Mathematics Stack Exchange
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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.

1252 questions
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Series expansion at $0$

Given: $X$, $Y$ iid random variables, $\mathbb E(X) = 0$, $\mathbb E(X^2) = 1$; $X+Y$ and $X-Y$ are independent; $\phi$ is the characteristic function of $X$ and $Y$ and $ \psi: t \rightarrow \dfrac{\phi(t)}{\phi(-t)}$. What is, in this case, the…
JohnD
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Characteristic Function and Expected Value

I have another question regarding the Indicator Function, namely understanding the following equality: $ 1 - E[\prod \limits_{i=1}^n (1 - 1_{A_i})] = \sum_{k=1}^{n} (-1)^{k+1} \sum_{1 \leq i_1 < ... < i_k \leq n} E[1_{A_{i_{1}}} * ... *…
user62487
  • 483
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Finding characteristic function and differentiate to get expectation

I was asked to find the characteristic function of a pdf and differentiate it to get the expectation. $p(x) = xe^{-x}$ for $x \ge 0$ I am doing this in the following way. Sorry that i don't know how to insert the correct symbol.... $$ϕw =…
Thomas
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Complex Conjugates of Characteristic Function

If $q(x)$ is the pdf, I can write it in terms of the characteristic function: $$q(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{-i\phi x} f_j(x,t,\phi)\, d\phi. $$ I see from the literature I'm reading that I can rewrite this as $$q(x) =…
SingularityFuture
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Characteristic function of a gaussian vector $R^4$

Let X : $\Omega \mapsto \mathcal{R}^4$ be a Gaussian vector with ${E}(X) = 0. $ Express: $$ \int_\Omega X_1(\omega)X_2(\omega)X_3(\omega)X_4(\omega) \mathcal{P}(d\omega) $$ as a function of $a_{ij}$ = Cov($X_i,X_j)$ with 1 $\leq i < j \leq…
Ricter
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Property of the characteristic function for events

We got to show the following equality: $1_{A_1 \cup...\cup A_n} = 1 - \prod \limits_{i=1}^n (1 - 1_{A_i})$ First I would like to ask for hints for how to proove this equation (no solution though, I would like to solve it myself). Secondly I was…
user62487
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Characteristic function of a standardized sum

Let ${X_n}$ be an iid binary random process with equal probability of $+1$ or $-1$ occurring at any time n.Now,if $Y_n$ is the standardized sum and equal to $\frac{1}{\sqrt{n}} \sum ^{n-1}_{k=0}X_{k}$,then please show that its characteristic…
XM551
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Finding a characteristic function of an exponential pdf

My pdf is defined as follows: $$f_X(x) = \frac{1}{\tau} e^{-x/\tau}$$ At first I started finding the characteristic function like so: $$\hat{f}_X(\xi) = \mathbb{E}[e^{i\xi X}] = \frac{1}{\tau}\int_{\mathbb{R}} e^{i\xi x}e^{-x/\tau}dx$$ I then wrote…
Naz
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What is the characterstic function?

Good evening, what is the characteristic function of a distribution with density function $$f(x) = \frac{1}{2\lambda}e^{-\frac{|x|}{\lambda}},$$ while $x\in\mathbb{R}$ with parameter $\lambda > 0$?
MathCracky
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How do I infer that $\lim_{k \rightarrow \infty} \chi_{B_k} =0$?

How do I infer that $\lim_{k \rightarrow \infty} \chi_{B_k} =0$? I know that $B_k$ measurable, And there's a set inclusion $ ... \subset B_2 \subset B_1$. And $m(B_k) \rightarrow 0$.
mavavilj
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Check Characteristic function

How to check whether $\sin(t)\over t$ is a characteristic function or not? If $\sin(t)\over t$ is a distribution function.
siddhartha kumar verma
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Proof with characteristic function

If f : X → N,g : X → N xA(x)= {1. if x E A; {0 if x E X\A Write the functions χAχB, χA +χB − χA∩B,χA + χB − 2χA∩B in the form χC. Ive managed to get χAχB. But I don't understand how to do the other two χA +χB − χA∩B,χA + χB − 2χA∩B Im…
M.Jones
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