# Tag Info

Accepted

• 57.3k
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### Prove that Kurtosis is at least one more than the square of the skewness

This demonstration uses the clever (yet elementary) method employed in Fisher's reference Burnside & Panton vol. II, section 142. It assumes a familiarity with the basics of determinants: their ...
• 323k

### Proof that variance is always greater than or equal to zero

As for your question regarding complex numbers, the variance is defined as being the expectation of the absolute value, or modulus, squared of the deviation from the mean. If the absolute value is not ...
• 5,236

### What is the correlation between a random variable and its probability integral transform?

If we assume $\mathbb E^F[X]=0$ then \begin{align} \mathbb E^F[XF(X)]&= \frac{1}{2}\int F(x)\{1-F(x)\}\,\text dx \end{align} Indeed, assuming the pdf $f$ is associated with the cdf $F$, \...
• 106k
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### Property of two independent Beta distribution

It does not seem to be a correct conjecture. It seems your condition is that the mode for $X$ is greater than the mode for $Y$. Since in non-symmetric Beta distributions, the mode is not equal to the ...
• 39.6k
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• 6,791

### How can I establish an inequality between $|\frac1n \sum_{i=1}^nX_i|$ and $\frac1n\sum^n_{i=1}|X_i|$ where $X_i \sim N(0,1)$?

Answer: Whatever the distribution of $X_1,...,X_n$, $$\mathbb{E} Y_2 \geq \mathbb{E} Y_1.$$ Details: For any $n$ numbers $X_1,..., X_n$ it is true that $$\sum_i |X_i| \geq |\sum_i X_i|$$ and ...
• 1,331
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### Upper bound for absolute third central moment

Let $p\ge 1.$ The $L^p$ norm of a random variable is defined as $$|X|_p = \left(E[|X|^p]\right)^{1/p}.$$ Minkowski's Inequality says this norm satisfies the triangle inequality. Apply it to the ...
• 323k
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### Is it true that $\langle X^4\rangle \ge 3 \langle X^2\rangle^2$?

Both of the inequalities you assert are false For a random variable with zero mean, the moment quantity $\langle X^4 \rangle /\langle X^2 \rangle^2$ is the kurtosis of the distribution, which has a ...
• 125k
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### Variance and covariance inequality

One way to do it is to proceed from the answer you linked in comments, at the second last line (at this point no properties of the normal have been used). Note that $Y$ is just a standardized $X$; i.e....
• 283k

### Prove that Kurtosis is at least one more than the square of the skewness

Consider the quadratic form \begin{align} Q(a, ~b,~c) & := \frac1n\sum \left(a + xb+ x^2 c\right)^2 \\ &= a^2\cdot v_0 + b^2\cdot v_2 + c^2\cdot v_4+ 2ab\cdot v_1+ 2bc\cdot v_3+ 2ca\cdot v_2,\...
• 8,292
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### Testing a Null Hypothesis Nested by the Alternative Hypothesis

By definition (see e.g. Jun Shao "Mathematical Statistics", 2nd edition, chapter 6, first paragraph) of statistical hypothesis tests, the null and the alternative hypothesis must be disjoint....
• 10.8k

### Is it true that $\langle X^4\rangle \ge 3 \langle X^2\rangle^2$?

An easy counterexample is a two point distribution $P(X = \pm 1) = 1/2$, for which $E[X] = 0$, $E[X^2] = E[X^4] = 1$. Hence $1 = E[X^4] < 3(E[X^2])^2 = 3$. This example also showed the related ...
• 18.8k
Accepted

### How does Chebyshev's inequality imply $P(X ≥ k) ≤ 1/(σk)^2$?

Your second form of the inequality is wrong. The initial form of Chebychev's inequality (for a random variable $X$ with zero mean) is: $$\mathbb{P}(|X| \geqslant k \sigma) \leqslant \frac{1}{k^2}.$$ ...
• 125k

### If $X < a$, $EX < a$?

\begin{align} EX &= \int x\cdot \mathbf{1}_{x< a} + x\cdot\mathbf{1}_{x\geq a} \ dP \\ &= \int x\cdot \mathbf{1}_{x< a}\ dP\\ &< \int a\cdot \mathbf{1}_{x< a}\ dP \\ &= a, \...
Accepted

### Variance of the reciprocal of a strictly positive random variable

Here is an answer (to a related question) which provides a valid generalization of $\mathbb{E}\left[\frac{1}{X}\right] \geq \frac{1}{\mathbb{E}[X]}$. What is the relation between $\mathbb{E}X^r$ and ...
• 13.4k

### Variance of the reciprocal of a strictly positive random variable

First, @Dilip Sarwate comments, for such ratio variables mean and variance often do not exist and then there is little to expect. For a detailed discussion of this see I've heard that ratios or ...
• 78.1k
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• 2,477
### How to calculate lower bound on $P \left[|Y| > \frac{|\lambda|}{2} \right]$?
Edit : This answer applies to the original question, that was : "for ANY random variable Y, what is the lower bound of (formula)" The lower bound is 0. Let's take $(Y_{n})$ a series ...