Questions tagged [characteristic-function]
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72
questions
0
votes
0answers
22 views
$lim_{n→∞}P((/1√n)\sum\limits_{i=1}^n(−1)^{i}X_{i}\leq$x) equals?
Let X1, X2, . . . be a sequence of i.i.d. random variables with variance 2.
Then for all x,
$lim_{n→∞}P((/1√n)\sum\limits_{i=1}^n(−1)^{i}X_{i}\leq$x)
equals
$(A) Φ(x√2)$
$(B) Φ(x/√2)$
$(C) Φ(x)$
$(D) ...
0
votes
0answers
12 views
Interpreting SAS output - Roots of AR Characteristic Polynomial
I urgently need help on interpreting the numbers from a SAS output on Characteristic Roots:
The VARMAX Procedure
The VARMAX Procedure Roots of AR Characteristic Polynomial
Index Real Imaginary Modulus ...
3
votes
1answer
80 views
Characteristic function and Fourier transform for a discrete random variable!
Let $\phi_{x}(t)= E [ e^{itx}]$ be the characteristic function
If X is a continuous random variable, then:
$\phi_{x}(t)= E [ e^{itx}] = \int e^{itx} f(x)dx$ (being $f(x)$ the probability density ...
0
votes
0answers
19 views
Goodness of fit using characteristic function
When assessing goodness of fit of a model, one can use, for example, a Q-Q plot of empirical vs theoretical distributions. But how does one perform a GoF assessment when there is no closed form ...
1
vote
0answers
35 views
Why do we use exponent in characteristic function?
A student who is attending probability 101, learned about normal distribution and generating functions recently.
We are given a "generating function" as follows: $$G(t)=<e^{itx}>=\int_{-\infty}^...
1
vote
1answer
75 views
What is the characteristic function of a rectified Normal distribution?
Rectified Normal distribution is a hybrid distribution with the following pdf:
$f(x;\mu ,\sigma ^{2})=\Phi (-{\frac {\mu }{\sigma }})\delta (x)+{\frac {1}{{\sqrt {2\pi \sigma ^{2}}}}}\;e^{{-{\frac ...
0
votes
1answer
39 views
How to perform van Zyl's (2018) ecf-based normality test in R [closed]
I have been reading quite a bit on empirical characteristic function (ecf) based normality tests, but cannot find any functions to perform such a test in R and lack the mathematical ability to figure ...
0
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0answers
29 views
How to find the characteristic function of a function related with Shannon entropy?
A random variable X is distributed with a known probability distribution $p(x)$. Suppose that $x$ is sampled in an independent and identically distributed process and with the results $\vec{x}=(x_1,...
3
votes
2answers
86 views
Expected of number of discrete uniform variables whose sum is bigger than k (from characteristic function of discrete Irwin–Hall distribution?)
The problem
Imagine we keep uniformly drawing $n$ integers $X_i$ from {0, 1, ..., 9} so that their sum is more than $10$. For instance, one draw would be {1, 0, 2, 5, 3}, hence $n=5$, and repeat this ...
1
vote
0answers
24 views
How can we write the below characteristic function?
Let us assume that $X$ is a random variable and $a$ is a constant. Now suppose $Y=a+bX$, what would the characteristic function of $Y$ would be? Is it?
\begin{eqnarray}
\mathbb{E}_X\left[\exp(iuY)\...
1
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0answers
27 views
On likelihood functions and characteristic functions
Let me preface this by saying that if someone manages to provide a solution to my problem, I will forever be indebted to them, as this problem has driven me crazy. Let us first assume that the ...
3
votes
1answer
40 views
A question about characteristic functions
A probability 101 question. We know that if two variables $X$ and $Y$ are independent then the characteristic function $\phi_{X+Y}(u)$ can be written as
\begin{equation}
\phi_{X+Y}(u)=\phi_{X}(u)\phi_{...
1
vote
0answers
71 views
A simple question about expectations
@psboonstra This is a valid point. After posting it, I too realized that the question is ill-posed. I attempted to oversimplify a problem that I had encountered in finding a characteristic function of ...
1
vote
0answers
80 views
Question about the log-normal distribution
The main object of my question is this: if $X$ has a log-normal distribution, $Y = X + Z$ and $Y$ has the same distribution as that of $Z^2$ (in other words, $F_{Z^2} = F_{X+Z}$) and $X, Z$ are ...
13
votes
1answer
388 views
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}}} .$$
Here the $X_k$'s are i.i.d. and have same distribution as $X$.
1. Update
I ...
1
vote
0answers
41 views
Difference between characteristic function and F-transform
I'm struggling to understand the difference between this two functions.
I have this condition: $P_j:=\mathbb{Q}(S_T>K):=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{+\infty}Re[\frac{e^{iuK}f_j(u,x,v)}{iu}]\...
1
vote
1answer
65 views
Finding the characteristic function of a simple linear PDF
X is a random variable with a pdf of $ f(x) =
\begin{cases}
x/2, & 0 \le x \le 2 \\
0, & \text{otherwise}
\end{cases}$
I tried finding the characteristic function of this but ended up with ...
1
vote
0answers
188 views
How to pass from {Probability density function, convolution} to {Probability density function, characteristic function}?
In Forsman, W.C. (1986) "Polymers in solution: theoretical considerations and newer methods of characterization", Springer, New York.
https://www.springer.com/la/book/9780306421464 page 24, it states:...
0
votes
2answers
349 views
How to modify the mean and variance/dispersion of a given distribution
I am trying to find a parametric adjustment that allows modifying the mean and variance/dispersion of a given distribution.
Ideally, this adjustment would be implemented through a parametric function ...
2
votes
2answers
76 views
Expected value of $e^{sP}$ where s is a complex number and $P$ is a Poisson rv
For each positive integer $N$, let $ B_N$ be a binomial $(N,1/3)$ random variable and $P$ be a Poisson(5) random variable. I am trying to understand the statistics of $B_P$. Could someone please hint ...
0
votes
0answers
19 views
Proving the characteristic function involving sequence of 2 iid variables [duplicate]
We have an iid sequence with: $X_1,X_2,...$ with mean $\mu$ and variance $\sigma^2$.
We have another similar sequence: $$Y_1,Y_2,...$$.
We have a sequence $U1,U_2,...$ where, $U_n=1/\sqrt(2(n\sigma^...
5
votes
3answers
221 views
Proof of Convergence in Distribution with unbounded moment
I posted the question here, but no one has provided an answer, so I am hoping I could get an answer here. Thanks very much!
Prove that given $\{X_n\}$ being a sequence of iid r.v's with density $|x|^{...
9
votes
2answers
1k views
When to prefer the moment generating function to the characteristic function?
Let $(\Omega, \mathcal{F}, P)$ be a probability space, and let $X : \Omega \to \mathbb{R}^n$ be a random vector. Let $P_X = X_* P$ be the distribution of $X$, a Borel measure on $\mathbb{R}^n$.
The ...
2
votes
0answers
23 views
finding process corresponding to laplace transform
I have a positive stochastic process $X(t)$ with Laplace transforms
$$
\mathbb{E}\left[\mathrm{e}^{-uX(t)}\right]=\left(\frac{a+u\mathrm{e}^{-\kappa t}}{a+u}\right)^{b}
$$
One can clearly see that the ...
2
votes
0answers
37 views
Good book on characteristic functions that includes the CF-proof of the CLT
The title basically says it all. I would like to learn about CF in order to understand the proof of the CLT that makes use of CF. Ideally I would like to read a book that does not only give proves of ...
3
votes
1answer
340 views
Sampling from characteristic/moment generating function
Suppose I am given a probability distribution only via its characteristic or moment generating function and I want to sample from that distribution to generate paths in a Monte Carlo simulation. Is ...
4
votes
0answers
72 views
Characteristic function inequality
Random variable $X$ and its characteristic function $\phi_X(t)$ then
$$\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 ...
5
votes
1answer
139 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
votes
1answer
193 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 ...
0
votes
0answers
97 views
Probability distribution from moments? [duplicate]
I would like to get the probability distribution (either pdf or cdf) for a variable, by knowing the first n-moments of the distribution.
I ask:
Is there a standard way to deal with this, and maybe a ...
1
vote
2answers
272 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 ...
2
votes
0answers
108 views
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 ...
1
vote
0answers
37 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
vote
1answer
123 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
237 views
Characteristic function of distribution
$$p(x)=e^{-2 |x|}$$ with x in [-inf, +inf].
I've calculated the characteristic function as $E[e^{ikx}]=\frac{1}{ik+2}-\frac{1}{ik-2}=\frac{4}{k^2+4}$. Now i'd like the moments.. so I suppose I should ...
2
votes
1answer
307 views
How to compute bernoulli distribution PDF from CF
The characteristic function for a Bernoulli distribution is
$$\phi(t) = (q+pe^{it}) \text{ where } p+q=1$$
I also know that the relationship between $\phi(t)$ and the pdf $f(k)$ is the Fourier ...
0
votes
0answers
223 views
Characteristic function of uniform random variable [duplicate]
I am trying to find out expectation of a function of a uniform random variable. I am given a random variable $x$ that is uniformly distributed over the interval $[0, a]$. I want to find out the ...
0
votes
1answer
48 views
$X \sim$ Poisson$(λ)$. What is the distribution of $X/c$? $(c > 0)$
Application: $X$ is the number of particles in a closed volume. $c$ is a constant that converts from particle count to ($>0$) molar concentration. For various reasons, I want to model $Y = X/c$ ...
0
votes
1answer
93 views
characteristic function of a linear function of a random variable
What are the broad steps required to solve a question like this?
Let $Y=aX+b$, where $X\sim\text{Exp}(\lambda)\,,\:\lambda>0$ and find the characteristic function of $Y$.
2
votes
1answer
95 views
How would you explain Characteristic Function in layman's terms?
What is a Characteristic Function?
Why is it needed?
Can you explain it in layman's terms and along with a simple & easy example?
Please, limit using formal math notations as far as possible.
11
votes
2answers
1k views
Central limit theorem proof not using characteristic functions
Is there any proof for the CLT not using characteristic functions, a simpler method?
Maybe Tikhomirov or Stein's methods?
Something self-contained you can explain to a university student (first year ...
3
votes
0answers
156 views
Sum of truncated Gammas and degenerate
I have a variable $X$ which I am modelling with a mixture model:
$$\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) &...
6
votes
2answers
2k 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. ...
3
votes
0answers
516 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
330 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:
$$\begin{alignat*}{1}
\hat{\phi}(t) & =e^{i\mu t}e^{-\frac{1}{2}\sigma^{2}t^{2}}\\
&...
1
vote
1answer
49 views
Characteristic function issue
As mentioned in a previous post, I've been trying to work through ALL of the problems in Jacod and Protter's Probability Essentials. The following problem has been giving me issues:
Let $Z \sim N(0,...
3
votes
1answer
1k views
Characteristic function example with Bernoulli and Poisson random variables
Let $\{X_n:n\ge 1\}$ be a sequence of i.i.d. Bernoulli random variables with probability of success $0<p<1$, i.e,
$$P\{X_1=1\}=1-P\{X_1=0\}=p.$$
The random variable $Y$ is independent of the ...
7
votes
2answers
1k 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 ...
7
votes
1answer
4k 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, ...
4
votes
1answer
176 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 $\...
