Linked Questions

1 vote
1 answer
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. ...
student.J's user avatar
0 votes
0 answers
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 ...
Ryan Zhou's user avatar
1 vote
1 answer
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.
Ann Height's user avatar
0 votes
0 answers
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.
ankit's user avatar
  • 421
0 votes
0 answers
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 $$E^p|X_n|\leq E |X_n|^p$$
Amin Roshani's user avatar
10 votes
3 answers
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 ...
Tricolor's user avatar
  • 307
7 votes
2 answers
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 ...
Harry's user avatar
  • 1,387
11 votes
2 answers
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 ...
Dave's user avatar
  • 63.7k
5 votes
1 answer
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 ...
safelyanonymous's user avatar
4 votes
1 answer
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 ...
purpleostrich's user avatar
3 votes
1 answer
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 ...
AvidLearner's user avatar
4 votes
2 answers
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?
Roman's user avatar
  • 594
3 votes
1 answer
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 ...
Caio C.'s user avatar
  • 303
3 votes
0 answers
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 ...
Parseltongue's user avatar
  • 1,010
3 votes
1 answer
86 views

$\mathbb E[|X_n|^r]<\infty$ 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|^...
Sayan Dutta's user avatar

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