Generating function Definition and 119 Threads

J.J. Sakurai Modern Quantum Mechanics p. 74 It says, [A,H] = 0; H|a'> = Ea' |a'> where H is the Hamiltonian A is any observable |a'> is eigenket of A then, exp ( -iHt/h)|a'> = exp (-iEa't/h)|a'> where h is the reduced Planck's constant. I want to know WHY ? and besides, I would...

Homework Statement Consider a branching process with branching probabilities given by P0=1/2 and Pj=1/3^{j} for j \geq 1 Find the probability generating function: \sum^{\infty}_{n=0} p_{n}x^{n} The Attempt at a Solution Now, the answer is supposed to be G(x) = (x+3)/(2(3-x)), but I...

Homework Statement The problem asks to use the Lagrange form of the remainder in Taylor's Theorem to prove that the Maclaurin series generated by f(x) = xex converges to f. From the actual answer, I'm guessing it wants me to use the Remainder Estimation Theorem to accomplish this...

Homework Statement Suppose RX(t) = E[(1 − tX)−1] is called the geometric generating function of X. Suppose the random variable Y has a uniform distribution on (0, 1); ie fY (y) = 1 for 0 < y < 1. Determine the geometric generating function of Y . Homework Equations The Attempt at...

Homework Statement Let f(x) = 2x 0<x<1 a) Determing the Moment Generating function M(t) of X b) Use the MGT to determine all moments about the origin c) Give the 3rd central moment called the skewness Homework Equations The Attempt at a Solution a) \int^1_0 e^{tx}2x dx =...

Homework Statement Find the mgf of 2/25*(5-y) fo 0<y<5 Homework Equations M(t) = INT e^yt f(y)dy The Attempt at a Solution = (2*(e^(5t)-5t-1))/25t Is this ok

1. Let X denote the mean of a random sample of size 75 from the distribution that has the pdf f (x) =1, 0<=x<=1. Calculate P (0.45 <X< 0.55). 2. Derive the moment-generating function for the normal density. 3. Let Y n (or Y for simplicity) be b (n, p). Thus, Y / n is approximately N [p, p (1...

The Legendre functions may be defined in terms of a generating function: g(x,t) = \frac{1}{\sqrt{1-2xt+t^2}} Of course, \frac{1}{\sqrt{1+x}} =\sum^{\infty}_{n=0} (\stackrel{-.5}{n})x^n . However, this series doesn't converge for all x. It only converges if |x| < 1. In our case, |t^2 -...

Homework Statement The Bessel function generating function is e^{\frac{t}{2}(z-\frac{1}{z})} = \sum_{n=-\infty}^\infty J_n(t)z^n Show J_n(t) = \frac{1}{\pi} \int_0^\pi cos(tsin(\vartheta)-n\vartheta)d\vartheta Homework Equations The Attempt at a Solution So far I...

Homework Statement Use the Legendre generating function to show that for A > 1, \int^{\pi}_{0} \frac{\left(Acos\theta + 1\right)sin\thetad\theta}{\left(A^{2}+2Acos\theta+1\right)^{1/2}} = \frac{4}{3A} Homework Equations The Legendre generating function \phi\left(-cos\theta,A\right) =...

Hi, I don't understand, in general, how am I supposed to find an appropriate generating function to a given canonical transformation. It seems to me like a lot of guesswork. Can anyone give me some guidelines? thanks.

I'm given the probability density function: f(x) = 3x^2 for x in [0, 1] f(x) = 0 elsewhere I want to find E[X^2] which is easy if I use the integral definition (I got 3/5). Yet, when I try and do this using Moment Generating Function (mgf) I cannot seem to get the same answer (in...

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 ...

(1-xt)/(1-2xt+t^2)=sum(Tn(x)t^n) How can i prove this equation? Could you give me a hint or suggestion?

Given that the moment generating function of a random variable is (e^t)/(2-e^t) is there a way I can go backwards and find the pdf, or could 2 different pdf's have the same mgf?

i have X_1,X_2,...X_n independant poisson-distributed variables with parameters: alfa_i and i=1,...k(unsure about this. however says so in the excercise) i am supposed to find the distribution of Y= SUM(from 1 to n) a_i*X_i where a_i>0 maybe one could use the "poisson paradigm" by...

I'm sure this is relatively easy, but after an hour or so googling, I can't seem to find the formula for generating terms of the http://steiner.math.nthu.edu.tw/chuan/123/test/euler.htm Is this known by some other name? Maybe that's why I can't find it? Thanks

I have absolutly no idea how to do this. so let X be a random variable with pdf fx(xy) = x for 0<=x<=1 2 - x for 1 <= 1 <= 2 0 otherwise. I"m looking through my book, and it doesn't give examples that resembles this. all I see is the moment is e^(tk) * the function... and tI...

Q: show that (1-4x)^(-1/2) generates the sequence 2n chooses n, n is defined as natural All the formulas I have requires integer exponent. I am not sure how to deal with (-1/2). Thanks for any input!