which for a given subset A of X, has value 1 at points of A and 0 at points of X − A.There is an indicator function for affine varieties over a finite field: given a finite set of functions
f
α
∈
F
q
[
x
1
,
…
,
x
n
]
{\displaystyle f_{\alpha }\in \mathbb {F} _{q}[x_{1},\ldots ,x_{n}]}
let
V
=
{
x
∈
F
q
n
:
f
α
(
x
)
=
0
}
{\displaystyle V=\left\{x\in \mathbb {F} _{q}^{n}:f_{\alpha }(x)=0\right\}}
be their vanishing locus. Then, the function
P
(
x
)
=
∏
(
1
−
f
α
(
x
)
q
−
1
)
{\textstyle P(x)=\prod \left(1-f_{\alpha }(x)^{q-1}\right)}
acts as an indicator function for
V
{\displaystyle V}
. If
x
∈
V
{\displaystyle x\in V}
then
P
(
x
)
=
1
{\displaystyle P(x)=1}
, otherwise, for some
f
α
{\displaystyle f_{\alpha }}
, we have
f
α
(
x
)
≠
0
{\displaystyle f_{\alpha }(x)\neq 0}
, which implies that
f
α
(
x
)
q
−
1
=
1
{\displaystyle f_{\alpha }(x)^{q-1}=1}
, hence
P
(
x
)
=
0
{\displaystyle P(x)=0}
.
The characteristic function in convex analysis, closely related to the indicator function of a set:
In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question:
{\displaystyle \operatorname {E} }
denotes expected value. For multivariate distributions, the product tX is replaced by a scalar product of vectors.
The characteristic function of a cooperative game in game theory.
The characteristic polynomial in linear algebra.
The characteristic state function in statistical mechanics.
The Euler characteristic, a topological invariant.
The receiver operating characteristic in statistical decision theory.
The point characteristic function in statistics.
In Hamilton's "on a general method in dynamics", he starts with varying the function ##U## and writes the equation:
$$\delta U=\sum m(\ddot x\delta x+\ddot y\delta y+\ddot z\delta z)$$
Then he defines ##T## to be:
$$T=\frac{1}{2}\sum m (\dot x^2+\dot y^2+\dot z^2)$$
Then by ##dT=dU##, he...
Problem summary
I have the characteristic function of a probability distribution but I'm having difficulty obtaining its derivative.
Background
I am reading the following paper: Schwartz, Lowell M. (1980). On round-off error. Analytical Chemistry, 52(7), 1141-1147. DOI:10.1021/ac50057a033.
The...
I have the characteristic function of the Cauchy distribution ##C(t)= e^{-(\mid t \mid)}##. Now, how would I show that the Cauchy distribution has no moments using this? I think you have to show it has no Taylor expansion around the origin. I am not sure how to do this.
Homework Statement
[/B]
I am trying to determien the characteristic function of the function:
$$ f(x)= ae^{-ax}$$
$$\therefore E(e^{itx}) =\int_0^\infty e^{itx}ae^{-ax} dx = a \cdot \frac{e}{it-a} |_0 ^ \infty $$
But I am not sure how to evaluate the integral.
Wolfram alpha suggests this...
Homework Statement
I am trying to understand the very last equality for (let me replace the tilda with a hat ) ##\hat{P_{X}(K)}=\hat{P(k_1=k_2=...=k_{N}=k)}##(1)
Homework Equations
I also thought that the following imaginary exponential delta identity may be useful, due to the equality of...
This thread will be a collection of multiple questions I asked before over different forums. I will start from the beginning, and I hope someone will follow the steps with me, because I did it before alone, and I ended with a numerical integration that is not finite, which doesn't make sense...
Is there a way to find the CDF of a random variable from its characteristic function directly, without first finding the PDF through inverse Fourier transform, and then integrate the PDF to get the CDFÉ
Suppose that ##Y=\sum_{k=1}^KX_{(k)}##, where ##X_{(1)}\leq X_{(2)}\leq\cdots X_{(N)}## and (##N\geq K##). I want to find the characteristic function of ##Y## as
\phi(jvY)=E\left[e^{jvY}\right]=E\left[e^{jv\sum_{k=1}^KX_{(k)}}\right]
In the case where ##\{X\}## are i.i.d random variables, the...
Homework Statement
Hi,
I have the probabilty density: ##p_{n}=(1-p)^{n}p , n=0,1,2... ##
and I am asked to find the characteristic function: ##p(k)= <e^{ikn}> ## and then use this to determine the mean and variance of the distribution.
Homework Equations
[/B]
I have the general expression...
Hello, guys. I am trying to solve for characteristic function of normal distribution and I've got to the point where some manipulation has been made with the term in integrands exponent. And a new term of t2σ2/2 has appeared. Could you be so kind and explain that to me, please...
Hello all,
I'm trying to find the characteristic function of the random variable ##X## whose PDF is ##f_X(x)=1/(x+1)^2## where ##X\in[0,\,\infty)##. I started like this:
\phi_X(j\nu)=E\left[e^{j\nu X}\right]=\int_0^{\infty}\frac{e^{j\nu X}}{(x+1)^2}\,dx
where ##j=\sqrt{-1}##. I searched the...
Homework Statement
I am trying to figure out following problem.
Let A ⊂ R. Then we can define the characteristic function:
\begin{align}
\chi_A : R → \{0, 1\}, x = \begin{cases}
1 & \text{if } x \in A \\
0 & \text{else }
\end{cases}
\end{align}
Let a be bigger than 0. I am...
Hi everyone,
in the course of trying to solve a rather complicated statistics problem, I stumbled upon a few difficult integrals. The most difficult looks like:
I(k,a,b,c) = \int_{-\infty}^{\infty} dx\, \frac{e^{i k x} e^{-\frac{x^2}{2}} x}{(a + 2 i x)(b+2 i x)(c+2 i x)}
where a,b,c are...
Homework Statement
I'm looking to find the characteristic function of
p(x)=\frac{1}{\pi}\frac{1}{1+x^2}[/B]
Homework Equations
The characteristic function is defined as
\int_{-\infty}^{\infty} e^{ikx}p(x)dx
3. The Attempt at a Solution
I attempted to solve this using integration by...
Homework Statement
1.) For N particles in a gravity field, the Hamiltonian has a contribution of external potential only (-mgh). Show that the particle density follows the barometric height equation (1).
2.) For N particles in a open system at constant pressure p and temperature T, let there...
Find the characteristic function for the PMF \(p_X[k] = \frac{1}{5}\) for \(k = -2, -1,\ldots, 2\).
The characteristic function can be found with
\begin{align*}
\phi_X(\omega) &= E[\exp(i\omega X)]\\...
Homework Statement
In order to determine the characteristic function of a random variable defined by: Z = max(X,0) where X is any continuous rv, i need to prove that:
F_{l,v}(g(l))=[ \phi_{X}(u+v)\phi_{X}(v) ] / (iv)
where F_{l,v}(g(l)) is the Fourier transform of g(l) and...
Homework Statement
The characteristic function of a set E is given by χe = 1 if x is in E, and χe = 0 if x is not in E. Let N be a natural number, and {an, bn} from n=1 to N, be any real numbers. Use the definition of the integral (Riemann) to show that \int \sum b_{n} X_{ \left\{ a_{n}...
Homework Statement
Let X,W,Y be iid with a common geometric density f_x(x)= p(1-p)^x for x nonnegative integer
and p is in the interval (0,1)
What is the characteristic function of A= X-2W+3Y ?
Determine the family of the conditional distribution of X given X+W?
Homework Equations...
Characteristic function and preimage?
Homework Statement
Let S be a nonempty subset of ℝ.
Define χs= { 1 if x is in S and 0 if x is not in S
Determine χs-1(Q) [where Q=set of all rational numbers]
and χs-1((0,∞))
We haven't really dealt much with this function, and I really...
Homework Statement
I must find the characteristic function of the Gaussian distribution f_X(x)=\frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{1}{2} \left [ \frac{(x- \langle x \rangle )^2}{\sigma ^2} \right ]}. If you cannot see well the latex, it's the function P(x) in...
Homework Statement
I must calculate the characteristic function as well as the first moments and cumulants of the continuous random variable f_X (x)=\frac{1}{\pi } \frac{c}{x^2+c^2} which is basically a kind of Lorentzian.Homework Equations
The characteristic function is simply a Fourier...
Homework Statement
Hey guys, I'm self studying some probability theory and I'm stuck with the basics.
I must find the characteristic function (also the moments and the cumulants) of the binomial "variable" with parameters n and p.
I checked out wikipedia's article...
Homework Statement
Prove that the characteristic function \chi_A: X\rightarrow R, \chi_A(x)=1,x\in A; \chi_A(x)=0, x\notin A, where A is a measurable set of the measurable space (X,\psi) , is measurable.
Homework Equations
a function f: X->R is measurable if for any usual measurable set...
This is inspired by Kardar's Statistical Physics of Particles, page 45, and uses similar notation.
Homework Statement
Find the characteristic function, \widetilde{p}(\overrightarrow{k}) for the joint gaussian distribution:
p(\overrightarrow{x})=\frac{1}{\sqrt{(2\pi)^{N}det...
I need some help. Is there a good way to do this type of question?
Homework Statement
Let X and Y be independent random Variables with exponential densities
fX(x) = Ωe-Ωx, if X≥0
0, otherwise
fY(y) = βe-βy, if y≥0
0, otherwise...
Constructing a "smooth" characteristic function
Suppose I'd like to construct a C^\infty generalization of a characteristic function, f(x): \mathbb R \to \mathbb R, as follows: I want f to be 1 for, say, x\in (a,b), zero for x < a-\delta and b > x + \delta, and I want it to be C^\infty on...
Hello,
I am trying to find a characteristic function (CF) of a Compound Poisson Process (CPP) and I am stuck :(.
I have a CPP defined as X(t) = SIGMA[from j=1 to Nt]{Yj}. Yj's are independent and are Normally distributed.
So, in trying to find the CF of X I do the following:
(Notation...
1. If \phi is a characteristic function, than is e^{\phi-1} also a characteristic function?
I know some general rules like that a product or weighted sum of characteristic functions are also characteristic functions, also a pointwise limit of characteristic functions is one if it's continuous...
Hello,
I considered a Binomial distribution B(n,p), and a discrete random variable X=\frac{1}{n}B(n,p). I tried to compute the characteristic function of X and got the following:
\phi_X(\theta)=E[e^{i\frac{\theta}{n}X}]=(1-p+pe^{i\theta/n})^n
I tried to compute the limit for n\to +\infty...
What exactly is a "joint characteristic function"? I want the characteristic function of the joint distribution of two (non-independent) probability distributions. I'll state the problem below for clarity. So my two distributions are the normal distribution with mean 0 and variance n, and the...
I am trying to compute the inverse Fourier transform numerically (using a DFT) for some complicated characteristic functions in order to compute their corresponding probability distribution functions. As a test case I thought I would invert the characteristic function for the simple exponential...
I have been thinking about this for quite some time now. When I look at the function that descibes the fat cantor set namely:
f(x) = 1 for x\inF and f(x) = 0 otherwise, where F is the fat cantor set.
I wonder, how do I prove that this is non-riemann integrable?
I have considered...
Hi there,
Recently I have come across a proof with application of characteristic function.
After some steps in the proof, it concluded that there is a neighborhood of 0 such that the characteristic function is constant at 1, then it said the characteristic function is constant at 1...
Hi.
Does anyone know a good source for learning about the characteristic function of a joint pdf. Is there any nice rules for that? For example assume having a waiting time density and a jump density which are independent (easy things first). Is there an elegant way to get the characteristic...
I was reading about it here:
http://mathworld.wolfram.com/CharacteristicFunction.html
very neat. But then I tried out of boredom integrating the expression by parts where u = the exponential term and v = f (x) (or P(x)). The integral came out nicely as I got a term similar to the left hand...
The characteristic function of the RATIONALS is a well-known example of a bounded function that is not Riemann integrable. But is the characteristic function of the IRRATIONALS (that is, the function that is 1 at every irrational number and 0 at every rational number) Riemann integrable on an...
Is it possible to exactly derive the mode of a probability distribution if you have the characteristic function? I cannot get the pdf of the distribution because the inverse Fourier transform of the characteristic function cannot be found analytically.
Any thoughts would be appreciated...
Homework Statement
Let (X,d) be a metric space, A subset of X, x_A: X->R the characteristic
function of A. (R is the set of all real numbers)
Let V_d(x) denote the set of neighbourhoods of x with respect the metric d.
Prove that x_A is continuous in x (x in X) if and only if there...
Homework Statement
Okay, so this is a three-part question, and I need some help with it.
1. I need to show that the function f(x) = e^{-1/x^{2}}, x > 0 and 0 otherwise is infinitely differentiable at x = 0.
2. I need to find a function from R to [0,1] that's 0 for x \leq 0 and 1 for x...
Hi all,
Im currently researching into Multivariate distributions, in particular I am trying to derive the characteristic function of the bivariate distribution of a gaussian. While knowing that a gaussian density function cannot be integrated how is it possible to find the characteristic...
This kind of bothers me:
our textbook does not explain (and the professor either) where characteristic function comes from, all it says is what it defined as, which is E[ejwX], where E is expectation of random variable X. But where is this e-term coming from?
Thanks in advance.
I need to calculate the characteristic function of an exponential distribution:
\phi _X \left( t \right) = \int\limits_{ - \infty }^\infty {e^{itX} \lambda e^{ - \lambda x} dx} = \int\limits_{ - \infty }^\infty {\lambda e^{\left( {it - \lambda } \right)x} dx}
I have arrived at the...