# Questions tagged [characteristic-function]

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### 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 ...
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### 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 ...
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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. ...
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### 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 $$E[X] = \exp(M + S^2/2)$$ ...
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### 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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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 $$\hspace{... 1answer 554 views ### How to compute the characteristic function of two random variables with different distributions? What are the steps to obtain the following result: Given that X have \Gamma(1,s) distribution; and that X=x, and Y have the Poisson distribution with parameter x. Then the characteristic function ... 1answer 409 views ### Deconvolution with fourier transform or characteristic function? Let us consider the following model:$$Y_j = X_j + \epsilon_j \hspace{15pt} j=1, ..., n$$Where Y_j is a noisy signal, \epsilon_j is the noise which is independend from the signal X_j. We have ... 0answers 23 views ### finding process corresponding to laplace transform I have a positive stochastic process X(t) with Laplace transforms$$ \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 ... 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 ... 0answers 104 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 ... 1answer 434 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. ... 1answer 48 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,... 2answers 245 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 ... 1answer 50 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 ... 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 ... 1answer 116 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 ... 1answer 2k views ### 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 ... 0answers 34 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}^...
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)\...