# Questions tagged [characteristic-function]

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### Analytic structure of the Dirichlet L functions with the Hurwitz zeta function

I am getting confused about how to get results from below by using the Hurwitz zeta function. Really don't know how to do it. "The Dirichlet L functions for non-principal Dirichlet characters are ...
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### Characteristic function of Weibull distribution

I am recently learning about characteristic functions. I find them very interesting, but I seem to be misunderstanding something. Wikipedia claims that they are the Fourier transform of the ...
1 vote
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### Characteristic function of random vector

Consider a bivariate probability distribution $$f(x,y)=\frac{a^2}{2\pi} \exp \left(-a\sqrt{x^2+y^2}\right)$$ where $a>0, -\infty<x<\infty, -\infty<y<\infty$. In this situation, I want ...
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### 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 ...
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### 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
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### 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 ...
### $X \sim$ 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 $Y = X/c$ ...
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 $Y$.