0
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0answers
16 views

Expected value and Differentiation of Characteristic function

Is there an example of random variable that has characteristic function to be differentiable at zero, but has no expected value?
1
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0answers
13 views

Quantitative version of Lévy's continuity theorem

Lévy's continuity theorem implies that if the sequence of characteristic functions $(\varphi_n)_n$ of a sequence of random variables $(X_n)_n$ converges pointwise to the characteristic function ...
0
votes
0answers
26 views

Inverting a difficult characteristic function.

Let $X$ be a random variable with characteristic function $$\phi_X(w)= \frac{1}{1-\frac{iw}{\lambda}e^{(\lambda-iw)\eta }}$$ where $\lambda$ and $\eta$ are constants. What is the pdf of $X$?
0
votes
1answer
55 views

the characteristic function of this distribution is equal to 0 everywhere except at the origin, mistake?

I wanted to compute the characteristic function of the distribution in question here: How to multiply a standard normal RV times a uniform{-1.1} RV? Let $X$ be standard $N(0,1)$, $Y$ be Uniform ...
0
votes
1answer
98 views

Characteristic function of Normal random variable squared

Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution While reviewing above, Why do you sub $X^2$ for the $Y$ in $e^{tY}$ and not the density of the normal ...
2
votes
1answer
44 views

Show that $\frac{1}{n}\sum_{j=1}^{n}X_{j}$ is Cauchy distributed when the $X_{i}$ are all Cauchy

Let $X_{1}, \cdots, X_{n}$ be i.i.d. Cauchy random variables with parameters $\alpha=0$ and $\beta=1$. (That is, their density is $f(x)=\frac{1}{\pi\,(1+x^{2})}$, $-\infty < x < \infty$.) Show ...
2
votes
0answers
80 views

Quantifying the “flatness” of functions which are the Fourier transforms of positive functions

I have a question which I admit is a little cumbersome for me to try to state succinctly, and which I fear may not have a simple answer, but I figured I'd give it a shot. In broad terms, I'm trying to ...
0
votes
1answer
258 views

Characteristic function of random variable $Z=XY$ where X and Y are independent non-standard normal random variables

I would like to find Characteristic function of random variable $Z=XY$ where X and Y are independent normal random variables, but they are not standard, i.e. $$X\sim N(\mu _x,\sigma_x)$$ $$Y\sim ...
4
votes
1answer
307 views

How to get PDF from characteristic function

I would appreciate if anybody could explain to me with a simple example how to find PDF of a random variable from its characteristic function. Thank you.
1
vote
1answer
120 views

Convergence in distribution of the log-Gamma distribution

Suppose $X$ has density $f(x)=\exp(kx-e^x)/\Gamma(k)$, $x>0$, for some parameter $k>0$. Then the moment-generating function of $X$ has the form $$ M_X(\theta)=\frac{\Gamma(\theta+k)}{\Gamma(k)}. ...
2
votes
1answer
84 views

X,Y are independent RVs with known characteristic functions. Find P(X+Y=2).

X,Y are independent random variables with the following characteristic functions: $ \phi_X(\theta) = \frac{1}{4}e^{i\theta}+\frac{3}{4}e^{i2\theta} \\ \phi_Y(\theta) = ...
2
votes
2answers
144 views

Which distribution has the moment-generating function $\frac{\pi t}{\sin \pi t}$

The distribution $F(x) = e^{-e^{-x}}$ has moment-generating function $M_F(t) = \Gamma(1-t)$. From this it follows that the distribution of $X-Y$ for independently $F$-distributed $X,Y$ has the ...
1
vote
1answer
59 views

Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution

I am trying to derive Chi-square distribution. The random variale is $$ U^2=\sum_{i=1}^k X_i^2 $$ where $X$ is a random variable with normal standard distribution. What is the distribution of ...
1
vote
1answer
147 views

Lower bounds of laplace transform of characteristic functions

I have the following integral: \begin{equation} f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt \end{equation} where $\varphi_X(t)$ is the characteristic function of some undetermined probability ...
1
vote
0answers
113 views

Probability density function of A = B + C via Joint Characteristic function of B and C

This problem is actually a subproblem of a longer derivation that I am trying to understand. I hope that I striped away all the unnecessary stuff that is not relevant. Please correct me if the ...
2
votes
2answers
3k views

Characteristic function of exponential and geometric distributions

I'm trying to derive the characteristic function for exponential distribution and geometric distribution. Can you guide me on getting them? Here is my solution so far: Exponential Dist ...
2
votes
0answers
340 views

Distribution of the sum of iid Beta-Negative-Binomial random variables

I am facing a problem when trying to calculate the distribution of the sum of iid Beta-Negative-Binomial random variables or for that matter if only parameter $r$ is different. To get a hint to how ...
2
votes
2answers
270 views

Obtaining cumulants using the characteristic function

If a random variable $x$ has a characteristic function $\phi(\omega)$, then the $n^{\mathrm{th}}$ moment of the distribution of $x$, $\mu_n$ can be calculated as: $$\mu_n = ...
2
votes
1answer
897 views

Characteristic function of a sum of Uniform random variables

Suppose I have $S=U_1+U_2+\dots+U_n$ where $U_i$ are distributed Uniform$[-1,1]$. I am trying to show a couple of things. First, what is the characteristic function. I can show this easily enough for ...
5
votes
2answers
563 views

Combinations of characteristic functions: $\alpha\phi_1+(1-\alpha)\phi_2$

Suppose we are given two characteristic functions: $\phi_1,\phi_2$ and I want to take a weighted average of them as below: $\alpha\phi_1+(1-\alpha)\phi_2$ for any $\alpha\in [0,1]$ Can it be proven ...
4
votes
3answers
710 views

Is it a characteristic function?

Can anyone explain, how can I prove either $\phi(x) = |\cos t|$ is characteristic function or not? And which random variable has this characteristic function? Thanks in advance.
5
votes
1answer
371 views

For symmetric stable distributions, why is $\alpha \le 2$?

I'm preparing a lecture on stable distributions, and I'm trying to find a simple explanation of the following fact. Suppose we are trying to come up with stable distributions. From the definition, ...