All Questions
Tagged with statistics stochastic-processes
713
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Optimal matching of Bernoulli random variables
Let $Z_1$, ..., $Z_n$ be a sequence of independent Bernoulli random variables such that
for all $i\in\left\{1,..,n\right\}$ $Z_i\sim\mathcal{B}(p_i)$ where $p_i < 1/2$.
Define $l(x_{1:n}, y_{1:n}) =...
- 2,150
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100
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Conditional expectation with random variables from different Probability spaces [closed]
Let $(X,\mathcal{F}_X,\mathbb{P})$ and $(Y,\mathcal{F}_Y,\mathbb{Q})$ be two probability spaces. I know that the expectation of random variable $Z:X\rightarrow \mathbb{R}$ is affected by the random ...
- 505
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44
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Renewal reward process's reward tail probability
Suppose we are given a dice with $K$ faces, denoted by $k=1,\dots,K$, where the probability of realizing a face $k$ is $p_k\in[0,1]$ with $\sum_{k=1,\dots,K}p_k=1$.
Now, we roll the dice repetitively. ...
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106
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Probability that a probability will be less than a certain value
Suppose I have a nonnegative random variable $X$ and I don't know its expected value, but I know that its expected value is less than or equal to $a$ with at least probability $p^*$. i.e, $\mathbb{P}(\...
- 372
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33
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Markov Chain Detailed Balance $\pi(x)*P(x, y) = \pi(y)*P(y, x)$
Let's say I have a Markov chain and it has a transition matrix denoted as $P$. The $(row, column)$ elements of the $P$ matrix are denoted as $P(i, j)$. Just by looking at the transition matrix $P$, ...
- 26.8k
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2
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46
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Probability 2 earthquakes happen in a period of time.
The amount of earthquakes that happen at island X follows the Poisson process with mean 2 . Given that 2 earthquakes have happened in this year, find the probability both the earthquakes happen ...
- 667
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1
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13
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Expectation of the process adapted to the filtration of the Wiener process
Suppose $\sigma_t$ is a stochastic process adapted to the filtration $\mathcal{F}_t$ generated by the Wiener process $W_t$.
I would like to know how to compute the following expectation:
$$E = \mathbb{...
- 351
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35
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Difference between compensator of point process under real parameter an its MLE estimator
Suppose we have some point process $N=N_{\theta_0}$ on the real line, driven by a conditional intensity $\lambda_{\theta_0}$ dependent on some finite-dimensional parameter $\theta_0\in\Theta\subset\...
- 3,318
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Convergence of weighted sum to Brownian Motion
Let $\{\varepsilon_t\}_{t = 1}^T$ be a sequence of iid random variables such that $\varepsilon_t \sim N(0, \sigma^2)$ and $\sigma^2 > 0$. Then it is known that (see 17.3.6 in James Hamilton's Time ...
- 174k
3
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965
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Two independent Poisson processes.
I am trying to prove the result that exactly $k$ occurrences of a Poisson process before the first occurrence of another independent Poisson process is a geometric random variable.
\begin{align}
&...
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What is the probability of getting the same side n times in a row in a coin toss
Assuming everything is fair what are the odds that one of the two sides in a coin toss wins 6 times in a row within the first 6 tosses?
Please also answer for the general case ...
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95
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Unbiased Cumulant Estimate - Fifth Cumulant
I am searching the definition of the $5^{th}$ unbiased cumulant estimate.
Let $K_j$, be the $j$-th unbiased cumulant estimate of a probability distribution, based on the sample moments.
Let $m_j$ be ...
- 11
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Sub-Gaussian $X_t$, prove $\mathbb{E}\left[\sup_{t\in T}X_t \right] \leq 2 \mathbb{E}\left[\sup_{\rho(t,s)\leq \delta}(X_t-X_s) \right]+J(\delta,T)$
This is a question-and-answer just for me, but if you have alternate answers or comments, feel free to share them.
Let $(T,\rho)$ be a metric space and $\{X_t\}_{t\in T}$ be a sub-Gaussian process ...
- 1,565
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46
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A Gaussian process and a Rademacher proecss are sub-Gaussian
This is a question-and-answer just for me, but if you have alternate answers or comments, feel free to share them.
Let $(T,\rho)$ be a metric space and $\{X_t\}_{t\in T}$ be a stochastic process ...
- 1,565
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When $X_t$ is conditionally normal distributed and has density $p_t$, how can we compute $\text E\left[\left\|\nabla\ln p_t(X_t)\right\|^2\right]$?
Let $d\in\mathbb N$ and $(X_t)_{t\ge0}$ be an $\mathbb R^d$-valued process. Assume $$\operatorname P\left[X_t\in\;\cdot\;\mid X_0\right]=\mathcal N(X_0,\Sigma_t)\tag1$$ for some covariance matrix $\...
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