1 vote
2k views

### If higher order moment is finite, then lower order moment finite? [duplicate]

I'm trying to prove the theorem below. If $E(|X|^n)<\infty$ for some positive integer $n$, then $E(|X^k|)<\infty$ for every positive integer $k$ such that $k<n$ Here's what I've tried. ...
732 views

### Finite Mean and Finite Variance [duplicate]

Let $X$ be a random variable. Suppose there exists a constant $c ∈ R$ such that $E(|X − c|^2) < ∞$. Show that the random variable $X$ has finite mean and variance. And I'm quite confused about the ...
1 vote
389 views

### Prove that E[x^n] >= (EX)^n for n = 2k [duplicate]

Prove that $E[x^n] \geq (Ex)^n$ for $n = 2k$ I only have the formula of E(x) but I don't know how to prove it.
33 views

### Is standard deviation always larger than mean deviation? [duplicate]

If yes why is it the case that it have this property? There must be some mathematical proof or intuitive understanding in that case.
• 421
21 views

### An inequality about expectation [duplicate]

If $X_n$ is a martingales with $sup E|X_n|^p<\infty$ where $p>1$, How can show that $$E^p|X_n|\leq E |X_n|^p$$
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427 views

### What's higher, $E(X^2)^3$ or $E(X^3)^2$

So I had a probability test and I couldn't really answer this question. It just asked something like this: "Considering that $X$ is a random variable, $X$ $\geqslant$ $0$, use the correct ...
• 307
1k views

### Prove that a distribution is symmetric using moments

Given, a random variable X whose mean , variance and fourth central moment are 0, 2 and 4 respectively. Now, how do I prove that (1) third moment is 0 (2) distribute is symmetric about 0 and (3) X ...
• 1,387
853 views

### Westfall says, "the proportion of the kurtosis that is determined by the central $\mu\pm\sigma$ range is usually quite small" but is the reverse true?

In his article that debunks the notion of kurtosis as measuring distribution peakedness, Peter Westfall writes, [T]he proportion of the kurtosis that is determined by the central $\mu\pm\sigma$ range ...
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715 views

### Upper bound for absolute third central moment

Suppose $X\in \mathbb{R}$ is a random variable with expected value $\mathbb{E}X = \mu$. I ran across a proof which uses the inequality $$\mathbb{E}[|X - \mu|^3] \leq 2^3 \mathbb{E}|X|^3.$$ Can ...
526 views

### Sufficient Conditions for the Central Limit Theorem

My understanding is that the central limit theorem applies as long as the variance of the random variable is less than infinity. Is this equivalent to saying that all moments are finite? If not, what ...
195 views

### Expectation of first of moment of symmetric r.v. in terms of variance

Let $X$ be a symmetric random variable with bounded moments and standard deviation $\sigma$. I want to lower-bound $\mathbb E[|X|]$ in terms of $\sigma$. Here is the formal conjecture; I wonder if ...
• 238
208 views

### Can a random variable be uncorrelated with its product with a correlated random variable?

I have a random variable $X.$ I want to find a random variable $Y$ such that $Y$ is correlated with $X,$ but $Y$ is not correlated with the product of $X$ and $Y.$ Is it always possible?
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205 views

### How to show that an m.d.s is not independent?

I have to prove that this Martingale Difference: $x_t = u_t u_{t-1}$ where $u_t \sim^{iid} (0, \sigma^2)$ is not serially independent, but am failing to do such thing. I also have to prove that it's ...
• 303
234 views

### What is the intuition behind taking the sum of square roots, squared

In a recent publication, the authors report the following transformation when aggregating across three different scales: Cognitive style level was used as a control variable and captured as ...
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### $\mathbb E[|X_n|^r]<\infty$ and $\mathbb E[|X_n|^r]\to \mathbb E[|X|^r]$ as $n\to \infty$
Let $\{X_n\}\xrightarrow{d}X$ and for some $p>0$, we have $$\sup_{n\ge 1} \mathbb E[|X_n|^p]<\infty$$ Show that for any $r\in (0,p)$, we have a. $\mathbb E[|X|^r]<\infty$ b. \$\mathbb E[|X_n|^...