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Questions tagged [probability-theory]

Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use [tag:probability-distributions] for specific distribution functions, and consider [tag:stochastic-processes] when appropriate.

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Concentration in Gauss space

This is a theorem called Concentration in Gauss space. Let $f$ be a real valued Lipschitz function on $\mathbb {R}^{n}$ with Lipschitz constant $K$, i.e. $\left|f(x)-f(y)\right|\leq K\|x-y\|_{2}$ for ...
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How much do tails contribute to a Gaussian's total variance?

H${}$ello, if $X\sim \mathcal{N}(0,I_{n\times n})$ what is a good upper bound for $\frac{1}{n}\int_{A} \|X\|^2 d\mathbb{P}$ when $\mathbb{P}(A)<\varepsilon$? Thanks!
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Is there a deeper reason why the simple symmetric random walk on $\Bbb Z^D$ turns transient when increasing $D$ from 2 to 3?

Polya proved the following very well-known Theorem: A simple random walk on $\Bbb Z^D$ is recurrent if and only if it is symmetric and $D\le2$. Dropping simplicity (i.e. allowing jumps to non-...
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Expectation of Gaussian r.v. conditioned on positive r.v.s with positive covariances is positive

Suppose that $\Delta_1,\dotsc,\Delta_K \sim \mathcal{N}(0, \Sigma)$, with $\mathrm{cov}(\Delta_i, \Delta_j) > 0$ for all $i,j$. Does $$ \mathbb{E}[\Delta_K \mid \Delta_1> 0, \dotsc, \Delta_{K-1}...
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1answer
26 views

Figuring out the variance of the birthday paradox

Given n people, if I want to estimate how many of them are likely to have an overlapping birthday with any other person, how do I calculate the variance? So far I have $E[X]=n\cdot p$ with $n$ as the ...
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23 views

Obtaining a probability distribution after a change of variables.

I have two random variables $x_1,x_2$ whose distributions are unknown. I define $y_1=g(x_1,x_2)$ and $y_2=f(x_1,x_2)$ where $f(\cdot)$ and $g(\cdot)$ are known and the probability distributions $p(y_1)...
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Does convergence in distribution imply convergence of regular versions of the conditional expectation?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(E,d)$ be a locally compact complete separable metric space $D([0,\infty),E):=\left\{x:[0,\infty)\to E\mid x\text{ is càdlàg}\right\}$...
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15 views

understanding transitional probabilities and measure

In this example 1.11, 1, could anyone just explain to me the way he has defined the $\mathcal P(x,\cdot)$? I know the definition of $\delta_x$ which indicates the probability centered at point mass. ...
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Martingality of a Doleans-Dade exponential local martingale.

A paper I read recently seems to make the following statement: if $\gamma_t$ is a progressively measurable process, and that $\exp\left(\int_0^T\gamma_s dW_s\right) \in L^p$ for some $p>1$, then ...
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7 views

Covariance operator of induced measure

Let $H$ be a real separable Hilbert space and $\mu$ be a real borel measure in $H$. Let $e$ be a unit vector of $H$. The induced measure in the direction $e$ is defined as the Borel measure $$\mu_e(E)...
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Continuity in mean of a stochastic process

If $X$ is a stochastic process, a.s. continuos and such that $\forall t \geq 0, X_t \in L^1_\omega$, is its mean function $t \rightarrow E[X_t]$ continuos? I can show it if $X \in L^1_\omega L^{\...
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20 views

Krylov-Bogolioubov Theorem

Could anyone kindly explain to me how $2\varphi$ comes in the calculation here in page $7$, 1Proof of Thm $1.10$? Thanks for helping. is thm 1.10 is same as 2here in thm 4.17?
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Interchange expectation and supremum?

In a book about statistical learning, I saw the following: enter image description here Intuitively, the formula in the middle is true, and the book says the "trivial" proof is left as an exercise (...
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1answer
21 views

Classify the 1000000 elements on the basis of 1000 tests [on hold]

There are 1000000 elements with parameters (a,b,c,d,e,f). I have to classify them into 2 classes (whether f(a,b,c,d,e,f) less than N or not). The calculations for ...
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10 views

Best way to visualize queueing theory for a Lecture on Markov Chains

Let the Markov Chain $X:=(X_{n})_{n \in \mathbb N_{0}}$ denote, for every $X_{n}$, the number of people waiting in a line at time $n$. Now note that $X$ is a Markov Chain living on $\mathbb Z_{+}$. ...
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the independence property of conditional expectation

I read a proof of following property: suppose $X$ is a random variable on the probability space $(\Omega,F,P)$, $A$ and $B$ are sub $\sigma$-field of $F$, $B$ is independent of $\sigma(X,A)$, then $E(...
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Given a càdlàg process $X$ and a measure μ on the Skorohod space, how can we show that μ is the law of $X$ up to the dependence on the initial value?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(E,d)$ be a locally compact complete separable metric space $D([0,\infty),E):=\left\{x:[0,\infty)\to E\mid x\text{ is càdlàg}\right\}$...
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24 views

A question about the proof of Theorem 5.21 in Van der Vaar(1998)

If we know that $\hat\theta_n\overset{p}\to\theta_0$, how does the following equation \begin{equation} \sqrt{n}V_{\theta_0}\cdot(\theta_0-\hat\theta_n)+\sqrt{n}o_p(|\hat\theta_n-\theta_0|)=G_n\psi_{\...
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1answer
48 views

“expectation of sum is sum of expectation”, is this claim true? if yes, how to justify this claim?

this post is saying linearity of expectation gives following equation $$\mathbb{E} [\sum_{j\neq i} Y_i Y_j] = \sum_{j\neq i} \mathbb{E} [Y_i Y_j]$$ per wiki, Linearity of Expected_value is ...
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1answer
16 views

For a sequence of experiments where each $X$ is the number of trials until success with varying $p$, is each $X$ independent?

Assume that, every time you buy a box of Wheaties, you receive a picture of one of the $n$ baseball player. Let $X_k$ be the number of additional boxes you have to buy, after you have obtained $k-1$ ...
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Intuition of TailVaR

As per the actuarial guide I have called the CMP - from Acted - tailVaR is the expected loss in excess of the benchmark value L. I don't really get that, so I tried splitting the equation into: $...
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1answer
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Sampling with replacement. $P[\upsilon_m>n]=\prod_{i=2}^{n}(1-\frac{i-1}{m})$.

Let $\{X_n,n \leq 1\}$ be i.i.d. and uniformly distributed on the set $\{1,...,m\}$. In repeated sampling, let $v_m$ be the time of the first coincidence; that is, the when we first get a repeated ...
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1answer
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Finding the probability of success that maximizes the variance of independent trials

A professor wishes to make up a true-false exam with n questions. She assumes that she can design the problems in such a way that a student will answer the jth problem correctly with probability $...
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37 views

Expectation of an inner product in an infinite dimensional Hilbert space

Let $\mathcal{H}$ be a Hilbert space with the Borel $\sigma$-algebra. Let $(\Omega, \mathcal{F}, P)$ be a probability space and $x,y$ two $\mathcal{H}$-valued random variables, i.e. measurable maps ...
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1answer
31 views

Expected value of the number of different numbers drawn in 37 rounds of roulette?

I need help with this problem. What is the expected value of the number of different numbers drawn in 37 rounds of roulette? Is this possible to interpret as the number of records? So the expected ...
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17 views

Proving $h(X_n)\implies h(X)$ in distribution

Let $X_n$ be a sequence of random variables on $(\Omega,\mathscr{F},P)$such that $X_n\implies X$ (converges in distribution). Let $h:\mathbb{R}\to\mathbb{R}$ be a function whose points of ...
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45 views

$X_i$, $i=1,2,..n$ independent R.Vs $P(X_i=1)=\frac{3}{4} ,\ P(X_i=-1)=\frac{1}{4}$. Prove $\sum_{i=0}^nX_i \to \infty$ a. s. as $n \to \infty$

I am asked to prove $X_1+X_2+X_3+...+X_n$ diverges almost surely as $n \to \infty$ Let $Y_n=X_1+X_2+...+X_n$ then what we want to prove is $P(Y_n=k)=1, \text{ as} (k,n) \to (\infty,\infty)$ Let us ...
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1answer
20 views

Probabilistic PCA: Derive $\mathbb{E}[z_n \mid x_n]$

Consider the probalistic pca setting, where $x \in \mathbb{R^d}$ is an input vector drawn from $p(x)$, $z \in \mathbb{R^m}$ is an explicit latent variable with $p(z) = \mathcal{N}(0,\mathbb{I}_m)$ ...
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How can we proof this implication in Corollary 4.8.7 in the book of Ethier and Kurtz?

I'm trying to understand the proof of the implication "(g) $\Rightarrow$ (f)" in Corollary 8.7 of Chapter 4 in the book Markov Processes: Characterization and Convergence by Stewart N. Ethier, Thomas ...
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1answer
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Distinct random variables in sampling

Let $X_1,X_2,\dots$ be i.i.d. random variables with $X_1 \sim U[0,1]$. Throwing out a null set all the variables are distinct. Can anyone explain this second sentence? What does he mean with "...
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1answer
28 views

It would be possible to define an uniform distribution on $\Bbb N$ using infinitesimals?

In standard analysis it is clear that it is impossible to define an uniform probability distribution on $\Bbb N$ because there is no constant $c\in\Bbb R$ such that $\sum_{k=1}^\infty c=1$. Using ...
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If $Y\sim\operatorname{Beta}(a,1-a)$ and $Z\sim\operatorname{Exp}(1)$, then $YZ\sim\operatorname{Gamma}(0,1)$?

I have two random variables $Y \sim \operatorname{Beta}(a, 1 - a)$ $Z \sim \operatorname{Exp}(1)$ If $Y$ and $Z$ are independent, why is the distribution of $X = YZ \sim \operatorname{Gamma}(...
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1answer
13 views

Actual meaning of the formula $E[\phi(X,Y)|\mathcal{G}]= E[\phi(x, \cdot)]|_{x=X}$

Let's suppose we have a probability space $(\Omega,\mathcal{F},P)$ and a sub sigma algebra $\mathcal{G} \subset \mathcal{F}$. Let's say we have $X$ that is $\mathcal{G}$-measurable and $Y$ that is $\...
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Strong Markov property and another stopping time

I'm trying to prove that given a regular continuous time Markov chain $X_t$ (pure jump process), its embedded chain given by $Y_n=X_{T_n}$ is a homogeneous Markov chain, where $T_n$ is the time of the ...
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25 views

Tricky information inequality $I(X;Z) \geq H(T)$

I am wondering whether $I(X; Z) \geq H(T)$ when the following conditions hold: $H(T | X) = H(T)$ $H(T | Y) = H(T)$ $H(T | X, Y) = 0$ $H(Y | Z) = H(T | Z) = 0$ $X, Y, Z, T$ are discrete. I know first ...
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1answer
23 views

about Random Variables Expectation

suppose $\mathcal{G} \space is \space \mathcal{F}'s \space sub \space \sigma-algebra, and \space X\in\mathcal{L^1}(\Omega, \mathcal{F}, P), \space\ $$\ Y\in\mathcal{L^1}(\Omega, \mathcal{G}, P)\ .$ ...
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1answer
20 views

Applying the definition of mutual independence to outcomes of random variables

My book defines mutual independence as: events ${A_1, A_2, ..A_n}$ are mutually independent if for any subset ${A_1, A_2, ..A_m}$ (where $m \leq n$) of these events we have: $$P(A_1 \cap A_2 \cap ......
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1answer
54 views

Suppose $X_1, \dots, X_n, Y$ are independent random variables. Prove that $X = (X_1, \dots, X_n)$ and $Y$ are independent variables.

Suppose $X_1, \dots, X_n, Y$ are independent random variables. Prove that $X = (X_1, \dots, X_n)$ and $Y$ are independent variables. My attempt: Fix $A \in \mathcal{R}$ (a Borel subset of the real ...
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1answer
33 views

What is the PMF of the product of two discrete independent random varibales? [on hold]

I would like know how can I multiply two independent discrete random variables. Let $X$, $Y$ be two discrete independent random variables. What is the probability mass function of $Z=X Y$? For two ...
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24 views

Show that a càdlàg function is uniformly right-continuous on compact intervals

Let $(E,d)$ be a locally compact separable metric space, $I\subseteq\mathbb R$ be an interval, $f:I\to E$ be càdlàg and $a,b\in I$ with $a<b$. How can we show that $\left.f\right|_{[a,\:b]}$ is ...
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1answer
24 views

Conditional Independence and product of random variables

I am stuck at the following situation: Let random variables $Y, X, W_1, W_2$. I know that $W_1$ and $W_2$ are each independent from $Y$ conditional on $X$: $$p\left(Y\mid \{X,W_1\}\right) = p\left(...
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Find a distribution. Wiener proccess [on hold]

Find the distribution of $$\frac{1}{t-s} \left(W_t^2 + W_s \left[ \frac{t}{s} W_s - 2W_t \right] \right), \qquad 0 < s <t. $$ How do i do this? Where $$ W_t, W_s $$ - Wiener process
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1answer
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Is $x \mapsto P_x(A)$ measurable for a measure $P_x$ determined by transition kernels $(\delta_x,P_i)_{i \ge 0}$

Given a sequence $P_i$ of transition kernels from $(E,\mathcal B(E)$ to $(E,\mathcal B(E))$ and $\delta_x$ the Dirac delta measure for the point $x\in E$, it follows from the Ionescu-Tulcea theorem ...
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1answer
51 views

If $f\in C_0$ and $\lambda>0$, how can we show that $x\mapsto\int_0^\infty e^{-\lambda t}f(x(t))\:{\rm d}t$ is continuous wrt the Skorohod topology?

Let $(E,d)$ be a locally compact separable metric space, $C_0(E)$ denote the space of continuous function from $E$ to $\mathbb R$ vanishing at infinity equipped with the supremum norm and $D([0,\infty)...
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1answer
11 views

Does time changed brownian motion have no-memory property?

Let $W=(W_t)_{t \geq 0}$ be a Browniwn motion. Do the processes $$X_t = W_{e^t} \quad \text{and} \quad Y_t = \exp \left(- \frac{t^2}{2} \right) W_{e^t}$$ have the no-memory property, i.e. are the sets ...
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19 views

calculate the probability of error in a array of bits

I need to calculate the probability in a certain problem. So there are 555 random bits [1 0 1 0 ... 1 0 0 1 1]. These 555 bits are divided in 37 parts of 15 bits each. Of the 555 bits, X bits flip at ...
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10 views

When is a linear recurrent process stationary?

Let’s call a sequence of random variables $\{X_n\}_{n = 1}^\infty$ stationary, if $\forall n, m, k \in \mathbb{N}$ $EX_n = EX_m$ and $Cov(X_n, X_m) = Cov(X_{n + k}, X_{m + k})$. Let’s call a sequence ...
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0answers
18 views

Covariance of two random variables, linear relationship and normalization of covariance

The covariance of two random variables $X$ and $Y$ is given by $$\displaystyle\operatorname{cov}\left[X,Y\right]=\mathbb{E}\left[\left(X-\mathbb{E}\left[X\right]\right)\left(Y-\mathbb{E}\left[Y\right]\...
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96 views
+300

If $X\sim \mathrm{lognormal}$ then $Y:=(X-d|x\geq d)$ has approximately a Generalized Pareto distribution.

Let $X$ be a random variable with lognormal distribution. Show that when sufficiently large then $Y:=(X-d|x\geq d)$ is approximately a random variable with generalized Pareto distribution. Hint: Use ...
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1answer
31 views

On the convergence in probability of a sequence of random variables.

Let $\{X_t \}_{t \in \mathbb{N}}$ be a sequence of independent random variables such that $E[X_t] = \theta E[X_{t-1}]$ for all $t \in \mathbb{N}$ where $|\theta|< 1$ and $E[X_0] = \mu > 0$. ...