Questions tagged [characteristic-function]

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Analytic structure of the Dirichlet L functions with the Hurwitz zeta function

I am getting confused about how to get results from below by using the Hurwitz zeta function. Really don't know how to do it. "The Dirichlet L functions for non-principal Dirichlet characters are ...
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Characteristic function of Weibull distribution

I am recently learning about characteristic functions. I find them very interesting, but I seem to be misunderstanding something. Wikipedia claims that they are the Fourier transform of the ...
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Find the characteristic function of a random variable involving a compoud poisson distribution

Let $\{\xi_i \}_{i=1}^{k}$ be a finite sequence of independent r.v. such that $\xi_{i} \sim F_i$. For each $i$, let: $\{\xi_{i,j}\}_{j=1}^{\infty}$ be a sequence of copies of $\xi_i$, that is: $\xi_{...
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Characteristic function of linearly transformed random variables with extracted factor

Given is a sequence of independent random variables $X_1, X_2,\ldots, X_n$ with characteristic function $\varphi_{X_i}(t)$. The characteristic function of $Y = \sum_{i=1}^n a_i X_i$, where the $a_i$ ...
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2 votes
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How to calculate the covariance matrix in (Xu and Knight 2010)?

Setup I'm reading (Xu and Knight 2010), which is a paper about estimating finite Gaussian mixture models using the CECF (Continuous Empirical Characteristic Function) method. The basic idea is to ...
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Stationarity of AR(2) model

I have one question, which is related to a solution presented in "A proof for the stationarity of an AR(2)" by Christoph Hanck. Since I still cannot write comments (I need at least 50 points)...
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3 votes
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Derivation of moment generating function for limiting distribution of sum of logbeta distributed variables

A sum of logbeta distributed variables occurs in this question Distribution with a given moment generating function Let, $X_j \sim Beta(j\sigma, 1-\sigma)$, $Y_j = -\log(X_j)$ and $S_n = \sum_{j=1}^n ...
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Characteristic function of random vector

Consider a bivariate probability distribution $$f(x,y)=\frac{a^2}{2\pi} \exp \left(-a\sqrt{x^2+y^2}\right)$$ where $a>0, -\infty<x<\infty, -\infty<y<\infty$. In this situation, I want ...
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2 votes
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Finding $\mathbb{E}|X|$ for variance-gamma random variable

Assume $X \sim f_{X}(n,\mu, \sigma)$ is a Variance-Gamma random variable. The density function involves a modified Bessel function, therefore is not that trivial to handle. I'm looking for $\mathbb{E}|...
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characteristic function of standard lognormal distribution

What is the characteristic function of standard lognormal distribution? By "$X$ yields the standard lognormal distribution", I mean $\log(X)\sim N(0,1)$. I understand that the moments of $X$ ...
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Characteristic Function and Quantile Function?

Characteristic functions (cf) are closely related to cdfs and pdfs of random variables, for example cf is the Fourier transform of the pdf Inversion formulae from Lévy and Gil-Pelaez Question: Is ...
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tails from cumulant generating function

I am trying to obtain finite-sample tail estimates using the mgf. I found a snippet in a google books preview of a formula giving the survivor function by some kind of inversion of the cumulant ...
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$lim_{n→∞}P((/1√n)\sum\limits_{i=1}^n(−1)^{i}X_{i}\leq$x) equals?

Let X1, X2, . . . be a sequence of i.i.d. random variables with variance 2. Then for all x, $lim_{n→∞}P((/1√n)\sum\limits_{i=1}^n(−1)^{i}X_{i}\leq$x) equals $(A) Φ(x√2)$ $(B) Φ(x/√2)$ $(C) Φ(x)$ $(D) ...
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Characteristic function and Fourier transform for a discrete random variable!

Let $\phi_{x}(t)= E [ e^{itx}]$ be the characteristic function If X is a continuous random variable, then: $\phi_{x}(t)= E [ e^{itx}] = \int e^{itx} f(x)dx$ (being $f(x)$ the probability density ...
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Goodness of fit using characteristic function

When assessing goodness of fit of a model, one can use, for example, a Q-Q plot of empirical vs theoretical distributions. But how does one perform a GoF assessment when there is no closed form ...
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Why do we use exponent in characteristic function?

A student who is attending probability 101, learned about normal distribution and generating functions recently. We are given a "generating function" as follows: $$G(t)=<e^{itx}>=\int_{-\infty}^...
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What is the characteristic function of a rectified Normal distribution?

Rectified Normal distribution is a hybrid distribution with the following pdf: $f(x;\mu ,\sigma ^{2})=\Phi (-{\frac {\mu }{\sigma }})\delta (x)+{\frac {1}{{\sqrt {2\pi \sigma ^{2}}}}}\;e^{{-{\frac ...
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How to perform van Zyl's (2018) ecf-based normality test in R [closed]

I have been reading quite a bit on empirical characteristic function (ecf) based normality tests, but cannot find any functions to perform such a test in R and lack the mathematical ability to figure ...
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How to find the characteristic function of a function related with Shannon entropy?

A random variable X is distributed with a known probability distribution $p(x)$. Suppose that $x$ is sampled in an independent and identically distributed process and with the results $\vec{x}=(x_1,...
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3 votes
2 answers
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Expected of number of discrete uniform variables whose sum is bigger than k (from characteristic function of discrete Irwin–Hall distribution?)

The problem Imagine we keep uniformly drawing $n$ integers $X_i$ from {0, 1, ..., 9} so that their sum is more than $10$. For instance, one draw would be {1, 0, 2, 5, 3}, hence $n=5$, and repeat this ...
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How can we write the below characteristic function?

Let us assume that $X$ is a random variable and $a$ is a constant. Now suppose $Y=a+bX$, what would the characteristic function of $Y$ would be? Is it? \begin{eqnarray} \mathbb{E}_X\left[\exp(iuY)\...
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On likelihood functions and characteristic functions

Let me preface this by saying that if someone manages to provide a solution to my problem, I will forever be indebted to them, as this problem has driven me crazy. Let us first assume that the ...
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4 votes
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A question about characteristic functions

A probability 101 question. We know that if two variables $X$ and $Y$ are independent then the characteristic function $\phi_{X+Y}(u)$ can be written as \begin{equation} \phi_{X+Y}(u)=\phi_{X}(u)\phi_{...
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A simple question about expectations

@psboonstra This is a valid point. After posting it, I too realized that the question is ill-posed. I attempted to oversimplify a problem that I had encountered in finding a characteristic function of ...
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Question about the log-normal distribution

The main object of my question is this: if $X$ has a log-normal distribution, $Y = X + Z$ and $Y$ has the same distribution as that of $Z^2$ (in other words, $F_{Z^2} = F_{X+Z}$) and $X, Z$ are ...
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13 votes
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Random variables $X, Z$ such that $Z$ and $\sqrt{X + Z}$ have the same distribution?

I am looking for the distribution of a random variable $Z$ defined as $$Z = \sqrt{X_1+\sqrt{X_2+\sqrt{X_3+\cdots}}} .$$ Here the $X_k$'s are i.i.d. and have same distribution as $X$. 1. Update I ...
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1 vote
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Difference between characteristic function and F-transform

I'm struggling to understand the difference between this two functions. I have this condition: $P_j:=\mathbb{Q}(S_T>K):=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{+\infty}Re[\frac{e^{iuK}f_j(u,x,v)}{iu}]\...
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1 vote
1 answer
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Finding the characteristic function of a simple linear PDF

X is a random variable with a pdf of $ f(x) = \begin{cases} x/2, & 0 \le x \le 2 \\ 0, & \text{otherwise} \end{cases}$ I tried finding the characteristic function of this but ended up with ...
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How to pass from {Probability density function, convolution} to {Probability density function, characteristic function}?

In Forsman, W.C. (1986) "Polymers in solution: theoretical considerations and newer methods of characterization", Springer, New York. https://www.springer.com/la/book/9780306421464 page 24, it states:...
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How to modify the mean and variance/dispersion of a given distribution

I am trying to find a parametric adjustment that allows modifying the mean and variance/dispersion of a given distribution. Ideally, this adjustment would be implemented through a parametric function ...
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Expected value of $e^{sP}$ where s is a complex number and $P$ is a Poisson rv

For each positive integer $N$, let $ B_N$ be a binomial $(N,1/3)$ random variable and $P$ be a Poisson(5) random variable. I am trying to understand the statistics of $B_P$. Could someone please hint ...
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Proving the characteristic function involving sequence of 2 iid variables [duplicate]

We have an iid sequence with: $X_1,X_2,...$ with mean $\mu$ and variance $\sigma^2$. We have another similar sequence: $$Y_1,Y_2,...$$. We have a sequence $U1,U_2,...$ where, $U_n=1/\sqrt(2(n\sigma^...
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5 votes
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Proof of Convergence in Distribution with unbounded moment

I posted the question here, but no one has provided an answer, so I am hoping I could get an answer here. Thanks very much! Prove that given $\{X_n\}$ being a sequence of iid r.v's with density $|x|^{...
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9 votes
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When to prefer the moment generating function to the characteristic function?

Let $(\Omega, \mathcal{F}, P)$ be a probability space, and let $X : \Omega \to \mathbb{R}^n$ be a random vector. Let $P_X = X_* P$ be the distribution of $X$, a Borel measure on $\mathbb{R}^n$. The ...
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finding process corresponding to laplace transform

I have a positive stochastic process $X(t)$ with Laplace transforms $$ \mathbb{E}\left[\mathrm{e}^{-uX(t)}\right]=\left(\frac{a+u\mathrm{e}^{-\kappa t}}{a+u}\right)^{b} $$ One can clearly see that the ...
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Good book on characteristic functions that includes the CF-proof of the CLT

The title basically says it all. I would like to learn about CF in order to understand the proof of the CLT that makes use of CF. Ideally I would like to read a book that does not only give proves of ...
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Sampling from characteristic/moment generating function

Suppose I am given a probability distribution only via its characteristic or moment generating function and I want to sample from that distribution to generate paths in a Monte Carlo simulation. Is ...
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Characteristic function inequality

Random variable $X$ and its characteristic function $\phi_X(t)$ then $$\Pr\left(|X|>\frac2T\right) \leq 2\left(1 - \frac1{2T}\int_{-T}^{T}\phi_X(t)dt\right) $$ I cannot find a way how to ...
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Noncentral chi² with a noncentral chi² noncentrality parameter

Denote by $\chi^2(\nu,\lambda)$ the noncentral chi-square distribution with degrees of freedom $\nu$ and noncentrality parameter $\lambda$. If $\Lambda \sim \chi^2(\nu, 2\theta)$ and $(X \mid \Lambda)...
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4 votes
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Application of Central Limit Theorem - Uniform Distribution

The following question I found on an old exam: Given $n$ i.i.d. random variables $X_k$, $1 \leq k \leq n$, with uniform distribution on $[-1,1]$, it is easy to compute the characteristic function of ...
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Probability distribution from moments? [duplicate]

I would like to get the probability distribution (either pdf or cdf) for a variable, by knowing the first n-moments of the distribution. I ask: Is there a standard way to deal with this, and maybe a ...
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1 vote
2 answers
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Dependent / Independent random variables with identical Cumulative distribution function

I'm stuck with an assignment, hope you guys can help. Question: Show, that there exist random variables $X,Y,X',Y'$ on a Probability Space $(\Omega, \mathscr{F},P)$, so that $X$ and $Y$ are not ...
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How can you determine if a function is heavy tailed from its characteristic function?

The question is given as follow: Let $N$ have a Poisson distribution with mean $\lambda$. $X_i$ is Cauchy distribution with mode 0 and and scaling parameter $1$. Find the characteristic function ...
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Sufficient statistic from characteristic function?

I have a density function $f(;\theta):{\mathbb R}\rightarrow{\mathbb R}_+$, where $\theta\in{\mathbb R}^d$. I know that there is a sufficient statistic $T$ of dimension $d$. If $\varphi_n$ is the ...
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Poisson binomial distribution-like problem

Given n trials, where, on each trial, you have a given probability of either winning or losing a set amount of money (with both the amount of money and the probability changing for each trial)- what ...
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3 votes
1 answer
318 views

Characteristic function of distribution

$$p(x)=e^{-2 |x|}$$ with x in [-inf, +inf]. I've calculated the characteristic function as $E[e^{ikx}]=\frac{1}{ik+2}-\frac{1}{ik-2}=\frac{4}{k^2+4}$. Now i'd like the moments.. so I suppose I should ...
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2 votes
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How to compute bernoulli distribution PDF from CF

The characteristic function for a Bernoulli distribution is $$\phi(t) = (q+pe^{it}) \text{ where } p+q=1$$ I also know that the relationship between $\phi(t)$ and the pdf $f(k)$ is the Fourier ...
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Characteristic function of uniform random variable [duplicate]

I am trying to find out expectation of a function of a uniform random variable. I am given a random variable $x$ that is uniformly distributed over the interval $[0, a]$. I want to find out the ...
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$X \sim$ Poisson$(λ)$. What is the distribution of $X/c$? $(c > 0)$

Application: $X$ is the number of particles in a closed volume. $c$ is a constant that converts from particle count to ($>0$) molar concentration. For various reasons, I want to model $Y = X/c$ ...
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characteristic function of a linear function of a random variable

What are the broad steps required to solve a question like this? Let $Y=aX+b$, where $X\sim\text{Exp}(\lambda)\,,\:\lambda>0$ and find the characteristic function of $Y$.
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