All Questions
Tagged with pr.probability st.statistics
1,135 questions
2
votes
2
answers
955
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Probability calculation, system uptime, likelihood of occurence.
A little stumped! This is probably a very basic probability question, but I am lost.
At work I was asked the probability of a user hitting an outage on the website. I have some following metrics. ...
- 123
3
votes
1
answer
2k
views
sum of order statistics
Suppose I have N real random variables with identical PDF. At every instance of these r.vs, I pick $K$ largest out of $N$. Lets call their sum as $S_K$. Alternatively, based on some criteria, I ...
- 33
2
votes
1
answer
447
views
MCMC with progressive demollification of delta distributions
Edit: I simplified the example to a canonical case for clarity.
Given an integral $\int_{\Omega}{g(\mathbf{x})}$ with a well-posed integrand $g(\mathbf{x})$ defined on some multidimensional space $\...
3
votes
1
answer
578
views
Why doesn't Stein effect happen for multinomial distributions?
(Medeen, et all, 1998)" show that Maximum Likelihood estimate is admissible for multinomial distribution under squared error. On other hand, James and Stein showed that arithmetic average is not an ...
- 2,252
3
votes
1
answer
188
views
Markov Chains based on sampled transition probabilities [closed]
If I have a process that transitions between states with some set, unknown probability, I can sample to find the transition probability. This probability is a sample average, with a well understood ...
- 191
2
votes
1
answer
422
views
Extending Wald's equation to two classes of i.d. random variables?
I try to adopt Wald's equation to a slightly more complex problem. In fact, after a full day, I found some solution now, but it has a confusing argument in the middle. Perhaps somebody can help me at ...
- 340
0
votes
1
answer
666
views
A Cauchy–Schwarz Type Inequality Involving Scaled Distributions
I have stumbled upon a rather intriguing inequality involving the product of the scaled distribution and the scaled density of a random variable. The inequality has a very attractive form, and it ...
- 197
9
votes
2
answers
674
views
Small crown probabilities (and infinite dimensional margin assumption)
My question is:
How do I find sharp upper bounds on $P(|q|\leq \epsilon)$ uniformly over a set of gaussian polynomes $q$ of degree two.
Notations and definitions (to make the question rigorous)
Let ...
2
votes
1
answer
256
views
Taking the partial derivative of the t-CDF with respect to the degrees of freedom
I am trying to find the maximum likelihood estimate of the parameters for the t-copula. Ideally I'd want to use a gradient-based method for optimization. However, I am having some difficulty in ...
- 41
1
vote
1
answer
742
views
proofs of stochastic boundedness
I'm looking at some statistical literature and trying to compare the results given there in probabilistic big-Oh notation with statements I'm more familiar with.
In particular, I'm trying to ...
- 922
2
votes
1
answer
2k
views
Statistical test comparing two relative frequencies
I'm working with four populations consisting of true/false events. They each have a different mean and variance. I have samples from each, with different sample sizes. Call the percentage of observed ...
- 597
2
votes
0
answers
98
views
Finding a general form of the density function when we have a four dimensional random variable
Consider a subject having time of the specific event $T_i$, which is a single sample from a
distribution $F_i$ with density $f_i$ and support
$[t_{\min},t_{\max}]$, for $i= 1,\ldots,n$. Let these ...
- 35
1
vote
0
answers
85
views
Maximum likelihood estimation with several distributions
My question concerns using Maximum likelihood to estimate unknown parameters used by several (poisson) distributions.
The parameters are the pairs $(a_1,b_1),\dots,(a_N,b_N)$, and for each pair $(i,j),...
- 11
3
votes
2
answers
921
views
Characteristic operator
Let $X_t\in\mathbb{R}$ be an Ito diffusion process given by $$ dX_t=a(b-X_t)dt+\sigma dW_t$$, then the characteristic operator of $X_t$ is given by $$L=a(b-x)\frac{\partial}{\partial x}+\frac{\sigma^...
- 29
3
votes
2
answers
453
views
What is this probability distribution?
Suppose we have a family $F_0,F_1,\dots$ of independent random variables which take the value $1$ with probability $p$ and $0$ otherwise; let $\delta$ be a number between $0$ and $1$. Let
$X_n = \...
- 1,180
3
votes
1
answer
528
views
Cover a line segment randomly with smaller line segments
Covering a circle randomly with arcs has been well studied in the past (Geometric Probability - Solomon).
But the problem when the circle is changed to a line segment doesn't seem to have been ...
- 161
1
vote
1
answer
282
views
Is an unbiased estimator with arbitrarily small variance necessarily consistent?
Given an unbiased estimator $\hat \theta_n$ of a parameter $\theta$, if the estimator has small variance (approaching $0$ as $n\to\infty$), it seems reasonable to expect that the estimator is ...
- 133
1
vote
1
answer
394
views
Conditional probability and independence
Suppose that we have vectors of events $\{H_1,...,H_n\}$ and $\{D_1,...,D_m\}$. Consider the following two sets of conditions:
Condition set 1
(1) $P(H_i H_j)=0$ for any $i\neq j$ and $\sum_iP(H_i)=...
- 2,619
1
vote
1
answer
281
views
A uniqueness proposition involving Erf, the error function
This is a generalization of a previous MO question, "Reducing system of equations involving Erf, Error Function".
Consider the system of equations:
$$1/2 + {\rm Erf}(x) - \alpha {\rm Erf}(\frac{x+y}{...
- 123
3
votes
1
answer
281
views
Deciding whether or not an exponentially distributed random variable exists in a set via the use of a "black box" function
I have some set of known size but with unknown elements, $(x_1, ..., x_N) \in X$, where the elements of $X$ are exponentially distributed random variables with unknown rate parameters, $(\lambda_1, ......
- 31
2
votes
0
answers
60
views
Consistency of M-estimators when the constraint set also has to be estimated
Let $K \subset \mathbb R^n$ compact and convex. Also let $H$, $G_i, \; i \in \{1,\dotsc,m\} $: $K \to \mathbb R$ be convex functions.
Assume we have the following convex optimization problem:
$$
\...
- 123
2
votes
0
answers
71
views
Asymptotic results for functions of order statistics
There are $n$ ($n \ge 3$) iid random variables $\{ {c_i}\} _{i = 1}^n$ on the interval $[\underline c,\bar c]$ ($\underline c>0$). The cdf $F(\cdot)$ and pdf $f(\cdot)$ are unkown to us, but we ...
- 29
7
votes
1
answer
804
views
Random, Linear, Homogeneous Difference Equations and Time Integration Methods for ODEs
Most methods (that I know of) of numerically approximating the solution of ODEs are "general linear methods". For this type of method, the so-called 'linear stability' is examined by applying the ...
- 103
0
votes
0
answers
213
views
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)$ ...
- 917
6
votes
0
answers
189
views
Pettis Integrability and Laws of Large Numbers
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space, and let $V$ be a topological vector space with a dual space that separates points. Let $v_n : \Omega \to V$ be a sequence of Pettis ...
- 8,512
4
votes
2
answers
714
views
Mathematical means of studying large and complex but finite systems?
I want a list of the sort of mathematics/mathematical tools that are applied to the study of complex and probabilistic systems in order to make quantitative and qualitative observations about their ...
- 2,383
2
votes
3
answers
403
views
On a randomized version of compressive sensing
The compressive sensing theory of Candes and Tao (See http://en.wikipedia.org/wiki/Compressed_sensing) relies highly on the fact that the underlying data (such as a signal or an image) is sparse or ...
- 21
1
vote
2
answers
1k
views
Can you explain a step in an expectation maximization algorithm in a Nature article?
I am currently going through the following article: http://www.nature.com/nbt/journal/v26/n8/full/nbt1406.html
In this article, how did they arrive at the values in the Estimation step (Figure 1 Step ...
- 19
7
votes
2
answers
404
views
Examples of Slowly Mixing Chains in Statistics
This should probably be community wiki, but I don't know how to set that myself.
I'm looking for examples or Markov chains that are used in statistics or statistical physics, and which are known to ...
- 71
6
votes
2
answers
545
views
Extension of copulas
Let $(X,Y)$ be a random vector. Suppose that the marginal distribution functions of $X$ and $Y$ are known (say $F_1$ and $F_2$). Then the joint law of $(X,Y)$ is given by the following formula:
$$F_{...
- 931
1
vote
1
answer
136
views
Transition time in finite voter model
I believe the following problem is related to something called the "voter model" in statistics. This is not my area of expertise so please forgive me if the answers turn out to be well known.
...
user32786
0
votes
1
answer
1k
views
Kernel width in Kernel density estimation
Hi,
I am doing some Kernel density estimation, with a weighted points set (ie., each sample has a weight which is not necessary one), in N dimensions.
Also, these samples are just in a metric space (...
- 481
1
vote
0
answers
70
views
Bounds on product of CDF or Beta function
I have functions of the form
\begin{align}
I_i = \int_0^\infty F_0(x)^aF_1(x)^b(1-F_0(x))^c(1-F_1(x))^ddF_i(x),~~~~i = 0,1.
\end{align}
$F_0(x)$ and $F_1(x)$ are CDFs corresponding to the random ...
- 11
3
votes
1
answer
663
views
Stationary non-isotropic spatial stochastic processes
I asked this question in math.stackexchange but got no response;
Are there any interesting examples of second order stationary processes on ${\mathcal R}^2$ or ${\mathcal R}^3$ that are not isotropic?...
- 549
0
votes
1
answer
577
views
Expectation of little o in probablity [closed]
If I have $Z=o_p(1)$ where $o_p$ is the little-o in probability. I'm interested in find some properties about $E(Z)$.
My first idea was
$E(Z)=E(Z (1_{Z>\varepsilon} + 1_{Z\leq\varepsilon}) ) \...
1
vote
2
answers
1k
views
what will be the distribution of ratio of correlated gamma distributed random variables?
If $X\sim \Gamma(a,\sigma_x^2)$ and $Y\sim \Gamma(b,\sigma_y^2)$. What will be the probability density function of R? Where $R=\frac{X+C}{X+Y}$, here $C$ is a positive constant, $\Gamma(.,.)$ denotes ...
- 133
0
votes
2
answers
429
views
E[log(Z_t^2)], proof of convergence with Law of Large Numbers
Hi all,
question:
Let $Z_t$ be an iid sequence with $$\mathbb{E}\log(Z_t^2)<0 $$
Show that $$\sum_{j=0}^\infty Z_t^2 Z_{t-1}^2 ... Z_{t-j}^2 < \infty$$ almost surely
I am supposed to use LLN ...
- 109
0
votes
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 ...
6
votes
1
answer
836
views
Peakedness of multimodal distributions
In Probability theory, does there exist some measures of peaked-ness for multi-modal distributions. I guess kurtosis as such would not be a good measure of peaked-ness for multimodal distributions. ...
- 99
1
vote
0
answers
245
views
Random walk conditioned on sum and last step
Can anyone help with the following (slightly weird) random walk question? I have a random walk starting at $X_0 = 1 = S_0$ where the steps $X_i$ are independent uniform random variates in $[-1,1]$. ...
- 747
4
votes
1
answer
213
views
Practical way to check for geometric convergence
Target distribution is multimodal, 24 dimensions, continuous state space. For MCMC integration (MH sampler) I use a manually tuned proposal distribution.
When I measure the convergence rate ...
- 101
1
vote
0
answers
186
views
Shrinkage (or Stein's phenomenon) in low dimensions, discrete contexts
I am trying to understand shrinkage, or the Stein phenomenon. As someone without a statistics background, the focus in most introductory presentations on normal distributions and squared error loss ...
- 203
6
votes
0
answers
262
views
Given that a conditional measure is Gaussian, how bad can the original measure be?
Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...
- 8,512
1
vote
0
answers
132
views
Eigen value distribution of autocorrelated Wishart matrix
Suppose the matrix W is constructed as $W=XX^T$ where $X_i(t) = \phi_i X_i(t-1) + a_i(t)$, and $a_i(t)$ ~ $N(0,1)$. I am interested in knowing the eigen value distribution of W. My google search on ...
- 11
1
vote
0
answers
104
views
Efficient evaluation of multidimensional kernel density estimate
Edit I have copied this discussion to the stats community site here, since I feel it is more relevant. Please feel free to close this in due course.
I've seen a reasonable amount of literature about ...
- 161
2
votes
1
answer
201
views
Moments of random matrices - when are they finite
I need to evaluate the moment
$$\mathbb{E} (AX)^n,$$ where A is an NxN Hermitian square matrix, and X is
$$X=ZZ^{\ast},$$ where
$Z=\mu+Y$, where $\mu$ is mean of $Z$ and $Y$ is a zero-mean complex ...
1
vote
0
answers
42
views
Adding weights to the Brier score
Fix $n > 0$, and consider the space $\cal P$ of probability functions defined over the Boolean closure of a fixed $\cal S = \{ s_1, \ldots, s_n \}$. The Brier score of $P \in \cal P$ at $s_i \in \...
- 111
4
votes
1
answer
151
views
Mean occurrences of letters in complete strings given by a Bernoulli scheme
Suppose one has an alphabet of $K$ letters, from which we draw sequentially letters; assume that the $n$-th letter occurs with a fixed probability $p_n$ independently of the others and of the previous ...
- 418
2
votes
0
answers
265
views
Expectation of a multivariate Gaussian over a plane
For a vector $X$ which follows a multinomial Gaussian distribution $N(\vec{0},\Sigma)$, a given vector $b$, and a known scalar value $c$, I would like to calculate the expectation :
$E[X|X^Tb = c]$
...
- 21
1
vote
0
answers
44
views
Validating a probability density distribution forecast model for a Markov process
Let's say we have a Markov process $X_t$, and we come up with a forecast model that takes some information from outside world and says: "value $X_{t+1}$ has probability density distribution $P_t(x)$". ...
- 93