All Questions
Tagged with characteristic-function probability
41 questions
3
votes
1
answer
95
views
Characteristic function of transformed random variable
Consider a random variable $X$ and a function $g(\cdot)$. Let $Y:=g(X)$, and let $\phi_X(\cdot), \phi_Y(\cdot)$ be the characteristic function (cf) of $X,Y$, respectively. Suppose that $\phi_X$ is non-...
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6
votes
2
answers
169
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 ...
- 65.8k
5
votes
2
answers
2k
views
Find mean and variance using characteristic function
Consider a random variable with characteristic function
$$
\phi(t)=\frac{3\sin(t)}{t^3}-\frac{3\cos(t)}{t^2}, \ \text{when} \ t \neq0
$$
How can I compute the $E(X)$ and $Var(X)$ by using this ...
- 21.2k
8
votes
1
answer
5k
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?
- 10.3k
3
votes
1
answer
94
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Name of PDF? - projecting uniform probability distribution on the unit circle to the x-axis
Consider a uniform probability distribution on a circle of radius r, i.e. $\{(x,y) \in \mathbb{R}^2: x^2 + y^2 = r^2 \}$.If we wish to project onto the x-axis, we can consider each point on the circle ...
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0
votes
0
answers
76
views
A problem of weak convergence of stochastic processes
Let $(X_1,X_2, ....)$ be a infinite sequence of random variables. Supoose that te sequence is strictly stationary. For all $n$, define the following infinite array:
$$
\begin{bmatrix}
Y_1 & Y_2 &...
- 1,007
2
votes
3
answers
178
views
If $X$ and $Y$ are independent random variables with $X+Y\stackrel{d}{=}X$, then show that $\mathbb P(Y=0)=1$ [closed]
Show that if $X$ and $Y$ are independent random variables with $X+Y\stackrel{d}{=}X$, then show that $\mathbb P(Y=0)=1$. Can the independence condition be dropped?
I could solve the first part using ...
- 133k
0
votes
1
answer
89
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Inverting a characteristic function if the integral of the modulus of the cf is infinity
I'm reading a lecture slide that starts by asking if there's a way to invert a characteristic function $\psi_X$ if $\int|\psi_X(t)|~\mathrm{d}t = \infty$. From my reading, the slide then provides a ...
- 10.3k
30
votes
2
answers
24k
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Link between moment-generating function and characteristic function
I am trying to understand the link between the moment-generating function and characteristic function. The moment-generating function is defined as:
$$
M_X(t) = E(\exp(tX)) = 1 + \frac{t E(X)}{1} + \...
- 129
0
votes
0
answers
35
views
Two non-obviously identical random variables that can be shown to be identical via their characteristic functions
It is well known that a characteristic function (CF) is uniquely associated with a probability density function (PDF). Knowing this, I was nonetheless intrigued by a remark in a video I have been ...
- 239
3
votes
2
answers
568
views
Show the Binomial distribution approaches a Normal distribution (using characteristic function)
Let $X_n = Bin(n,p)$. We fix $p$, and we want to show that as $n \to \infty$, $\frac{X_n-np}{\sqrt{np(1-p)}}$ converges to $N(0,1)$ in distribution. And I want to show this using characteristic ...
- 31.6k
1
vote
1
answer
281
views
Density from characteristic function: Durrett example 3.3.8 and 3.3.9
Letting $\varphi(t)$ be the characteristic function for the probability measure $\mu$, we know if $\int \left|\varphi(t)\right|dt < \infty$, then $\mu$ has density function
$$f(y) = \frac{1}{2\pi} \...
- 10.3k
10
votes
1
answer
9k
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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, ...
- 1
3
votes
1
answer
111
views
Partials of PDF with no closed form solution
I need to estimate partial derivatives for all N parameters denoted $\theta_{N}$ of a probability density function(PDF) $\mathcal{f}$.
This PDF $\mathcal{f}$ has no closed form solution and is ...
- 133k
1
vote
0
answers
125
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Find the characteristic function of a random variable involving a compoud poisson distribution
Let $\{\xi_i \}_{i=1}^{k}$ be a finite sequence of independent r.v. such that $\xi_{i} \sim F_i$.
For each $i$, let:
$\{\xi_{i,j}\}_{j=1}^{\infty}$ be a sequence of copies of $\xi_i$, that is: $\xi_{...
- 1,007
5
votes
4
answers
623
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|^{...
- 86.4k
2
votes
1
answer
97
views
Finding $\mathbb{E}|X|$ for variance-gamma random variable
Assume $X \sim f_{X}(n,\mu, \sigma)$ is a Variance-Gamma random variable. The density function involves a modified Bessel function, therefore is not that trivial to handle.
I'm looking for $\mathbb{E}|...
- 4,505
2
votes
1
answer
478
views
Characteristic Function and Quantile Function?
Characteristic functions (cf) are closely related to cdfs and pdfs of random variables, for example
cf is the Fourier transform of the pdf
Inversion formulae from Lévy and Gil-Pelaez
Question: Is ...
- 21.5k
3
votes
1
answer
3k
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 ...
- 58.2k
4
votes
1
answer
702
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 ...
- 82.8k
1
vote
0
answers
227
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,...
- 11
8
votes
2
answers
5k
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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. ...
- 82.8k
1
vote
0
answers
43
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)\...
- 1,226
2
votes
0
answers
130
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 ...
- 1,226
4
votes
1
answer
205
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_{...
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1
vote
0
answers
75
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 ...
- 1,226
1
vote
0
answers
108
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
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,...$$.
We have a sequence $U1,U_2,...$ where, $U_n=1/\sqrt(2(n\sigma^...
3
votes
0
answers
34
views
finding process corresponding to laplace transform
I have a positive stochastic process $X(t)$ with Laplace transforms
$$
\mathbb{E}\left[\mathrm{e}^{-uX(t)}\right]=\left(\frac{a+u\mathrm{e}^{-\kappa t}}{a+u}\right)^{b}
$$
One can clearly see that the ...
- 528
5
votes
1
answer
2k
views
Characteristic function example with Bernoulli and Poisson random variables
Let $\{X_n:n\ge 1\}$ be a sequence of i.i.d. Bernoulli random variables with probability of success $0<p<1$, i.e,
$$P\{X_1=1\}=1-P\{X_1=0\}=p.$$
The random variable $Y$ is independent of the ...
- 5,257
4
votes
0
answers
1k
views
What is the distribution of an affine transformation of a log-normal variable?
Let $X$ be a log-normal variate and $Y = aX + b$ is the affine transformation X.
Is $Y$ log-normal? I suspect it is not.
Since $X$ is log-normal, its expected value is
$$
E[X] = \exp(M + S^2/2)
$$
...
- 5,257
69
votes
13
answers
13k
views
What is the most surprising characterization of the Gaussian (normal) distribution?
A standardized Gaussian distribution on $\mathbb{R}$ can be defined by giving explicitly its density:
$$ \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$$
or its characteristic function.
As recalled in this ...
- 5,257
2
votes
1
answer
775
views
How to compute the characteristic function of two random variables with different distributions?
What are the steps to obtain the following result:
Given that X have $\Gamma(1,s)$ distribution; and that X=x, and Y have the Poisson distribution with parameter x. Then the characteristic function ...
- 5,257
50
votes
5
answers
18k
views
What is the purpose of characteristic functions?
I'm hoping that someone can explain, in layman's terms, what a characteristic function is and how it is used in practice. I've read that it is the Fourier transform of the pdf, so I guess I know what ...
- 5,257
3
votes
1
answer
706
views
Deconvolution with fourier transform or characteristic function?
Let us consider the following model:
$$Y_j = X_j + \epsilon_j \hspace{15pt} j=1, ..., n$$
Where $Y_j$ is a noisy signal, $\epsilon_j$ is the noise which is independend from the signal $X_j$. We have ...
- 5,257
2
votes
1
answer
852
views
Get Characteristic Function from MGF? [duplicate]
I've been calculating characteristic functions and MGF's and was wondering whether we can always get the characteristic function simply by substituting $it$ instead of $t$ in the resulting equation. ...
- 5,257
15
votes
1
answer
9k
views
Characteristic function and Fourier transform
I understand the definition of characteristic functions used in
probability theory:
For a random Variable $X$ with probability density function $f_X$ the characteristic function is defined as:
$$\...
- 5,257
1
vote
1
answer
168
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 ...
- 1,277
3
votes
1
answer
164
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
1
vote
1
answer
104
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,...
- 3,230
12
votes
1
answer
9k
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 ...
- 334k