Questions tagged [characteristic-function]
The characteristic-function tag has no usage guidance.
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1answer
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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 ...
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0answers
48 views
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:...
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2answers
118 views
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 ...
2
votes
2answers
56 views
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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0answers
17 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^...
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3answers
158 views
Proof of Convergence in Distribution with unbounded moment
I posted the question here, but no one has provided an answer, so I am hoping I could get an answer here. Thanks very much!
Prove that given $\{X_n\}$ being a sequence of iid r.v's with density $|x|^{...
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2answers
418 views
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 ...
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0answers
23 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 ...
2
votes
0answers
34 views
Good book on characteristic functions that includes the CF-proof of the CLT
The title basically says it all. I would like to learn about CF in order to understand the proof of the CLT that makes use of CF. Ideally I would like to read a book that does not only give proves of ...
3
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1answer
147 views
Sampling from characteristic/moment generating function
Suppose I am given a probability distribution only via its characteristic or moment generating function and I want to sample from that distribution to generate paths in a Monte Carlo simulation. Is ...
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0answers
58 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 ...
6
votes
1answer
88 views
Noncentral chi² with a noncentral chi² noncentrality parameter
Denote by $\chi^2(\nu,\lambda)$ the noncentral chi-square distribution with degrees of freedom $\nu$ and noncentrality parameter $\lambda$.
If $\Lambda \sim \chi^2(\nu, 2\theta)$ and $(X \mid \Lambda)...
4
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1answer
100 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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0answers
79 views
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 ...
1
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2answers
157 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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0answers
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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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0answers
32 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 ...
1
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1answer
98 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 ...
3
votes
1answer
208 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 ...
3
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1answer
235 views
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 ...
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0answers
48 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 ...
0
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1answer
45 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 $Y = X/c$ ...
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1answer
75 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 $Y$.
2
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1answer
70 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.
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2answers
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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 ...
3
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0answers
131 views
Sum of truncated Gammas and degenerate
I have a variable $X$ which I am modelling with a mixture model:
$$\begin{aligned}
(X|A) &\sim \mathbb{1}_{0 \leq x < w \cdot m} \cdot \frac{\text{Gamma}(\alpha,0,\beta / m)}{k_1} \\
(X|B) &...
5
votes
2answers
1k views
Characteristic Function of a Compound Poisson Process
The definition of a compound Poisson process and its characteristic function I have are the following:
Let $\lambda>0$ and $N\sim\text{Poisson}(\lambda T)$. Also, $\{X_i\}_{i=1}^N$ are i.i.d. ...
2
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0answers
325 views
What is the distribution of an affine transformation of a log-normal variable?
Let $X$ be a log-normal variate and $Y = aX + b$ is the affine transformation X.
Is $Y$ log-normal? I suspect it is not.
Since $X$ is log-normal, its expected value is
$$
E[X] = \exp(M + S^2/2)
$$
...
0
votes
1answer
228 views
Plot the density function of a normal random variable knowing only the characteristic function in R
The characteristic function of a normal random variable with mean $\mu$ and
standard deviation $\sigma$ is:
$$\begin{alignat*}{1}
\hat{\phi}(t) & =e^{i\mu t}e^{-\frac{1}{2}\sigma^{2}t^{2}}\\
&...
1
vote
1answer
40 views
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
votes
1answer
886 views
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 ...
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votes
2answers
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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 ...
5
votes
1answer
2k views
What is “t” in generating functions?
I am studying generating functions applied to probability (moment generating functions, probability generating functions and characteristic functions). I perfectly see their purposes and usefulnesses, ...
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1answer
167 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 $\...
4
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2answers
622 views
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}} - ...
2
votes
1answer
71 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{\...
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0answers
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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 \...
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1answer
168 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 - ...
2
votes
2answers
92 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, ...
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0answers
203 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 ...
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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?
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1answer
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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?
3
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1answer
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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$?
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votes
2answers
177 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$$
for some constant $c$.
What I get so ...
2
votes
1answer
267 views
Get Characteristic Function from MGF? [duplicate]
I've been calculating characteristic functions and MGF's and was wondering whether we can always get the characteristic function simply by substituting $it$ instead of $t$ in the resulting equation. ...
3
votes
1answer
105 views
Prove that $E[g(X)] = \int_{-\infty}^{\infty}G(t)\phi(t) dt$
Let $X$ denote a real-valued random variable with characteristic function $\phi$. Suppose that $g$ is a real-valued function on $\mathbb{R}$ that has the representation
$\hspace{25mm}g(x) = \int_{-\...
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1answer
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Characteristic Function proof
Let $\phi_1,\ldots,\phi_n$ denote characteristic functions for distributions on the real line. Let $a_1,\ldots,a_n$ denote nonnegative constants such that $a_1+\ldots+a_n = 1$. Show that
$$\hspace{...
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votes
1answer
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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 ...
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vote
0answers
595 views
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 ...
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1answer
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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 ...
