# What is Characteristic function: Definition and 46 Discussions

In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

The indicator function of a subset, that is the function

1

A

:
X

{
0
,
1
}
,

{\displaystyle \mathbf {1} _{A}\colon X\to \{0,1\},}

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:

χ

A

(
x
)
:=

{

0
,

x

A
;

+

,

x

A
.

{\displaystyle \chi _{A}(x):={\begin{cases}0,&x\in A;\\+\infty ,&x\not \in A.\end{cases}}}

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:

φ

X

(
t
)
=
E

(

e

i
t
X

)

,

{\displaystyle \varphi _{X}(t)=\operatorname {E} \left(e^{itX}\right),}

where

E

{\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.

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1. ### I How did Hamilton derive the characteristic function V in his essay?

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...
2. ### I Finding the derivative of a characteristic function

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...
3. ### I How to find the moments using the characteristic function?

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.
4. ### Characteristic Function Integrand Evaluation

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...
5. ### Characteristic function of the sum of random variables

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6. ### I The CDF from the characteristic function

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7. ### I The CDF from the Characteristic Function

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
8. ### I The characteristic function of order statistics

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...
9. ### B The characteristic function

Suppose I have a random variable whose moments are not defined, can I still use the characteristic function to find the CDF of that random variable?
10. ### Moments from characteristic function geometric distribution

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...
11. ### A Gaussian distribution characteristic function

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...
12. ### B How can I find this characteristic function

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...
13. ### Convolution of characteristic function

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...
14. ### How can the difficult Gaussian integral be solved using standard tricks?

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15. ### Finding a characteristic function (integral help)

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...
16. ### Statistical Thermodynamics (multiple questions)

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...
17. ### MHB Discrepency with book's answer to characteristic function

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)]\\...
18. ### Calculate Fourier transform for the characteristic function of a rv

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...
19. ### Riemann integral of characteristic function.

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}...
20. ### Characteristic function of Sum of Random Variables

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21. ### Characteristic function and preimage?

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...
22. ### Characteristic function of a Gaussian

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...
23. ### Characteristic function of a continuous random variable

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24. ### Characteristic function of the binomial distribution

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...
25. ### Simple question about measurable characteristic function

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...
26. ### Characteristic Function of Joint Gaussian Distribution

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...
27. ### Characteristic function of z (Joint?)

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...
28. ### Constructing a smooth characteristic function

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29. ### Characteristic Function of a Compound Poisson Process

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...
30. ### Is this a characteristic function?

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...
31. ### Characteristic function of binomial distribution.

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...
32. ### Characteristic function of joint distribution

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...
33. ### Fourier transform of the exponential characteristic function

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...
34. ### Proving Fat Cantor Function is Non-Riemann Integrable

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...
35. ### Is the characteristic function constant everywhere if it is constant at 0?

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...
36. ### Characteristic function ?

Prove or disprove that function \phi(t)=\frac{1}{1+|t|} is charcteristic function of some random variable.
37. ### Exploring the Characteristic Function of Joint PDFs: Tips and Techniques

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...
38. ### Characteristic function in prbability

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...
39. ### Is the characteristic function of the irrationals Riemann integrable on [a,b]?

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...
40. ### Calculating the mode of a distribution from the characteristic function

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...
41. ### Characteristic function

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...
42. ### Approximation of the characteristic function of a compact set

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...
43. ### How can I find the characteristic function of a bivariate gaussian distribution?

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...
44. ### What is the Origin of the e-Term in the Characteristic Function?

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.
45. ### Characteristic function of an exponential distribution

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...
46. ### Characteristic function (Probability)

How can I show that if \phi(t) is a characteristic function for some distribution, then |\phi(t)|^2 is also a characteristic function?