGT.
Homework Equations
The Attempt at a Solution...
If i know that Pr(X_t>b)=0, where X_t>0 and b is positive finite, then should the expectation of X_t be finite? Is there any case where it is infinite?
Okay, so I'm now reviewing ladder operators (no, not homework).
While reviewing a quantum problem involving the L_z operator at this website (http://quantummechanics.ucsd.edu/ph130a/130_notes/node219.html#example:expectLz"), I found myself confused.
Okay, here's my question: don't we need to...
Homework Statement
Let X denote a random variable with the following probability mass function:
P(j)= 2^(-j), j=1,2,3,...
(a) Compute the moment generating function of X.
(b) Use your answer to part (a) to compute the expectation of X.
Homework Equations
m.g.f of X is M (t) =...
Good Evening:
I'm given this problem:
A device that continuously measures and records seismic activity is placed in a remote
region. The time, T, to failure of this device is exponentially distributed with mean
3 years. Since the device will not be monitored during its first two years of...
Hello everybody,
I have two questions on conditional expectation w.r.t (Polynomial) OLS:
Let X_t be a random variable and F_t the associated filtration, Vect_n{X_t} the vector space spanned by the polynomials of order {i, i<=n }, f(.) one function with enough regularity. I am wondering how...
Hi:
e, z, mu are vectors of size N
I need to show that E(e|z+mu) = E(e|mu) or at least E(e|z+mu) converges in probability to E(e|mu) as N goes to infinity, under the assumption that Z is not correlated with e.
My guess is that to get this result I also need z to be orthogonal to mu...
The distribution function of a random variable X is given by:
F(x) =
0 if x <-3
3/8 if -3 <= x < 0
1/2 if 0 <= x < 3
3/4 if 3 <= x <4
1 if x => 4
Calculate E(X) and E(X2 - 2|X|)
Well I'm at a loss of E(X) although once I know this the other should be fairly simple..
Ive got...
Homework Statement
Find the expectation value <x> if:
from 0 <= x <= a, psi = A x/a
from a <= x <= b, psi = A(b-x)/(b-a)
Normalizing gives me that A = sqrt(3/b) (verified correct)Homework Equations
The Attempt at a Solution
<x> = \int_0^b x \psi^2 dx = \int_0^a \frac{A^2 x^3}{a^2}dx + \int_a^b...
Just a quick question.
I finished an expectation value sum and noticed that the given solution had me stumped.
Ive attached a quick picture of the simplifying process which was given as the solution.
The only thing i don't understand is how to get the value iCm/(pi*hbar)^1/2.
I don't know...