Characteristic function Definition and 46 Threads

In Hamilton's "on a general method in dynamics", he starts with varying the function ##U## and writes the equation: $$\delta U=\sum m(\ddot x\delta x+\ddot y\delta y+\ddot z\delta z)$$ Then he defines ##T## to be: $$T=\frac{1}{2}\sum m (\dot x^2+\dot y^2+\dot z^2)$$ Then by ##dT=dU##, he...

Problem summary I have the characteristic function of a probability distribution but I'm having difficulty obtaining its derivative. Background I am reading the following paper: Schwartz, Lowell M. (1980). On round-off error. Analytical Chemistry, 52(7), 1141-1147. DOI:10.1021/ac50057a033. The...

I have the characteristic function of the Cauchy distribution ##C(t)= e^{-(\mid t \mid)}##. Now, how would I show that the Cauchy distribution has no moments using this? I think you have to show it has no Taylor expansion around the origin. I am not sure how to do this.

Homework Statement [/B] I am trying to determien the characteristic function of the function: $$ f(x)= ae^{-ax}$$ $$\therefore E(e^{itx}) =\int_0^\infty e^{itx}ae^{-ax} dx = a \cdot \frac{e}{it-a} |_0 ^ \infty $$ But I am not sure how to evaluate the integral. Wolfram alpha suggests this...

Homework Statement I am trying to understand the very last equality for (let me replace the tilda with a hat ) ##\hat{P_{X}(K)}=\hat{P(k_1=k_2=...=k_{N}=k)}##(1) Homework Equations I also thought that the following imaginary exponential delta identity may be useful, due to the equality of...

This thread will be a collection of multiple questions I asked before over different forums. I will start from the beginning, and I hope someone will follow the steps with me, because I did it before alone, and I ended with a numerical integration that is not finite, which doesn't make sense...

Is there a way to find the CDF of a random variable from its characteristic function directly, without first finding the PDF through inverse Fourier transform, and then integrate the PDF to get the CDFÉ

Suppose that ##Y=\sum_{k=1}^KX_{(k)}##, where ##X_{(1)}\leq X_{(2)}\leq\cdots X_{(N)}## and (##N\geq K##). I want to find the characteristic function of ##Y## as \phi(jvY)=E\left[e^{jvY}\right]=E\left[e^{jv\sum_{k=1}^KX_{(k)}}\right] In the case where ##\{X\}## are i.i.d random variables, the...

Suppose I have a random variable whose moments are not defined, can I still use the characteristic function to find the CDF of that random variable?

Homework Statement Hi, I have the probabilty density: ##p_{n}=(1-p)^{n}p , n=0,1,2... ## and I am asked to find the characteristic function: ##p(k)= <e^{ikn}> ## and then use this to determine the mean and variance of the distribution. Homework Equations [/B] I have the general expression...

Hello, guys. I am trying to solve for characteristic function of normal distribution and I've got to the point where some manipulation has been made with the term in integrands exponent. And a new term of t2σ2/2 has appeared. Could you be so kind and explain that to me, please...

Hello all, I'm trying to find the characteristic function of the random variable ##X## whose PDF is ##f_X(x)=1/(x+1)^2## where ##X\in[0,\,\infty)##. I started like this: \phi_X(j\nu)=E\left[e^{j\nu X}\right]=\int_0^{\infty}\frac{e^{j\nu X}}{(x+1)^2}\,dx where ##j=\sqrt{-1}##. I searched the...

Homework Statement I am trying to figure out following problem. Let A ⊂ R. Then we can define the characteristic function: \begin{align} \chi_A : R → \{0, 1\}, x = \begin{cases} 1 & \text{if } x \in A \\ 0 & \text{else } \end{cases} \end{align} Let a be bigger than 0. I am...

Hi everyone, in the course of trying to solve a rather complicated statistics problem, I stumbled upon a few difficult integrals. The most difficult looks like: I(k,a,b,c) = \int_{-\infty}^{\infty} dx\, \frac{e^{i k x} e^{-\frac{x^2}{2}} x}{(a + 2 i x)(b+2 i x)(c+2 i x)} where a,b,c are...

Homework Statement I'm looking to find the characteristic function of p(x)=\frac{1}{\pi}\frac{1}{1+x^2}[/B] Homework Equations The characteristic function is defined as \int_{-\infty}^{\infty} e^{ikx}p(x)dx 3. The Attempt at a Solution I attempted to solve this using integration by...

Homework Statement 1.) For N particles in a gravity field, the Hamiltonian has a contribution of external potential only (-mgh). Show that the particle density follows the barometric height equation (1). 2.) For N particles in a open system at constant pressure p and temperature T, let there...

Find the characteristic function for the PMF \(p_X[k] = \frac{1}{5}\) for \(k = -2, -1,\ldots, 2\). The characteristic function can be found with \begin{align*} \phi_X(\omega) &= E[\exp(i\omega X)]\\...

Homework Statement In order to determine the characteristic function of a random variable defined by: Z = max(X,0) where X is any continuous rv, i need to prove that: F_{l,v}(g(l))=[ \phi_{X}(u+v)\phi_{X}(v) ] / (iv) where F_{l,v}(g(l)) is the Fourier transform of g(l) and...

Homework Statement The characteristic function of a set E is given by χe = 1 if x is in E, and χe = 0 if x is not in E. Let N be a natural number, and {an, bn} from n=1 to N, be any real numbers. Use the definition of the integral (Riemann) to show that \int \sum b_{n} X_{ \left\{ a_{n}...

Homework Statement Let X,W,Y be iid with a common geometric density f_x(x)= p(1-p)^x for x nonnegative integer and p is in the interval (0,1) What is the characteristic function of A= X-2W+3Y ? Determine the family of the conditional distribution of X given X+W? Homework Equations...

Characteristic function and preimage? Homework Statement Let S be a nonempty subset of ℝ. Define χs= { 1 if x is in S and 0 if x is not in S Determine χs-1(Q) [where Q=set of all rational numbers] and χs-1((0,∞)) We haven't really dealt much with this function, and I really...

Homework Statement I must find the characteristic function of the Gaussian distribution f_X(x)=\frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{1}{2} \left [ \frac{(x- \langle x \rangle )^2}{\sigma ^2} \right ]}. If you cannot see well the latex, it's the function P(x) in...

Homework Statement I must calculate the characteristic function as well as the first moments and cumulants of the continuous random variable f_X (x)=\frac{1}{\pi } \frac{c}{x^2+c^2} which is basically a kind of Lorentzian.Homework Equations The characteristic function is simply a Fourier...

Homework Statement Hey guys, I'm self studying some probability theory and I'm stuck with the basics. I must find the characteristic function (also the moments and the cumulants) of the binomial "variable" with parameters n and p. I checked out wikipedia's article...

Homework Statement Prove that the characteristic function \chi_A: X\rightarrow R, \chi_A(x)=1,x\in A; \chi_A(x)=0, x\notin A, where A is a measurable set of the measurable space (X,\psi) , is measurable. Homework Equations a function f: X->R is measurable if for any usual measurable set...

This is inspired by Kardar's Statistical Physics of Particles, page 45, and uses similar notation. Homework Statement Find the characteristic function, \widetilde{p}(\overrightarrow{k}) for the joint gaussian distribution: p(\overrightarrow{x})=\frac{1}{\sqrt{(2\pi)^{N}det...

I need some help. Is there a good way to do this type of question? Homework Statement Let X and Y be independent random Variables with exponential densities fX(x) = Ωe-Ωx, if X≥0 0, otherwise fY(y) = βe-βy, if y≥0 0, otherwise...

Constructing a "smooth" characteristic function Suppose I'd like to construct a C^\infty generalization of a characteristic function, f(x): \mathbb R \to \mathbb R, as follows: I want f to be 1 for, say, x\in (a,b), zero for x < a-\delta and b > x + \delta, and I want it to be C^\infty on...

Hello, I am trying to find a characteristic function (CF) of a Compound Poisson Process (CPP) and I am stuck :(. I have a CPP defined as X(t) = SIGMA[from j=1 to Nt]{Yj}. Yj's are independent and are Normally distributed. So, in trying to find the CF of X I do the following: (Notation...

1. If \phi is a characteristic function, than is e^{\phi-1} also a characteristic function? I know some general rules like that a product or weighted sum of characteristic functions are also characteristic functions, also a pointwise limit of characteristic functions is one if it's continuous...

Hello, I considered a Binomial distribution B(n,p), and a discrete random variable X=\frac{1}{n}B(n,p). I tried to compute the characteristic function of X and got the following: \phi_X(\theta)=E[e^{i\frac{\theta}{n}X}]=(1-p+pe^{i\theta/n})^n I tried to compute the limit for n\to +\infty...

What exactly is a "joint characteristic function"? I want the characteristic function of the joint distribution of two (non-independent) probability distributions. I'll state the problem below for clarity. So my two distributions are the normal distribution with mean 0 and variance n, and the...

I am trying to compute the inverse Fourier transform numerically (using a DFT) for some complicated characteristic functions in order to compute their corresponding probability distribution functions. As a test case I thought I would invert the characteristic function for the simple exponential...

I have been thinking about this for quite some time now. When I look at the function that descibes the fat cantor set namely: f(x) = 1 for x\inF and f(x) = 0 otherwise, where F is the fat cantor set. I wonder, how do I prove that this is non-riemann integrable? I have considered...

Hi there, Recently I have come across a proof with application of characteristic function. After some steps in the proof, it concluded that there is a neighborhood of 0 such that the characteristic function is constant at 1, then it said the characteristic function is constant at 1...

Prove or disprove that function \phi(t)=\frac{1}{1+|t|} is charcteristic function of some random variable.

Hi. Does anyone know a good source for learning about the characteristic function of a joint pdf. Is there any nice rules for that? For example assume having a waiting time density and a jump density which are independent (easy things first). Is there an elegant way to get the characteristic...

I was reading about it here: http://mathworld.wolfram.com/CharacteristicFunction.html very neat. But then I tried out of boredom integrating the expression by parts where u = the exponential term and v = f (x) (or P(x)). The integral came out nicely as I got a term similar to the left hand...

The characteristic function of the RATIONALS is a well-known example of a bounded function that is not Riemann integrable. But is the characteristic function of the IRRATIONALS (that is, the function that is 1 at every irrational number and 0 at every rational number) Riemann integrable on an...

Is it possible to exactly derive the mode of a probability distribution if you have the characteristic function? I cannot get the pdf of the distribution because the inverse Fourier transform of the characteristic function cannot be found analytically. Any thoughts would be appreciated...

Homework Statement Let (X,d) be a metric space, A subset of X, x_A: X->R the characteristic function of A. (R is the set of all real numbers) Let V_d(x) denote the set of neighbourhoods of x with respect the metric d. Prove that x_A is continuous in x (x in X) if and only if there...

Homework Statement Okay, so this is a three-part question, and I need some help with it. 1. I need to show that the function f(x) = e^{-1/x^{2}}, x > 0 and 0 otherwise is infinitely differentiable at x = 0. 2. I need to find a function from R to [0,1] that's 0 for x \leq 0 and 1 for x...

Hi all, Im currently researching into Multivariate distributions, in particular I am trying to derive the characteristic function of the bivariate distribution of a gaussian. While knowing that a gaussian density function cannot be integrated how is it possible to find the characteristic...

This kind of bothers me: our textbook does not explain (and the professor either) where characteristic function comes from, all it says is what it defined as, which is E[ejwX], where E is expectation of random variable X. But where is this e-term coming from? Thanks in advance.

I need to calculate the characteristic function of an exponential distribution: \phi _X \left( t \right) = \int\limits_{ - \infty }^\infty {e^{itX} \lambda e^{ - \lambda x} dx} = \int\limits_{ - \infty }^\infty {\lambda e^{\left( {it - \lambda } \right)x} dx} I have arrived at the...

How can I show that if \phi(t) is a characteristic function for some distribution, then |\phi(t)|^2 is also a characteristic function?