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.

learn more… | top users | synonyms

2
votes
1answer
412 views

Characteristic function of vector-valued random variables

I just begins my self-study on Brownian motion. I got stuck on the part about random-vector and characteristic function. Here are my questions: I'm not quite get about how characteristic function of ...
0
votes
1answer
22 views

Calculate characteristic function

$p(n)=(1-r)^2nr^{n-1},n=1,2,...$ $f(z)=1/(1-z)$ has derivative $f'(z)$ with convergent power series $f'(z)=1/(1-z)^2=1+2z+3z^2+...$ the answer I have got is $(1-r)^2e^{it}(1-re^{it})^{-2}$ , I am not ...
0
votes
1answer
23 views

Calculate the characteristic function $\varphi_W$ of W

$p(x)=xe^{-x}$ for $x\geq 0$ or $0$ otherwise. I tried to substitute $e^{-x}$ but then i found there is still a $x$ in front.
1
vote
1answer
26 views

If $A$ is unitary and $f_A(x)=f_B(x)$ and $m_A(x)=m_B(x)$ then $A$ is similar to $B$

Given $A_{n\times n},B_{n\times n} \in \mathbb C$ then: if $A$ is unitary and the characteristic polynomial $f_A(x)=f_B(x)$ then $B$ is also unitary. if $A$ is normal and $f_A(x)=f_B(x)$ ...
0
votes
1answer
34 views

If the characteristic polynomial of $A,B$ is the same does it mean that $A,B$ are similar?

If the characteristic polynomial of $A,B$ is the same does it mean that $A,B$ are similar? So I read that it's true only if $A,B$ are diagonalizable, but why? if the characteristic polynomial is ...
1
vote
1answer
80 views

Characteristic function and moment generating function: differentiating under the integral

In order to justify the interchange of the derivative and integral when differentiating a characteristic function, one can use the dominated convergence theorem: $$\frac{d}{dt} \int e^{itx} P(dx) ...
-2
votes
2answers
37 views

What probability mass, density or distribution function might corresponds to this moment generating function? [duplicate]

I have somehow come up with a random variable $X$ with moment generating function (assuming it exists) $$M_{X}(t) = -t (1 - e^t)$$ What is the probability mass, density or distribution function? It ...
1
vote
0answers
33 views

Solve IVP with the method of characteristics (quasilinear PDE) (+shockwaves)

The question i just can't figure out one bit is: Solve for the continious function $u(x,t)$: $$u_t+(1-2u)u_x=0 $$ $$-\infty < x < \infty, t>0$$ $$u(x,0)= \left\{ \begin{matrix} \frac{1}{4} ...
3
votes
1answer
43 views

Abstract machines that compute primitive recursive functions

What it the simplest (least powerful) abstract machine that can compute primitive recursive sets, i.e. sets whose characteristic or indicator function is primitive recursive? ...
0
votes
1answer
19 views

Probability with Indicators Textbooks

I am new to using indicator functions (although I am quite familiar with undergrad-level probability and what an indicator function is). I am trying to relearn probability using indicator functions ...
2
votes
0answers
47 views

Proofs of some characteristic function properties

I saw these properties in the wikipedia page but I was unable to prove them. I had an idea to construct a random variable with the desired characteristic function, yet I haven't managed to that so ...
3
votes
1answer
53 views

The partial derivative of a characteristic function (exercise).

Assume that you have a probability space $(\Omega, \mathcal{F},P)$ and a random varaible $X: (\Omega, \mathcal{F})\rightarrow(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$. Define the characteristic ...
6
votes
0answers
51 views

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 ...
0
votes
0answers
25 views

what are some drawbacks of the characteristic method for solving ODE?

for those who are unfamiliar with this method: http://imgur.com/a/kefvI What are some differential equations that can't be solved using this method? What are eventual disadvantages using methods. I ...
0
votes
0answers
19 views

$p$-adic digits via character sums

Let $p$ be a prime and let $n = \sum_{k=0}^\infty n_k p^k$ be a $p$-adic integer with each $0 \leq n_k \leq p-1$. Fix $0 \leq c \leq p-1$. Is there a way to check whether the $i$-th digit $n_i$ equals ...
0
votes
1answer
39 views

equivalence of properties of characteristic function of a random variable

I would like to prove that for a random variable $X$ and its characteristic function $\phi_X$ the following three properties are equivalent. $i) \ \phi_X(s) = 1$ for some $s \neq 0$ $ii) \ \phi_X$ ...
5
votes
1answer
86 views

Tensor product for vector bundles is commutative, associative, and has an identity element?

How do I see that the tensor product operation for vector bundles over a fixed base space is commutative, associative, and has an identity element?
2
votes
1answer
62 views

Does $\varphi_{X+Y}(t)=\varphi_X(t)\varphi_Y(t)$ imply independence of $X$ and $Y$? [duplicate]

It shouldn't, but I am blanking on a counterexample. ETA: Note that the $t$ is shared on both sides - which differentiates this from this question. Similarly $F_{X,Y}(x,y)=F_X(x)F_Y(y)$ implies ...
2
votes
0answers
29 views

Characteristic function of $\chi^2$ distribution with $n$ degrees of freedom

I'm computing the formula for the characteristic function of the random variable $X \sim \chi^2(n), $ $n\in\mathbb{N}$. After some substitutions in the integral and some messing around with certain ...
2
votes
1answer
31 views

Why $(1 - a|x|) \cdot I(|x| \leqslant \frac{1}{a})$ is a characteristic function?

I'm trying to prove Pòlya's theorem, but got stuck at the very first step (I was given a plan of the proof) This step is to proof, that $\phi_a(x) = (1 - a|x|) \cdot I(|x| \leqslant \frac{1}{a})$ is ...
0
votes
1answer
27 views

Calculating characteristic function of random variable

I would like to calculate the characteristic function of $Z_{\beta, n}=(1-\beta^2)^{1/2}\sum_{k=0}^n\beta^kX_k$, where $X_i$ are independent random variables with $P(X_i = 1)=P(X_i=-1) = 1/2$ and ...
1
vote
1answer
22 views

Two matrices $A, B$ satisfying in characteristic polynomial of $B$ and $A$, respectively.

Let matrices $A,B\in M_n(\mathbb{C})$ such that $A$ satisfy in characteristic polynomial of $B$, and $B$ satisfy in characteristic polynomial of $A$. Can we say that: $A$ is diagonalizable if and ...
1
vote
1answer
39 views

Characteristic function notation

Having a rather basic understanding of probabilities I would like to ask you what exactly means the following notation. I am looking at the Gardiner's handbook for stochastic methods and am interested ...
0
votes
0answers
17 views

Proving two representations of a simple function are equivalent

Let's allow $\phi(x)=\sum^{n}_{j=1}a_{j}\chi_{A_{j}}(X)$ and also lets let $\phi(x)=\sum^{m}_{k=1}b_{k}\chi_{B_{k}}(X)$ be two different representations of the same simple function $\phi(x)$. Prove ...
0
votes
0answers
47 views

Calculation of infinite product

My question is to prove the identity: $$ \prod_{n=1}^{\infty}\left(\frac{\cos t-1}{n}+1\right)=\exp\left(-\int_0^1x^{-1}(1-\cos xt)dx\right) $$ which arises as a product of characteristic functions of ...
2
votes
0answers
41 views

Joint characteristic function of $X$ and $F(X)$

$X$ is a random variable. Its distribution function and characteristic function are $F_X$ and $\phi_X$, respectively. Then, we know, $F_X(X)$ follows uniform distribution. Let's say, $U=F_X(X)$. My ...
1
vote
1answer
38 views

Characteristic function using conjugate property

To prove that $e^{−i|x|}$ is not a characteristic function: $$e^{−i|x|} =\cos|x|-i \sin|x|.$$ Its conjugate will be $\cos|x|+i \sin|x|$ which is not equal to $\phi(-x)$. Is my solution correct?
14
votes
1answer
6k views

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 ...
0
votes
1answer
611 views

How can you plot straight lines in Matlab using only values on x axis and the gradient of each line?

I'm trying to plot characteristics for the Burgers Equations. But I have to plot them only using the inbuilt function plot. It seems like a fairly straight forward problem but I still cant solve it
2
votes
0answers
47 views

Distribution of $aX+bX^2+cX^3$ where $X$ is standard normal

I am looking for some distributional characteristic (for example a characteristic function) of a random variable which is defined as $aX+bX^2+cX^3$, where $X$ is a standard normal variable. Is there ...
2
votes
1answer
44 views

Characteristic exponent of $\alpha$-stable Levy process

I'm studying the book "Probability and Stochastics" by Erhan Cinlar - the probability class that I took covered until the beginning of chapter 7, and I'm now trying to do some of the exercises but ...
2
votes
2answers
40 views

Sam and Jane play with indicators.

There are $30$ red balls and $50$ white balls. Sam and Jane take turns drawing balls until they have drawn them all. Sam goes first. Let $N$ be the number of times Jane draws the same color ball as ...
3
votes
1answer
44 views

Why is a characteristic function continuous at $0$?

My lecture notes say: $t \mapsto \exp(-t^2/2)$ is a characteristic function (of $\mathcal{N}(0,1)$), so it is clear that it is continuous at $0$. So why does "being a characteristic function" ...
12
votes
1answer
3k views

The partial expectation $\mathbb{E}(X;_{X>K})$ for an alpha-stable distributed random variable

The partial expectation $\mathbb{E}(X;_{X>K})$ for an alpha-stable distributed random variable: By playing with convolutions of Characteristic Functions of alpha-Stable distributions $S(\alpha, ...
0
votes
0answers
42 views

Characteristic function of Cantor distribution

In Wiki, it provided the Characteristic function of Cantor distribution. That is, $e^{\mathrm{i}\,t/2}\prod_{i= 1}^{\infty} \cos{\left(\frac{t}{3^{i}} ...
7
votes
1answer
49 views

How to show the following characteristic function is positive definite

Let $\phi$ is the characteristic function of a probability measure on $\mathbb{R}$, how can I prove $\sum_{i,j=1}^{n}{\phi(t_i-t_j)\bar{\phi}(t_i-t_j)\xi_i\bar{\xi_j}}\geq 0$ for all $n\in\mathbb{N}$, ...
3
votes
1answer
22 views

difference of characteristic function for measure and random variable

Suppose random variable $X$ follow a certain (known) distribution. And I denote the probability measure $\mu$ as the distribution (pushforward measure) of $X$. Is there any difference between ...
1
vote
1answer
44 views

Convolution of indicator functions with values in a finite field

This is something I haven't seen online yet, indicator functions with values in a finite field. Probably for a good reason, but I would like to know why, and if there are still things that can be ...
0
votes
2answers
19 views

Characteristic and Principal Ideal.

This might be a simple question for some of you, but I am quite confused on the whole concept of principal ideals. Question 1: What is the characteristic of $\mathbb{Z}_2[X,Y]$ where it is the ring ...
1
vote
0answers
45 views

Suppose that $Y_{\lambda}=^{d}P(\lambda)$. Prove that $[Y_{\lambda}-\lambda]/{\sqrt{\lambda}}\to^{d}N(0,1)$

Suppose that $Y_{\lambda}=^{d}P(\lambda)$. Prove that $[Y_{\lambda}-\lambda]/{\sqrt{\lambda}}\to^{d}N(0,1)$ when $\lambda \to \infty$ using characteristic functions. So $$\phi(t)=\sum_{k=0}^{\infty} ...
4
votes
1answer
62 views

How to calculate $\int_0^1\cdots\int_0^1\frac{1}{x_1+x_2+\cdots+x_n+1}dx_1\cdots dx_n$

I met an integral $$\int_0^1\cdots\int_0^1\frac{1}{x_1+x_2+\cdots+x_n+1}dx_1\cdots dx_n$$ I calculated $n=1,2,3$ and made an induction! then i got the result: ...
3
votes
1answer
34 views

$X\sim\mathcal N(0,1)$, Why is $\Phi_X^{(j)}(0)=0$ for $j$ odd?

If $X\sim\mathcal N(0,1)$ Why is $\Phi_X^{(j)}(0)=0$ for $j$ odd ? ($\Phi_X^{(j)}(0):j^{th}$ derivative of the characteristic function of the r.v. $X$) We computed ...
0
votes
1answer
14 views

Expected value of number of “good” triples in the set of vectors

Given m vectors $v_1, v_2, ...,v_m$ in $\mathbb{R}^n$ with coordinates $0$ or $1$. Each vector is a result of $n$ Bernoulli trials. It may be that $v_i = v_j, i \neq j$. Consider three vectors: ...
1
vote
2answers
36 views

Aymptotic Convergence of Mean Estimator

I want to show: Let $x_{i}$ be an iid random variable with support $x_{i} \in [0,1]$. Prove $n^{1/3}\frac{1}{n} \sum\limits_{i=1}^{n} (x_{i} - \mathbb{E}[x_{i}] ) \xrightarrow{p} 0$. From ...
1
vote
0answers
53 views

integral of $e^{-\imath x}$ from $a$ to $\infty$ and characteristic functions

In the following result: $$\frac{1}{2 \pi} \int_{-\infty}^{\infty} \Phi(u) \left[\int_a^{\infty} e^{-i u x}dx\right] du = \frac{1}{2} + \frac{1}{ \pi} \int_0^{\infty} \Re\left[\frac{e^{-i u a} ...
-1
votes
1answer
37 views

Let $M$ be an $n\times n$ diagonalizable matrix with characteristic polynomial, find the matrix

Let $M$ be an $n\times n$ diagonalizable matrix with characteristic polynomial $a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$, Find the matrix $a_nM^n+a_{n-1}M^{n-1}+...+a_1M+a_oI$, where $I$ is the ...
1
vote
1answer
35 views

Find the characteristic polynomial of $(M^{-1})^3$

Given that M is a square matrix with characteristic polynomial $p_{m}(x) = -x^3 +6x^2+9x-14$ Find the characteristic polynomial of $(M^{-1})^3$ My attempt: x of $(M^{-1})^3$ is $1^3$, $(-2)^3$ , ...
1
vote
1answer
40 views

Convolution of characteristic function

I am trying to figure out following problem. Let A ⊂ R. Then we can define the characteristic function: Let a be bigger than 0. I am trying to find a following convolution: \begin{align} ...
4
votes
0answers
49 views

Exsistence of random variable $U[-1,1]$

I have problems with the following question: Let X be the random variable such that $X\sim U[-1,1]$. Does there exsist a random variable Y, independent to X, such that $X+Y\sim 2Y$? I thought that ...
1
vote
0answers
23 views

PDE -how to find characteristic function

How to find characteristic of the following equation $\mathbf{e^{-x}u_{xx}+2 \hspace{1pt} e^yu_{xy}+e^x \hspace{1pt} u_{x}=0}$ would the characteristic derived from the below equation? ...