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Questions tagged [characteristic-function]

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Random variables $X, Y$ such that $X$, $Y$ and $\sqrt{X + \sqrt{Y}}$ belongs to the same family of distributions?

Is there a family of positive distributions such that if $X$ has the distribution in question, then $\sqrt{X}$ also has a distribution from the same family. Ideally, it would be great if $X+Y$ also ...
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Question on excluding one point on the support of a continuous random variable

Suppose that I have a continuous random variable $X$ which has a support equal to $\mathbb{R}$. Can I construct a new random variable $Y$ such that it is equal in distribution to $X$ on the support of ...
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Classification of random variables with (non-)vanishing characteristic function

What is a necessary and sufficient condition for a random variable to have non-vanishing characteristic function? I wonder how should one check whether a random variable's characteristic function is ...
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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 momenta

I would like to get the probability distribution (either pdf or cdf) for a variable, by knowing the first n-momenta of the distribution. I ask: Is there a standard way to deal with this, and maybe a ...
171 views

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 ...
226 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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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 ...
170 views

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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How would you explain Characteristic Function in layman's terms?

What is a Characteristic Function? Why is it needed? Can you explain it in layman's terms and along with a simple & easy example? Please, limit using formal math notations as far as possible.
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Central limit theorem proof not using characteristic functions

Is there any proof for the CLT not using characteristic functions, a simpler method? Maybe Tikhomirov or Stein's methods? Something self-contained you can explain to a university student (first year ...
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I came across a question which asked to obtain the probability function of $X$ (a discrete random variable) with its characteristic function given as follows: $${\phi _X}(t) = {e^{\lambda ({e^{it}} - ... 1answer 74 views Finding the characteristic function of Y \sim U(-1,1) I know that \phi_Y(t) = E(e^{itY})=E(\cos(tY))+iE(\sin(tY)) After integration I have found that E(\cos(tY))= \frac{\sin(t)}{t} and E(\sin(tY))=0. So is the characteristic function just \frac{\... 0answers 26 views Joint cumulants of Zn2 characters Let f_{c}:Z_2^n \rightarrow \{-1,1\} be the character defined as f_c(x) = (-1)^{<x,c>}, where c,x \in Z_2^n. It is easy to see that since f_{c_1}\cdot\ldots\cdot f_{c_k} = f_{c_1 \oplus \... 1answer 183 views Characterizing clusters by separate feature vector scores Say I have a medium amount of dependent variables in a study. These are scores from questionnaires that have been standardized so all are on a scale from 0 to 1. I have clusters of my patients - ... 2answers 93 views Show that Y_1 X_1 + Y_2 X_2 \,{\buildrel d \over =}\, (Y_1^2+Y_2^2)^{1/2}X_1 I would like verification of my solution to the following problem. QUESTION: Let X_1, X_2 \,{\buildrel iid \over \sim }\, N(0,1)  and let Y_1, Y_2 be two independent random variables (X_1, ... 0answers 226 views Characteristic functions can establish stochastic dominance? In an answer to the question here What is the purpose of characteristic functions? people answered the general question about characteristic functions. One answer mentioned that one can use it to ... 1answer 2k views characteristic functions and symmetry If the characteristic function of a random variable is a real-valued function, does this imply that the random variable must be symmetric about zero? 1answer 2k views Moment-generating function or characteristic function of univariate skew-t distribution Is there a moment-generating function or a characteristic function for a univariate skew-t distribution y\sim ST\left(\xi,\omega^2,\alpha,\nu\right) as defined by Azzalini? 1answer 1k views Characteristic function of the Dirac delta? What is the characteristic function of the Dirac delta function? Is it e^{i*0}=1? 2answers 195 views Characteristic function problem Suppose X_1 and X_2 are independent random variables and suppose also that X_1 and X_1-X_2 are independent. Show that$$\mathbb{P}_{X_1}[X_1=c]=1 for some constant $c$. What I get so ...
I've been calculating characteristic functions and MGF's and was wondering whether we can always get the characteristic function simply by substituting $it$ instead of $t$ in the resulting equation. ...
Prove that $E[g(X)] = \int_{-\infty}^{\infty}G(t)\phi(t) dt$
Let $X$ denote a real-valued random variable with characteristic function $\phi$. Suppose that $g$ is a real-valued function on $\mathbb{R}$ that has the representation \$\hspace{25mm}g(x) = \int_{-\...