Questions tagged [characteristic-function]
The characteristic-function tag has no usage guidance.
96
questions
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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 ...
2
votes
2
answers
128
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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 ...
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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}]\...
- 111
1
vote
1
answer
89
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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
422
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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:...
- 549
1
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2
answers
3k
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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 ...
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^...
4
votes
1
answer
407
views
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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9
votes
2
answers
2k
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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 ...
- 13.9k
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
4
votes
0
answers
103
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 ...
- 2,160
5
votes
1
answer
2k
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Characteristic function example with Bernoulli and Poisson random variables
Let $\{X_n:n\ge 1\}$ be a sequence of i.i.d. Bernoulli random variables with probability of success $0<p<1$, i.e,
$$P\{X_1=1\}=1-P\{X_1=0\}=p.$$
The random variable $Y$ is independent of the ...
- 5,179
3
votes
1
answer
163
views
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_{-\...
- 5,179
13
votes
2
answers
2k
views
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 ...
- 5,179
4
votes
0
answers
1k
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What is the distribution of an affine transformation of a log-normal variable?
Let $X$ be a log-normal variate and $Y = aX + b$ is the affine transformation X.
Is $Y$ log-normal? I suspect it is not.
Since $X$ is log-normal, its expected value is
$$
E[X] = \exp(M + S^2/2)
$$
...
- 5,179
67
votes
14
answers
12k
views
What is the most surprising characterization of the Gaussian (normal) distribution?
A standardized Gaussian distribution on $\mathbb{R}$ can be defined by giving explicitly its density:
$$ \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$$
or its characteristic function.
As recalled in this ...
- 5,179
3
votes
1
answer
3k
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The mgf and cf of Student's t distribution
A student's t distributed rv $X$ has characteristic function but no moment generating function. I wonder if cf(X)=$E[e^{itX}]$, why we cannot take $t=-iu$ to get the mgf $E[e^{uX}]$? (This question ...
- 5,179
2
votes
1
answer
773
views
How to compute the characteristic function of two random variables with different distributions?
What are the steps to obtain the following result:
Given that X have $\Gamma(1,s)$ distribution; and that X=x, and Y have the Poisson distribution with parameter x. Then the characteristic function ...
- 5,179
48
votes
5
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What is the purpose of characteristic functions?
I'm hoping that someone can explain, in layman's terms, what a characteristic function is and how it is used in practice. I've read that it is the Fourier transform of the pdf, so I guess I know what ...
- 5,179
1
vote
0
answers
680
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Time series (stochastic process) estimating parameters using characteristic function
I have a time series of assets ${A_1, A_2, ..., A_n}$, which is described by a sophisticated distribution having the following characteristic function: $\phi(u; t;\theta)$, where $\theta$ is a vector ...
- 5,179
3
votes
1
answer
681
views
Deconvolution with fourier transform or characteristic function?
Let us consider the following model:
$$Y_j = X_j + \epsilon_j \hspace{15pt} j=1, ..., n$$
Where $Y_j$ is a noisy signal, $\epsilon_j$ is the noise which is independend from the signal $X_j$. We have ...
- 5,179
2
votes
1
answer
839
views
Characteristic Function proof
Let $\phi_1,\ldots,\phi_n$ denote characteristic functions for distributions on the real line. Let $a_1,\ldots,a_n$ denote nonnegative constants such that $a_1+\ldots+a_n = 1$. Show that
$$\hspace{...
- 5,179
2
votes
1
answer
806
views
Get Characteristic Function from MGF? [duplicate]
I've been calculating characteristic functions and MGF's and was wondering whether we can always get the characteristic function simply by substituting $it$ instead of $t$ in the resulting equation. ...
- 5,179
14
votes
1
answer
9k
views
Characteristic function and Fourier transform
I understand the definition of characteristic functions used in
probability theory:
For a random Variable $X$ with probability density function $f_X$ the characteristic function is defined as:
$$\...
- 5,179
1
vote
2
answers
669
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 ...
- 323k
2
votes
1
answer
111
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{\...
- 78.1k
1
vote
0
answers
57
views
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 ...
- 11
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votes
1
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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?
- 78.1k
0
votes
0
answers
229
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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 ...
- 323k
1
vote
1
answer
166
views
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 ...
- 1,257
3
votes
1
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482
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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 ...
- 13k
1
vote
1
answer
79
views
$X \sim$ 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 $Y = X/c$ ...
- 283k
4
votes
1
answer
2k
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Characteristic function of the Dirac delta?
What is the characteristic function of the Dirac delta function?
Is it $e^{i*0}=1$?
Tim♦
- 138k
0
votes
1
answer
222
views
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 $Y$.
- 283k
3
votes
1
answer
161
views
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.
CommunityBot
- 1
4
votes
0
answers
205
views
Sum of truncated Gammas and degenerate
I have a variable $X$ which I am modelling with a mixture model:
$$\begin{aligned}
(X|A) &\sim \mathbb{1}_{0 \leq x < w \cdot m} \cdot \frac{\text{Gamma}(\alpha,0,\beta / m)}{k_1} \\
(X|B) &...
- 525
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vote
1
answer
574
views
Plot the density function of a normal random variable knowing only the characteristic function in R
The characteristic function of a normal random variable with mean $\mu$ and
standard deviation $\sigma$ is:
$$\begin{alignat*}{1}
\hat{\phi}(t) & =e^{i\mu t}e^{-\frac{1}{2}\sigma^{2}t^{2}}\\
&...
- 301
4
votes
2
answers
2k
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Derivation of PMF of Poisson Distribution from its Characteristic Function
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}} - ...
- 7,663
1
vote
1
answer
101
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Characteristic function issue
As mentioned in a previous post, I've been trying to work through ALL of the problems in Jacod and Protter's Probability Essentials. The following problem has been giving me issues:
Let $Z \sim N(0,...
- 3,150
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votes
2
answers
2k
views
Why does multiplication in the frequency domain equal convolution in the time domain?
This question came in the context of understanding how to get a distribution of a sum of two iid random variables. I'm working through the top answer to this question Consider the sum of $n$ uniform ...
CommunityBot
- 1
5
votes
1
answer
310
views
Maximise expectation of exponential given mean and variance
The problem is as follows:
Suppose that $X$ is a random variable with $\mathbb{E}X=0$, $\mathbb{D}X = \sigma^2$ and having a finite support: $P(|X|\leq a)=1$. What is the maximum possible value of $\...
- 339
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vote
0
answers
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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 \...
- 537
1
vote
1
answer
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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 - ...
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3
votes
2
answers
132
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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, ...
- 19.2k
12
votes
1
answer
8k
views
How to find a density from a characteristic function?
A distribution has the characteristic function
$$\phi(t) = (1-t^2/2)\exp(-t^2/4),\ -\infty \lt t \lt \infty$$
Show that the distribution is absolutely continuous and write the density function of ...
- 323k
4
votes
2
answers
372
views
Characteristic function problem
Suppose $X_1$ and $X_2$ are independent random variables and suppose also that $X_1$ and $X_1-X_2$ are independent. Show that
$$\mathbb{P}_{X_1}[X_1=c]=1$$
for some constant $c$.
What I get so ...
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