All Questions
Tagged with expected-value probability-inequalities
35
questions
2
votes
1
answer
36
views
Expectation under convex order by multiplying
I am trying to understand if the following statement is true, or the conditions under it is satisfied. Let $M,N$ and $X>0$ be random variables. If the following inequality holds for any concave non-...
- 157
2
votes
1
answer
48
views
Expectation under convex order
I am trying to understand if the following statement is true. Let $M,N$ and $X$ be random variables. If the following inequality holds for any concave non-decreasing function $u$
\begin{equation}
\...
- 157
2
votes
1
answer
44
views
Tight upper bound on the function of expected value
Let $R$ be a positive integer, $\mathcal{X}$ be the sample space and $x \in \mathcal{X}$ be an event of the sample space; $P(x)$ denotes the probability of occurrence of event $x$. The problem is to ...
- 319
1
vote
0
answers
85
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How to find $\mathbb{E} \left[\frac{\bar{\mu}}{\bar{\sigma}^2}\right]$?
I asked the same question on math stacks: MathStacks:, and some user suggest to ask it here for better insight. So this question has found interest in many research problems, but there have been no ...
- 11
5
votes
1
answer
584
views
Equality in Gaussian Poincare Inequality
The Gaussian Poincare inequality states that: for $f: \mathbb{R}^n \to \mathbb{R}$ and $Z\sim \mathcal{N}(0,I)$, we have that
\begin{align}
Var(f(Z)) \le E[ \| \nabla f(Z)\|^2].
\end{align}
My ...
- 195
7
votes
2
answers
178
views
Bounding sum of quartic deviations from sample mean
[Cross-posted here with no answers for a few days]
I came - to the very best of my knowledge from reading the source - across the following statement in The Jackknife and Bootstrap, Shao and Tu, p. 87:...
- 34.2k
2
votes
0
answers
58
views
Concrete bound of expected value of a difference of I.I.D. Uniform Random Variables
In the following, $X,X_1,X_2,\dots X_n$ are I.I.D. uniform random variables in $[0,1]^d$ in $\mathbb{R}^d$. The problem I am attempting to solve is Exercise 2.4 from Gyorfi's "A distribution free ...
- 21
3
votes
1
answer
49
views
Show $(E|X|^2)/(E|X^2|) \leq P(X \not =0)$
I'm looking to show this inequality is true, and in turn use it to conclude the second moment method's bound.
Show that $\frac{E|X|^2}{E|X^2|} \leq P(X \not =0)$.
Again, I'm not supposed to use second ...
- 185
3
votes
2
answers
99
views
Boundary of $E\left[\frac{\prod_{i=1}^n x_i}{\prod_{i=1}^n x_i+\prod_{i=n+1}^m x_i}\right]$
Suppose $X_i$ are i.i.d. In addition, $X_i>0$ and $E[X_i]>1$. Suppose $E[X_i]$ is known, could we find upper bound or lower bound for the following expectation:
$$ E\left[\frac{\prod_{i=1}^n x_i}...
- 51
2
votes
1
answer
92
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Prove that $E[e^{2(m−1)X^2}]\le m$
I'm reading "Understanding Machine Learning: From theory to algorithms".
The problem is as follows, which is Exercise 31.1 of the book on page 416.
Let $X$ be a random variable that satisfies
$P[X \...
- 323
2
votes
1
answer
39
views
Does this expectation inequality holds?
Let $X\in L_p(P), p>1$. Is the following result true?
$$E[\lvert X\rvert I(\lvert X\rvert>C)]\leq C^{1-p}E\lvert X\rvert^p.$$
where $C>0$.
It can be found in the proof of Corollary A.1 (...
- 1,097
2
votes
1
answer
810
views
Unusual Markov inequality for normal distribution
I'm trying to answer the following question from Larry Wassermans book on statistical inference.
My question is how did they arrive at the Markov bound, it does not seem like the normal form of the ...
- 467
1
vote
1
answer
1k
views
How can I prove $ P(X> 0) \geq \frac{(E[X])^2}{E[X^2]}$ for a random variable $X$?
For a random variable $ X \geq 0 $ and $E[X^2] < \infty $, I'm asked to prove the following: $$ P(X> 0) \geq \frac{(E[X])^2}{E[X^2]}$$
It makes intuitive sense to me that it must be the case, ...
- 113
3
votes
1
answer
12k
views
General solution of expected value of E(f(X))?
This is maybe a trivial question I came up while solving a few examples and understanding Markov/Chebyshev inequalities and subsequently in evaluating Chernoff bounds. Suppose $X$ is a random variable ...
- 33
1
vote
0
answers
254
views
Tight upper bound on the expectation of a concave function
N is a random variable whose sample space is [0,$\infty$). I have an expression in terms of the expectation of this variable and I want to find a tight upper bound on the whole expression. The ...
- 183