All Questions
Tagged with pr.probability st.statistics
1,134 questions
0
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Absolute continuity of probability measures determined by dependence structure
We are on $\mathbb{R}^d$ with Borel $\sigma$-algebra. Let $\mu_1, ..., \mu_d$ be probability measures on $\mathbb{R}$ and $\Pi(\mu_1, \mu_2, ..., \mu_d)$ be the set of probability measures on $\mathbb{...
- 1,095
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268
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Does the linear combination of the quantile $\alpha F^{-1}(\tau)+\beta G^{-1}(\tau)$ still a quantile
$F(x)$ and $G(y)$ are distribution functions.
Define the $\tau$th quantile for cdf $F(x)$, $G(y)$ as
$$\xi_\tau\equiv F^{-1}(\tau)=\inf\{x:F(x)\ge \tau\}$$
and
$$\eta_\tau\equiv G^{-1}(\tau)=\inf\{y:...
- 141
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votes
0
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124
views
Which sub-sequence selection rules preserve the iid property?
Let $\xi_1,\ldots,\xi_n$ be an iid sequence of random variables. If we take a sub-sequence $\xi_{i_1},\ldots,\xi_{i_k}$ with constant indices $1\leq i_1 <\ldots <i_k\leq n$, then the sub-...
- 251
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0
answers
102
views
Probability of random variable being lesser than the other
Say there are two independent random variables, $X$ and $Y$, and we have samples $\{x_1,\dots x_n\},\{y_1,\dots y_n\}$. I am interested in bounding the probability of the event $C = \mathbb{1}_{X<Y}...
- 197
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0
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322
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Comparison of Parameter estimation using maximum likelihood and Maximum entropy
I am not sure if the question is appropriate but I want to try my luck. One can estimate a parameter using maximum likelihood and we know it is optimal. On the other hand there are methods which uses ...
- 495
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216
views
Hoeffding's lemma for unbounded r.v with bounded exponential map
Let $X$ be a real r.v with $E[e^{\lambda X}] < \infty $ for all $\lambda \in [-c,c]$.
Is it possible to get an Hoeffding's lemma like bound on $E[e^{\lambda(X-EX)}]$. That is, an upper bound: $$E[...
- 9
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444
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How to decide a value of learning rate for Stochastic Gradient Descent?
I'd like to know how to decide a value of learning rate for Stochastic Gradient Descent (SGD), such as $\eta$ on the following parameter update iteration equation,
$w_{i+1} = w_i + -\eta \nabla E_n(...
0
votes
1
answer
408
views
Generating independent random variable from two correlated random variables
Suppose two random variables $X$ and $V$ are given. I am wondering what kind of condition we need to impose on joint distribution of $V$ and $X$ to make sure that there exists a random variable $Z$ ...
- 1,109
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213
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Behavior of the sum of the exponents of chi-squared random variables normalized by their maximum
Let $X_1,X_2,\ldots,X_n$ be a sequence of $n$ i.i.d. chi-squared random variables with $k$ degrees of freedom, and denote by $X_\max$ the maximum of this sequence. Furthermore, let $k=\omega(1)$ ...
- 907
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160
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Two Different Representations of Multivariate Bernstein Polynomials
In the literature the multivariate Bernstein polynomial of a function $f:[0,1]^m\rightarrow\mathbb{R}$ is often defined as the following:
$$B_{f,n}(x_1,\dots,x_m)=\sum_{\mathbf{k}\in \{0,\dots,n\}^m}...
- 103
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0
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112
views
Markov renewal process with failure?
I hope this question is not too elementary for this site, and that it contains a sufficient degree of detail.
I have a problem where I want to model sequences of variable length $\boldsymbol{e}_i = (...
- 101
0
votes
1
answer
107
views
Can one combine (join) probabilities from 2 aspects of a related process?
Consider 2 related aspects of a process for prices in a financial market:
time &
return.
Time
Say I've identified a distribution that reasonably models the occurrence of the lengths of price ...
- 111
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0
answers
352
views
prewhitening (whitening transform) in terms of expected-value-wr-sigma-algebra
I'm trying to understand the mathematics of prewhitening a little better. (See http://en.wikipedia.org/wiki/Whitening_transformation, e.g.)
Taking the conditional expectation of an RV with respect to ...
0
votes
1
answer
207
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Copulas and marginals thereof
Hello everyone,
I recently became aware of the existence of the copula concept.
So, I have been reading a few things about copulas lately, but
I cannot seem to find information on the following ...
- 103
0
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0
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319
views
Estimating a multinomial sum
I have the following sum
\begin{equation}
\sum_{r_1=q+1}^{\tau}\dots\sum_{r_\lambda=q+1}^{\tau}{\tau\choose r_1,\dots,r_\lambda,\tau-r_1-\dots -r_\lambda} (\Lambda-\lambda)^{\tau-r_1-\dots-r_\lambda}
\...
0
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0
answers
343
views
Can KL divergence go to 0, but $E[\log(p/q)^2]$ diverge in certain cases?
Let $p(x)$ be a fixed distribution over a discrete space.
Let $A, C > 0$ be constants.
Let $\epsilon > 0$. Can we find an example of a distribution
$q_{\epsilon}$ such that $\mathrm{KL}(p||q_{\...
- 1
0
votes
0
answers
138
views
Why do I not use post hoc tests with a 2 x 2 factorial?
I know this is an obvious answer. I am probably over thinking what I'm doing, but I cannot recall. Does it have to do with not having enough variables to compare the various means?
-1
votes
6
answers
2k
views
Chances to win an election
Let's say that tomorrow national president election is held. A poll asks 1100 persons which of the two candidates, A or B, will he or she will vote. 750 say will vote A, and 250 say will vote B. What ...
- 330
-1
votes
1
answer
2k
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how to prove that the real part and the modulus of a characteristic function is still a characterisitc function? [closed]
this is a problem from Durret's probability textbook.
Show that if $\varphi$ is a ch.f., then $Re\varphi$ and $|\varphi|^2$ are also ch.f.
I am wondering how to prove this. Actually I'm not even sure ...
- 11
-1
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2
answers
1k
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Does the variance of a strictly monotonically increasing function of a random variable have anything to do with the variance of the random variable? [closed]
Assume that there is a continuous random variable x, and its variance is var(x). Furthermore, there is a strictly monotonically increasing function f. Can anybody prove that the larger the var(x), the ...
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-1
votes
1
answer
545
views
probability mass function fitting [closed]
I have a probability mass function of some experimental data who's log looks like the following: (please ignore the fact that it is not normalized)
![alt text][1]
[image shack image removed]
(...
- 307
-1
votes
1
answer
199
views
How can one quantify the convergence of relative frequency to probability?
I have already asked this on stackexchange but did not get any answer.
Say I run a simple Bernoulli trial a number of times and compute the relative frequency for success. Clearly the relative ...
-1
votes
1
answer
75
views
Finiteness of "novel variance" from a kernel on a compact space [closed]
Let $c(i,i')$ be a kernel function on a reasonable index space $I$. Choose a dense sequence of points $\{i_1, i_2, \cdots \} \subseteq I$, and define the one-point kernel functions $k_n := c(\cdot, ...
- 8,512
-1
votes
1
answer
297
views
The distribution of the sum of values from a normal and a truncated normal distribution
Using R to extract truncated normal distribution samples and normal distribution samples separately, when they are combined, the image drawn by the hist function is very similar to a normal ...
-1
votes
1
answer
113
views
Approximating expectation of exponential of Wishart matrix
I am trying to obtain an Approximating expectation of exponential of Wishart matrix $X (N,N)$ with $\operatorname{rank} (X)=K$defined as:
\begin{align}
J = E[{e^{{v^H}Xv}}]
\end{align}
where $v$ is $...
- 377
-1
votes
1
answer
312
views
expectation of upper quantile proportion
(edited considerably following comments)
We have a collection $\boldsymbol{S}$ of $n$ discrete random variables $X_1$, $X_2$, $\dots$, $X_n$ $\overset{\small \text{i.i.d.}}{\small \sim}$ $\mathcal{D}$...
- 121
-1
votes
2
answers
512
views
Deriving the joint distribution of multivariate normal transformed into Bernoulli
Given a covariance matrix $\sum_{ij}$ and a mean vector $\mu$ I have sampled $N$ multivariate normal vectors $Z = (z_1,...z_n)$ My goal is to create a vector of Bernoulli random variables $Y = (y_1,......
-1
votes
1
answer
1k
views
Rank of covariance matrix whose diagonal elements are same [closed]
Suppose A is a covariance matrix whose diagonal elements are same, i.e. $A_{1,1}=A_{2,2}=\cdots=A_{N,N}$, can we conclude that A is full rank?
Suppose the absolute values of the off-diagonal elements ...
-2
votes
1
answer
43
views
$E(\mathbf{y}|\mathbf{x}+\mathbf{z})=g(\mathbf{x})$ almost surely, if $\mathbf{z}\perp \!\!\! \perp \{\mathbf{y},\mathbf{x}\}$ jointly?
Let $\mathbf{y},\mathbf{x}$ and $\mathbf{z}$ be real-valued random vectors with possibly different dimensions.
If $\mathbf{z} \perp\!\!\!\perp \{\mathbf{y},\mathbf{x}\}$ (i.e., $\mathbf{z}$ is ...
- 193
-2
votes
1
answer
347
views
Forms of multivariate CLT [closed]
I am looking for a good reference for differnt kinds of multivariate central limit theorems. I was wondering how far the i.i.d. condition of the standard multivariate clt can be relaxed, as in can the ...
-2
votes
2
answers
2k
views
probability of subset sum after rolling dice 4 times [closed]
If we roll 4 dices (fair), what is the probability of "sum of subset" being 5. e.g. 1432,1121, 2344, 2354 have a subset sum of 5. Can you illustrate how to calculate this.
- 1
-2
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1
answer
92
views
Existence or impossibility of Gaussian factory
Gaussian factory problem: given an iid sequence $x_i \sim \mathcal{N}(\mu,\sigma^2)$, $i=1,2,\dots$, with $\mu$ and $\sigma^2$ both unknown, construct a realization $y \sim \mathcal{N}(0,1)$.
-3
votes
1
answer
123
views
Are the first 4 statistical moments independent? [closed]
Are the first 4 statistical moments independent? Is there a mathematical demonstration that can show independence one from each other? Can the value of one moment influence the value of another? If so,...
- 11
-3
votes
2
answers
450
views
Expected values of two random variables related to a simple urn problem
In an urn there are $u$ balls, $b$ of which are black.
If we perform $n$ trials of one ball at a time with replacement, the probability of the event $E$ to get $n$ times a black ball is $P(E)=\left(\...
- 145