The probability-inequalities tag has no wiki summary.
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0answers
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Correction term in variance of mixture [duplicate]
In the answer to this question, the correction term is claimed to be nonnegative:
$$p_A \mu_A^2 + p_B \mu_B^2 - p_A^2 \mu_A^2 - p_B^2 \mu_B^2 - 2 p_A p_B \mu_A \mu_B \geq 0$$
where $p_A + p_B = 1$, $0 ...
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1answer
55 views
Is Jensen's inequality applicable for two variables?
This is the Jensen's inequality I saw in my textbook:
$$E{ f(X) } \geq f( E(X) ),$$
where $f$ is a convex function.
Is this also applicable for two random variables--independent or otherwise--like ...
1
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1answer
74 views
Intuition behind Dvoretzky Kiefer Wolfowitz inequality
I've been reading about the Dvoretzky-Kiefer-Wolfowitz inequality, in the context of confidence bands on empirical distribution functions.
I think I understand the inequality at face value: that the ...
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1answer
39 views
Why does KL divergence show up in the proof of Hoeffding's inequality?
In some textbook the KL divergence shows up in the proof of Hoeffding's inequality (e.g., eq. (5) of this material). In contrary, most other textbooks seem not mention this fact. I know that KL ...
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Why Hoeffding's inequality has a sharper bound than Markov's inequality?
Basically, I can understand the proofs' details for both inequalities, but still I have no idea why the bound of Hoeffding is sharper than that of Markov? Is there any underlying intuitive that can be ...
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1answer
71 views
Berry-Esseen theorem
In the Berry-Essen theorem, why is the standard normal distribution used in the context of the closedness of two distributions? Why can't we use the general normal distribution where will things will ...
4
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1answer
117 views
What can we conclude about the distribution of the sum of two random variables?
If we know, for independent random variables $X$ and $Y$,
$P(X>x)\leq0.05$, and $P(Y>y)\leq0.05$, can we say anything about $P(X+Y>x+y)$? Can we be certain that it is less than $0.05$? Under ...
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22 views
About a exponential bound
Let's consider $ X $ a random variable of magnitude of a $n-vector$ with i.i.d. coordinates each with exponential distribution. What is a good bound on its tail distribution, of the form
$P(X_n>t) ...
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0answers
44 views
About Galton watson process
My question is about a homework question that I found interesting. It gives another proof (without using martingales) for that the critical Galton Watson tree dies out eventually. But it has given a ...
4
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0answers
82 views
About tail distribution of a sum
Do we know anything about the tail distribution of sum of squares of a limited number of i.i.d exponentially distributed random variables? I'm looking for a good bound.
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2answers
152 views
Where can I use Chebychev's inequality?
What are some real-world applications where I can leverage Chebychev's inqueality? All the examples I find are either related to coin tosses or some school scores related problems. Are there any ...
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0answers
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Tail inequality on sum of product of normal variables
For independent random variables $ x_1,..,x_n$ and $y_1,...,y_n$ following normal distribution $N(0,1)$, I need a simple estimate formula for
$P(| \sum_1^n x_iy_i | \leq nt ) \leq e^{(?)}$ for ...
4
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1answer
163 views
Inequality for bivariate normal distribution
Let $X_1$ and $X_2$ be bivariate normal with mean $\mu=(0,\mu_2)$, for any $\mu_2$, and correlation $\rho$.
Consider the following inequality:
\begin{align*}
Pr\left\{|X_1| \ge ...
3
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1answer
174 views
Order statistics of absolute value of bivariate normal distribution
Suppose $X_1$ and $X_2$ are bivariate normal and let $|X|_{(1)}$ and $|X|_{(2)}$
be the ordered version of their absolute value. I am interesting in finding the following probabilities or some bounds ...
3
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1answer
474 views
How to compare or normalize two sets of probabilities?
I'm a stats newbie so I hope this question isn't too easy for you! :) I'm not sure how to ask it so I've written two versions: the short version is a pure stats question and the longer version ...
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14 views
Inequality distribution function [duplicate]
Possible Duplicate:
Inequality distribution function
I am trying to understand a proof (no homework).
If $\frac{\overline{F}(x-y)}{\overline{F}(x)} \rightarrow 1$ for $x \rightarrow ...
2
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1answer
141 views
Inequality convolution distribution function
$F$ is the distribution function of a positive random variable.
Why does the following hold for every $x \in (0,\infty)$:
$F^{*2}(x) \leq F^2(x)$, where
$F^{*2}=F*F=\int\limits_0^x F(x-y)dF(y)$
2
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1answer
238 views
An instance of the Cauchy–Schwarz inequality
In the proof of Theorem 6.5 from the book by Devroye et al., how is the last inequality derived?
$$
\begin{aligned}
\mathbb{E}\left\{|\eta(X)-1/2|\mathbf{I}_{\{g(X)\ne g^*(X)\}}\right\}
&\leq ...
4
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1answer
303 views
Bounded in probability ($o_{P}$)
In Larry Wasserman's lecture notes on $o_{P}$ and $O_{P}$, I am not able to follow the derivation of the following example in page 9.
Consider $m$ coins with probabilities $p_{1}, \ldots ,p_{m}$. ...
2
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1answer
238 views
Understanding proof of McDiarmid's inequality
I am working through Wasserman's lecture notes set 2 and I am unable to fill in the missing steps in the derivation of McDiarmid's inequality (p.5). Just like my previous question in the forum, I am ...
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1answer
374 views
Bound on moment generating function
This question arises from the one asked here about a bound on moment generating functions (MGFs).
Suppose $X$ is a bounded zero-mean random variable taking on values in
$[-\sigma, \sigma]$ and let ...
4
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1answer
422 views
Understanding proof of a lemma used in Hoeffding inequality
I am studying Larry Wasserman's lecture notes on Statistics which uses Casella and Berger as its primary text. I am working through his lecture notes set 2 and got stuck in the derivation of lemma ...
3
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1answer
299 views
One sided Chebyshev inequality for higher moment
Is there an analogue to the higher moment Chebyshev's inequalities in the one sided case?
The Chebyshev-Cantelli inequality only seem to work for the variance, whereas Chebyshevs' inequality can ...
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1answer
447 views
Probability inequalities
I am looking for some probability inequalities for sums of unbounded random variables. I would really appreciate it if anyone can provide me some thoughts.
My problem is to find an exponential upper ...
12
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3answers
654 views
What is a tight lower bound on the coupon collector time?
In the classic Coupon Collector's problem, it is well known that the time $T$ necessary to complete a set of $n$ randomly-picked coupons satisfies $E[T] \sim n \ln n $,$Var(T) \sim n^2$, and $\Pr(T ...
