# Questions tagged [characteristic-function]

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### Name of PDF? - projecting uniform probability distribution on the unit circle to the x-axis

Consider a uniform probability distribution on a circle of radius r, i.e. $\{(x,y) \in \mathbb{R}^2: x^2 + y^2 = r^2 \}$.If we wish to project onto the x-axis, we can consider each point on the circle ...
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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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### 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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### 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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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 $(A) Φ(x√2)$ $(B) Φ(x/√2)$ $(C) Φ(x)$ $(D) ... 3 votes 1 answer 3k views ### Characteristic function and Fourier transform for a discrete random variable! Let$\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 ... 0 votes 0 answers 75 views ### 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 ... • 553 1 vote 0 answers 54 views ### Why do we use exponent in characteristic function? [closed] 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)=\langle e^{itx}\... • 11 4 votes 1 answer 663 views ### 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 ... 0 votes 1 answer 56 views ### 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 ... • 45 1 vote 0 answers 221 views ### 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,... 4 votes 2 answers 417 views ### 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 ... • 311 1 vote 0 answers 43 views ### 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)\... • 1,226 2 votes 0 answers 120 views ### 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 ... • 1,226 4 votes 1 answer 175 views ### 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 \phi_{X+Y}(u)=\phi_{X}(u)\phi_{... • 1,226 1 vote 0 answers 74 views ### 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 ... • 1,226 1 vote 0 answers 91 views ### 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 ... 13 votes 1 answer 487 views ### 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}}} .$$Here the X_k's are i.i.d. and have same distribution as X. 1. Update I ... 1 vote 0 answers 103 views ### Difference between characteristic function and F-transform I'm struggling to understand the difference between this two functions. I have this condition: P_j:=\mathbb{Q}(S_T>K):=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{+\infty}Re[\frac{e^{iuK}f_j(u,x,v)}{iu}]\... 1 vote 1 answer 89 views ### Finding the characteristic function of a simple linear PDF X is a random variable with a pdf of f(x) = \begin{cases} x/2, & 0 \le x \le 2 \\ 0, & \text{otherwise} \end{cases} I tried finding the characteristic function of this but ended up with ... • 35 1 vote 0 answers 425 views ### How to pass from {Probability density function, convolution} to {Probability density function, characteristic function}? In Forsman, W.C. (1986) "Polymers in solution: theoretical considerations and newer methods of characterization", Springer, New York. https://www.springer.com/la/book/9780306421464 page 24, it states:... 1 vote 2 answers 3k views ### How to modify the mean and variance/dispersion of a given distribution I am trying to find a parametric adjustment that allows modifying the mean and variance/dispersion of a given distribution. Ideally, this adjustment would be implemented through a parametric function ... • 317 2 votes 2 answers 129 views ### Expected value of e^{sP} where s is a complex number and P is a Poisson rv For each positive integer N, let B_N be a binomial (N,1/3) random variable and P be a Poisson(5) random variable. I am trying to understand the statistics of B_P. Could someone please hint ... 0 votes 0 answers 21 views ### Proving the characteristic function involving sequence of 2 iid variables [duplicate] We have an iid sequence with: X_1,X_2,... with mean \mu and variance \sigma^2. We have another similar sequence:$$Y_1,Y_2,...$$. We have a sequence U1,U_2,... where, U_n=1/\sqrt(2(n\sigma^... 5 votes 4 answers 592 views ### Proof of Convergence in Distribution with unbounded moment I posted the question here, but no one has provided an answer, so I am hoping I could get an answer here. Thanks very much! Prove that given \{X_n\} being a sequence of iid r.v's with density |x|^{... • 535 9 votes 2 answers 3k views ### When to prefer the moment generating function to the characteristic function? Let (\Omega, \mathcal{F}, P) be a probability space, and let X : \Omega \to \mathbb{R}^n be a random vector. Let P_X = X_* P be the distribution of X, a Borel measure on \mathbb{R}^n. The ... • 4,067 3 votes 0 answers 33 views ### finding process corresponding to laplace transform I have a positive stochastic process X(t) with Laplace transforms$$ \mathbb{E}\left[\mathrm{e}^{-uX(t)}\right]=\left(\frac{a+u\mathrm{e}^{-\kappa t}}{a+u}\right)^{b} $$One can clearly see that the ... • 528 3 votes 1 answer 77 views ### Good book on characteristic functions that includes the CF-proof of the CLT The title basically says it all. I would like to learn about CF in order to understand the proof of the CLT that makes use of CF. Ideally I would like to read a book that does not only give proves of ... 6 votes 2 answers 1k views ### Sampling from characteristic/moment generating function Suppose I am given a probability distribution only via its characteristic or moment generating function and I want to sample from that distribution to generate paths in a Monte Carlo simulation. Is ... • 528 6 votes 2 answers 159 views ### Characteristic function inequality Random variable X and its characteristic function \phi_X(t) then$$\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 ... • 483 5 votes 1 answer 260 views ### 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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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 ...