Mathematics

Questions tagged [characteristic-functions]

Questions about characteristic functions, of a set (which gives $1$ if the element is on the set and $0$ otherwise) or of a random variable (its Fourier transform). Do not use this tag if you are asking about the method of characteristics in PDE or the characteristic polynomial in linear algebra.

Given a set $A \subseteq X$, the characteristic function of $A$ is the function $\chi_A : X \to \mathbb{R}$ given by

$$\chi_A(x) = \begin{cases} 1 & x \in A\\ 0 & x \notin A. \end{cases}$$

Characteristic function defined as above is a synonym for indicator function.

1290 questions
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Characteristic function of a standard normal random variable

The characteristic function of a random variable $X$ is given by $$\Phi_X(\omega) = \mathbb{E}e^{j\omega X}=\int_{-\infty}^\infty e^{j\omega x}f_X(x) dx.$$ One can easily capture the similarity between this integral and the Fourier transform . For a…
Federico Magallanez
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Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent.

This is solely a reference request. I have heard a few versions of the following theorem: If the joint moment generating function $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ whenever the expectations are finite, then $X,Y$ are…
nullUser
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A random variable is symmetric if and only if its characteristic function is real-valued

Quick summary: I am stuck on the implication: $\phi_X$ real-valued $\rightarrow$ $X$ symmetric. Assume you have a probability space $(\Omega, \mathcal{F},P)$, and a random varaiable $X: \Omega \rightarrow \mathbb{R^d}$, it is measurable with respect…
user119615
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A criterion for independence based on Characteristic function

Let $X$ and $Y$ be real-valued random variables defined on the same space. Let's use $\phi_X$ to denote the characteristic function of $X$. If $\phi_{X+Y}=\phi_X\phi_Y$ then must $X$ and $Y$ be independent?
Jeff
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Characteristic function of exponential and geometric distributions

I'm trying to derive the characteristic function for exponential distribution and geometric distribution. Can you please guide me on getting them? Here is my solution so far: Characteristic function of exponential distribution: $\phi(t) =…
user48405
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Distribution of the sum of absolutes values of T-distributed random variables

Where $X$ is a r.v. following a symmetric $T$ distribution with $0 $ mean and tail parameter $\alpha$ . I am looking for the distribution of the $n$ -summed independent variables $ \sum_{1 \leq i \leq n}|x_i|$ . $Y=|X|$ has for PDF $\frac{2…
Nero
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How to get PDF from characteristic function

I would appreciate if anybody could explain to me with a simple example how to find PDF of a random variable from its characteristic function. Thank you.
May
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Intuitions about positive definite functions

I am looking for further intuitions about positive definite functions, and have several related questions on this matter. I know this isn't the most specific question, but I find that speaking intuitions can be very helpful and can actually at times…
rem
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Does "independence" of moments imply independence?

Suppse you have two random variables $X,Y$ and you are given that for any $m,n$ that: $$E(X^n Y^m) = E(X^n)E(Y^m)$$ Does this imply that $X$ and $Y$ are independent? Are there some condtions on how fast the moments grow can be added to help? Attempt…
Mihai Nica
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Is this a new characteristic function for the primes?

I recently found the following function, that returns $1$ if its argument is a prime number, and $0$ otherwise. Let $\chi_{\mathbb{P}}: \mathbb{Z}\rightarrow\{0,1\}$ for $n>4$ be defined as follows: $$…
Klangen
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uniform convergence of characteristic functions

Assume that a sequence of probability measures $\mu_n$ converges weakly to $\mu$. Let $\phi_n$ and $\phi$ denote respetively the characteristic function of $\mu_n$ and $\mu$. Prove that $\phi_n$ converges uniformly to $\phi$ on any bounded interval.…
Shine
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$A$ and $A^2$ have same characteristic polynomial

Is it possible to have a non-identity $2 \times 2$ diagonalizable, invertible, complex matrix $A$ s.t characteristics polynomials of $A$ and $A^2$ are the same? I am not getting any hint even how to create one. I can start with two different…
Ri-Li
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Determining if something is a characteristic function

Suppose $X$ is a continuous random variable with pdf $f_X(x)$. We can compute its characteristic function as $\varphi_X(t)=\mathbb{E}[e^{itX}].$ Question: Given a function, say $\psi(t)$, how does one show that it is a characteristic function?…
eeeeeeeeee
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Show a set E is measurable iff its characteristic function is measurable.

I am trying to show that a set $E$ is measurable iff its characteristic function is measurable. I saw a proof online that someone wrote as a homework solution, however it doesn't make sense to me. They wrote: Suppose $E$ is a measurable set in a…
ai.jennetta
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Can $\log(1-U)-\log(U)+W$ be normally distributed, with $U$ uniform on $(0,1)$ and $W$ independent of $U$?

Assume that $U$ and $V$ are independent random variables with values in $(0,1)$ and that $U$ is uniformly distributed. Can it happen that $$L=\log\left(\frac{(1-U)V}{U(1-V)}\right)$$ is normally distributed? As a motivation, note that $L$ is the log…
Did
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