Linked Questions
19 questions linked to/from Prove that $E(X^n)^{1/n}$ is non-decreasing for non-negative random variables
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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.
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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 ...
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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.
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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.
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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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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 ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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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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 ...
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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|^...
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