0
votes
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?
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 ...
2
votes
1answer
97 views

Characteristic function of an integer-valued distribution, inversion formula

I am working on the following: Show that if $\varphi$ the characteristic function of an integer-valued distribution then \begin{align*} \mathbb P(X = k) = \frac{1}{2\pi} \int_{-\pi}^\pi e^{-itk} ...
0
votes
0answers
32 views

Characteristic function of a exponential random variable, problems with complex integral.

I tried to compute the characteristic function of a random variable, which is exponential distributed with parameter $\lambda$: \begin{align*} \varphi(t) &= \mathbb E[e^{itX}] = ...
0
votes
0answers
34 views

Limit of the expectation of the sum

Show that for $g(t)= E \left\{\sum_{n=3}^{\infty}\frac{(iut)^{n}}{n!}\right\}$ that $\lim_{t \to 0} \frac{|g(t)|}{t} =0$. I think I should bound it and then use LDCT, but I'm having trouble doing ...
2
votes
1answer
83 views

Prove that $ \mathsf{E}[g(X)] = \int_{- \infty}^{\infty} G(t) \varphi(t) \, d{t} $.

Problem Let $ X $ be a real-valued random variable with characteristic function $ \varphi $. Suppose that $ g: \mathbb{R} \to \mathbb{R} $ satisfies $$ \forall x \in \mathbb{R}: \quad g(x) = ...
0
votes
1answer
101 views

Using characteristic function to deduce convergence of Bernoulli random variables

Let $Y_1, Y_2,...$ be a sequence of independent Bernoulli(0.5) random variables and $X_n = \sum_{i=1}^{n} Y_i 2^{-i}$ I need to use the characteristic function to deduce that $X_n$ converges in ...
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
303 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.
2
votes
0answers
170 views

Why the probability characteristic function is always exist but moment generation function is not always exist?

I know that the characteristic function is always exist and moment generation function is not always exist instantly but don't know exactly mathematically.
1
vote
1answer
173 views

Conditional Characteristic Function

Given I know the joint characteristic function of the random variables $X,Y$ and the characteristic function of $Y$, is there a way to recover the characteristic function of $X|Y$ without inverting ...
8
votes
2answers
120 views

Does “independence” of moments imply independence?

Suppse you have two random variables $X,Y$ and you are given that for any $m,n$ that: $$E(X^n Y^m) = E(X^n)E(Y^m)$$ Does this imply that $X$ and $Y$ are independent? Are there some condtions on how ...
0
votes
1answer
111 views

How do you invert a characteristic function, when integral does not converge?

I need to find the probability density of some distribution with characteristic function given by: $$\frac{1}{9} + \frac{4}{9} e^{iw} + \frac{4}{9} e^{2iw}$$ I know the formula for inverting a ...
3
votes
2answers
305 views

Characteristic function for positive part of random variables

I need your help in solving the following problem: I have to calculate the characteristic function for the positive side of a random variable - simplest case: let $Y$ be standard normal and $Y^+ =\max ...
1
vote
1answer
75 views

Proof by Characteristic Function

If $ X_1, X_2, \ldots,X_n$ are independent random variables variables with expectation $0$ and finite third moments. Show that $$E((X_1+X_2+\cdots+X_n)^3) = EX_1^3+EX_2^3+\cdots+EX_n^3$$ using the aid ...
0
votes
3answers
95 views

Prove the double angle formula

Let $X$ and $Y$ be independent random variables, where $X\in U(-1,1)$ and $Y$ assumes the values of $+1$ and $-1$ with probabilities $1/2$. Show first that $Z=X+Y = U(-2,2)$ by finding 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 ...
1
vote
1answer
491 views

relationship between non-central moments and the characteristic function of a distribution?

Given a random variable $X$, consider the k-th (non-central) moment about $a$, $E \left[ ( X - a )^k \right]$ Is there any relation of this value to the chracteristic function of variable's ...
0
votes
1answer
46 views

Question on proof of Schoenberg correspondence from Lévy Process and Stochastic Calculus by Applebaum

I quote the proof here from Applebaum's Lévy Processes and stochastic calculus (and the things before it to present the full picture) We say $\phi:\mathbb{R}^d\rightarrow\mathbb{C}$ is conditionally ...
1
vote
2answers
115 views

Mean and deviation of characteristic function of an event

If I am given two events $A$ and $B$ and the probabilities $P(A), P(A|B), P(B), P(B|A)$, and am told that the random variables $X$ and $Y$ are defined as $$X(a) = 1\ \text{if}\ a \in A\text{, else}\ ...
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.