# Tagged Questions

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### Show that if X has a density f such that f’ exists and is integrable?

Show that if $X$ has a density $f$ such that $f'$ exists and is integrable, then its characteristic function has the property : $\phi(t)=ο(t^{-1} )$ as $t\to \infty$. Hint: If $X$ has a density ...
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### What would be the simplified form of this expression?

I'm working on a Homework problem involving Convergence of Random variables and I've arrived at an expression which looks like follows:  M_{X_n}(ju)= ...
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### 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 ...
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### How does what I did imply that $X$ is Normal $N(0,1)$?

Let $X$, $Y$ be i.i.d, that $X+Y$ and $X-Y$ are independent, and that $\varphi_{X}(2u)=(\varphi_{X}(u))^{3}\varphi_{X}(-u)$. Also, let $E\{X\}=0$ and $E\{X^{2}\}=1$. Show that $X$ is Normal $N(0,1)$. ...
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### Show that $|1-\varphi_X (u)|\leq E\{ |uX| \}$

Show that $|1-\exp\{ix\}|^{2}=2(1-\cos x) \leq x^{2}$ for all $x \in \mathbb{R}$. Use this to show that $|1-\varphi_X(u)|\leq E\{|uX|\}$, where $\varphi_X(u) =E\{\exp(i\langle u,X\rangle)\}$ is the ...
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### Let X, Y be i.i.d, X+Y and X-Y independent, show that the characteristic function E{exp(i<2u,x>)}= …

Let $X$ and $Y$ be i.i.d. Suppose further that $X+Y$ and $X-Y$ are independent. Show that $\varphi_{X}(2u)=(\varphi_{X}(u))^{3}\varphi_{X}(-u)$. What I tried to do was work backwards, starting with ...
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### For X, Y real valued and independent, and X and X+Y having the same distribution, Y=0 a.s.

Let X, Y be real valued and independent. Suppose X and X+Y have the same distribution. Show that Y is a constant r.v. equal to 0 almost surely. Here's what I have so far: By the uniqueness of ...
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### 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 ...
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### Even numbered moments of N(0,1) using characteristic functions

Let $X$ be $N(0,1)$. Show that $E\{X^{2n+1}\}=0$ (Easy - calculate it directly using the definition of expectation, and you're taking the integral of an odd function over a symmetric interval, so =0), ...
Suppose $E{|X|^{2}}<\infty$ and $E{X}=0$. Show that Var(X)$= \sigma^{2}<\infty$ (done), and that $\varphi_{X}(u)=1-\frac{1}{2}u^{2}\sigma^{2}+o(u^{2})$ (what I can't figure out how to find, ...