Generating function Definition and 119 Threads

Based off of these MIT Notes: https://ocw.mit.edu/courses/8-09-classical-mechanics-iii-fall-2014/f00f7f68ac7ba346a0868efb7430582c_MIT8_09F14_Chapter_4.pdf 1) This set of notes starts with the premise that ##L’ = L + \frac{dF(q,t)}{dt} = L + \frac{\partial F}{\partial q} \dot q + \frac{\partial...

I am reading the Lancaster & Blundell, Quantum field theory for gifted amateur, p.225 and stuck at understanding some derivations. We will calculate a generating functional for the free scalar field. The free Lagrangian is given by $$ \mathcal{L}_0 = \frac{1}{2}(\partial _\mu \phi)^2 -...

Using the generating function for the legendre polynomial: $$ \sum_{n=0}^{\infty} P_{n}(x) t^{n}=\frac{1}{\sqrt{1-2 x t+t^{2}}} $$ It's possible to expand the coulomb potential in a basis of legendre polynomials (or even spherical harmonic ) like this: $$ \begin{aligned} &\frac{1}{\left.\mid...

My questions: 1) What about if t = 2? Is there a certain meaning to ##G_X (2)## ? 2) PGF for uniform distribution is ##G_X (t)=\frac{t(1-t^n)}{n(1-t)}## and for t = 1 ##G_X (1)## is undefined so ##G_X (1) =1## is not true for all cases? Thanks

(a) I find the geometric distribution $$X~G0(3/8)$$ and I find p to be 0.375 since the mean 0.6 = p/q. So p.g.f of X is $$(5/8)/(1-(3s/8))$$. (b) Not sure how to find the p.g.f of Y once we know there are 6 customers?

Homework Statement [/B] A random variable x has a probability function ##G(t)##. Show that the probability that ##x## takes an even value is ## \frac 1 2 ( 1+G(-1))##Homework EquationsThe Attempt at a Solution [/B] ##G(t)= \sum_{k=0}^\infty p_k t^k ##... ## 1=P(X=even)+ P(X=odd)##...1 ##G(-1)=...

Homework Statement Suppose the potential in a problem of one degree of freedom is linearly dependent upon time such that $$H = \frac{p^2}{2m} - mAtx $$ where A is a constant. Solve the dynamical problem by means of Hamilton's principal function under the initial conditions t = 0, x = 0, ##p =...

Homework Statement Long and thin sample of silicon is stationary illuminated with an intensive optical source which can be described by a generation function ##G(x)=\sum_{m=-\infty}^\infty Kδ(x-ma)## (Dirac comb function). Setting is room temperature and ##L_p## and ##D_p## are given. Find the...

Let me show you part of a book "Mechanics From Newton’s Laws to Deterministic Chaos" by Florian Scheck. I do not understand why these integrands can differ by more than time derivative of some function M. Why doesn't it change the value of integrals? It seems this point is crucial for me to...

Homework Statement Given a system with a Lagrangian ##L(q,\dot{q})## and Hamiltonian ##H=H(q,p)## and that the Lagrangian is invariant under the transformation ##q \rightarrow q+ K(q) ## find the generating function, G. Homework EquationsThe Attempt at a Solution ##\delta q = \{ q,G \} =...

Homework Statement a) I have to find and expression for sequence of $b_n$ in terms of generating functions of the sequence of $a_n$ $$b_n = (-1)^{n}(n+1)a_0 +(-1)^{n-1}n a_1+...+(-1)2a_{n-1}+a_n$$ with $$a_n = a_{n-1} +8a_{n-2} -12a_{n-3} +25(-3)^{n-2} + 32n^2 -64$$ b) I have to use the...

Generating function for Bessel function is defined by G(x,t)=e^{\frac{x}{2}(t-\frac{1}{t})}=\sum^{\infty}_{n=-\infty}J_n(x)t^n Why here we have Laurent series, even in case when functions are of real variables?

I have proved (8.1). However I am trying to prove that ##\bar{X},X_i-\bar{X},i=1,...,n## has a joint distribution that is multivariate normal. I am trying to prove it by looking at the moment generating function: ##E(e^{t(X_i-\bar{X})}=E(e^{tX_i})E(e^{-\frac{t}{n}\sum_{i=1}^n X_i})## I am...

I'm working this problem for my math stat class. Here is what I have for it. First of all, is this the correct method for finding MGF? I thought it was but I don't understand the answers I am getting. How do I determine my values for t? For both I have t not equal to 0 because t is in the...

Homework Statement Prove that ##\psi_n## in Eq. 2.85 is properly normalized by substituting generating functions in place of the Hermite polynomials that appear in the normalization integral, then equating the resulting Taylor series that you obtain on the two sides of your equation. As a...

Homework Statement Homework Equations and in chapter 1 I believe that wanted me to note that The Attempt at a Solution For the first part of this question, as a general statement, I know that P[2 n + 1](0) = 0 will be true as 2n+1 is an odd number, meaning that L is odd, and so the Legendre...

In this thread we are looking at the following generating function $$\sum_{n=1}^\infty [\psi_1(n)]^2 y^n$$ We know that this is as hard as evaluating $$\sum_{n=1}^\infty [H_n^{(2)}]^2 y^n$$ This is not a tutorial as I have no idea how to solve for a general formula. I'll keep posting my...

Homework Statement Show that the Bessel functions Jn(x) (where n is an integer) have a very nice generating function, namely, G(x,t) := ∑ from -∞ to ∞ of tn Jn(x) = exp {(x/2)((t-T1/t))}, Hint. Starting from the recurrence relation Jn+1(x) + Jn-1(x) = (2n/x)Jn(x), show that G(x,t)...

How do you know that if two random variables have the same moment generating function then they have the same probability distribution.

Homework Statement given one each of u types of candy, two each of v types of candy, and three each of of w types of candy, find a generating function for the number of ways to select r candies. The Attempt at a Solution I am not sure if I understand this correctly, but this is what I came...

It's about equation (6.5) I'm not entirely getting the reasoning explained by the author so I came up with the following, can anyone confirm or refute. One way to look at equation (6.5) would be: We create variations on the ##q## variables, in the form of ##\delta q(t)##. Since ##Q=Q(q,p,t)##...

To make my explanation easier open the ''Generating function approach'' section on this wiki article: http://en.wikipedia.org/wiki/Canonical_transformation The function ##\frac{dG}{dt}## represents the function that always can be added to the Lagrangian without changing the mechanical...

Given a Hamiltonian ##H##, with a spectrum of eigenvalues ##\lambda##, you can define its zeta function as ##\zeta_H(s) = tr \frac{1}{H^s} = \sum_{\lambda}^{} \frac{1}{\lambda^s}##. Subsequently, the log determinant of ##H## with a spectral parameter ##m^2## acts as a generating function for...

Homework Statement a man draws balls from an infinitely large box containing either white and black balls , assume statistical independence. the man draws 1 ball each time and stops once he has at least 1 ball of each color . if the probability of drawing a white ball is p , and and q=1-p is...

$$F(z) = \sum_{n=0}^\infty a_n x^n $$ $$\partial_zF(z) = \sum_{n=0}^\infty (n+1)a_{n+1}x^n $$ So, we can begin to piece together some differential equations in terms of generating functions in order to satisfy some discrete recursion relation (which is the desired problem to solve). However I...

Homework Statement Let X ~ Normal(μ,σ2). Define Y=eX. a) Find the PDF of Y. b) Show that the moment generating function of Y doesn't exist. Homework EquationsThe Attempt at a Solution For part a, I used the fact that fy(y) = |d/dy g-1(y)| fx(g-1(y)). Therefore I got that fy(y)=...

Homework Statement Write the integral that would define the mgf of the pdf, f(x) = \frac 1{\pi} \frac 1{1+x^2} dx Homework Equations The moment generating function (mgf) is E e^{tX}[\itex]. The Attempt at a Solution My question really has to do with improper integrals. I must...

Hi, Please I need you help to solve this problem: ---------- Consider a planar tree with $n$ non-root vertices (root edge selected). 1. Give a generating function for vertices distance $d$ from the root. 2. Proof that the total number is $$\displaystyle...

Homework Statement Homework Equations The Attempt at a Solution The problem is the integral is non-elementary, so now what? Part (b) follows trivially from part (a). But is there some kind of shortcut I have to take, because no matter what substitution I do, the integral...

What is the difference between generating function and Rodrigue's formula? Some says that from generating function you can generate required polynomial (say for example from generating function of Legendre polynomial you can get Legendre polynomial.), but in that case,as far as i know, Rodrigues...

With mean = 2 with exponential distribution Calculate E(200 + 5Y^2 + 4Y^3) = 432 E(200) = 200 E(5Y^2) = 5E(Y^2) = 5(8) = 40 E(4Y^3) = 4E(Y^3) = 4(48) = 192 E(Y^2) = V(Y) + [E(Y)]^2 = 2^2+2^2= 8 E(Y^3) = m_Y^3(0) = 48(1-2(0))^{-4} = 48 is this right?

I am given that The kth moment of an exponential random variable with mean mu is E[Y^k] = k!*mu^k for nonnegative integer k. I found m^2 (0) = (-a)(-a-1)(-beta)^2. The answer I found is however mu^2+a*beta^2 which is different from the k! From the given formula. Could someone help me figure it...

Prove the generating function $$e^{\frac{x}{2}\left(z-z^{-1}\right)}=\sum_{n=-\infty}^{\infty}J_n(x)z^n$$

Assume X is a random variable under a probability space in which the sample space ?= {a,b,c,d,e}. Then if I am told that: X({a}) = 1 X({b}) = 2 X({c}) = 3 X({d}) = 4 X({e}) = 5 And that: P({a}) = P({c}) = P({e}) = 1/10 P({b}) = P({d}) = 7/20 Find the C.D.F of X, the density of X...

Hi everyone, So I am taking a statistics course and finding this concept kinda challenging. wondering if someone can help me with the following problem! Let X be a random variable with probability density function $$f(x)=\begin{cases}xe^{-x} \quad \text{if } x>0\\0 \quad \text{ }...

Could anybody give me some examples of generating function in physics, it's application, and it's use? Thank you

Estimation of x i.e. E(x) = Ʃx.p(x) ... p(x) is probabiltiy of x Now my book defines another function mgf(x) i.e. moment generating function of x which is defined as: - mgf(x) = E(etx) I don't understand why was this function defined. Basically we included etx in our function because then...

Let X1,X2,…,Xn be independent random variables that all have the same distribution, let N be an independent non-negative integer valued random variable, and let SN:=X1+X2+⋯+XN. Find an expression for the moment generating function of SN so all i know is that it is i.i.d but i am not sure what...

Suppose a gambler starts with one dollar and plays a game in which he or she wins one dollar with probability p and loses one dollar with probability 1 - p. Let fn be the probability that he or she first becomes broke at time n for n = 0, 1, 2... Find a generating function for these...

Homework Statement Hi everyone! Me and my colleague are working our way through Harold J Larson's "Introduction to Probability Theory and Statistical Inference: Third Edition", and we found something interesting. We both have the same edition of the text, but mine is slightly newer?, and...

Suppose I have the MGF moment generating function mx(t) = (e^t -1)/t How can I find EX?

Question A moment-generating function of X is given by M(t) = 0.3e^t + 0.4e^(2t) + 0.2e^(3t) + 0.1e^(5t) Find the pmf of X My attempt x f(x) 1 0.3 2 0.4 3 0.2 4 0 5 0.1 I am just wondering whether it is correct to say "0" for 4 or is it more correct to say x f(x) 1 0.3 2 0.4 3 0.2 5 0.1 or...

I'm trying to define Bessel's function by using the generating function, I know i need to go through a recursion formula to get there. $e^{(\frac{x}{2}(t-\frac{1}{t})}=\displaystyle\sum_{n=-\infty}^{\infty}J_n(x)t^n$ if this or anyone has latex that's the generating function. Any...

Homework Statement Let X be a random variable with a Laplace distribution, so that its probability density function is given by f(x) = \frac{1}{2}e^{-|x|} Sketch f(x). Show that its moment generating function MX (θ) is given by M_{X}(\theta) = \frac{1}{1 - \theta^2} and hence find...

I am suppose to use the generating function for e_{m}(x_{1} . . . . x_{n}) to solve a problem. I have tried looking for it but I can not seem to find any information on it. Does anyone know what it is?

I am working on a probabilty theory problem: Let (X,Y) be distributed with joint density f(x,y)=(1/4)(1+xy(x^2-y^2)) if abs(x)≤1, abs(y)≤1; 0 otherwise Find the MGF of (X,Y). Are X,Y independent? If not, find covariance. I have set up the integral to find the mgf ∫∫e^(sx+ty)f(x,y)dx dy with...

Homework Statement X is distributed exponentially with λa=2. Y is distributed exponentially with λb = 3. X and Y are independent. Let W=max(X,Y), the time until both persons catch their first fish. Let k be a positive integer. Find E(W^k). Also, find P{(1/3)<X/(X+Y)<(1/2)}...

given m(t) = (1-p+p*e^t)^5 what is probability P(x<1.23) i know that m(t) = e^tx * f(x) m'(0) = E(X) and m''(0) , can find the var(x) should i calculate it using a normal table?

For any integer valued RV X Summation n=0 to infinity of s^n P(X=<n) = (1-s)^-1 * Summation k=0 to infinity of P(x=k)s^k Why does Sum k=0 to infinity P(x=k)s^k = sum n=0 to infinity of P(X=< n)

I attached a pdf. The questions are not really what is stumping me. Its the wording of the question I don't understand. When it says, "Assume that 0 < λ < 1. Note that your answers will be in terms of the constant λ." and "Assume that λ > 0. Note that your answers will be in terms of the...