Tagged Questions

2
votes
1answer
42 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
50 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
73 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
564 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
204 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
39 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
77 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
428 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.