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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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'Knowing' and 'Learning' a Random Variable

When using basic ideas from probability to think about information (e.g. entropy etc.), some commonly used jargon includes phrases such as: Learning a discrete random variable (r.v.) $Y$ Knowing a ...
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Real Borel measure such that induced measure has covariance but not a mean operator

Kuo's example $6$ in Gaussian Measures in Banach Spaces Let $\mu$ be a Borel measure in $\mathbb R$ such that $\int_\mathbb{R}t^2\mu(dt)<\infty$ and $\int_\mathbb{R} t\mu(dt)=\infty$. Then $S_{\...
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Gaussian product - posterior probability distribution

I am with Elements of Statistical Learning 8.4 Relationship between the bootstrap and bayesian inference. We observe a single observation $z$ from a normal distribution $z \sim N(\theta,1)$ We ...
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1answer
26 views

Is $P(A \mid B) = P(A \mid B,C) + P(A \mid B , C^c)$?

I am confused with these statements: \begin{align} P(A\mid B) &= P(A\mid B \cap \Omega)\\ &= P(A\mid B \cap (C \cup C^c) \\ &= P(A\mid B \cap C) + P(A\mid B \cap C^c) \quad\text{because $...
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Stability of trivial solution of the $dx_t = 5x_tdt + 2019x_tdW_t$

Let $x_t = 0$ be a trivial random process. Check if it is a solution for the SDE $dx_t = 5x_tdt + 2019x_tdW_t$ (where $W_t$ is Wiener process). If it is - then check wheather is it stable. It is ...
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1answer
20 views

How do I solve given only 1 of 3 probability values

I am given 3 disjoint events A, B, and C. And the following expression to solve when the P(A) = 2/5. $P(A \cup ( B^c \cup C^c)^c )$ From De Morgen's Law: I can transform it to $P(A \cup (B \cup C))$...
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Probability Question Replaced and Not Replaced

There are 3 blue balls, 3 Green balls, and 6 Red balls. We take two balls one after another. If the first ball is red or green, we replace the red ball with another red and a blue. Then take the ...
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Optimal tail bounds for the sample mean

Given a bounded random variable $X$ such that $|X| < M$, I want to know how many iid samples $x_1, \dots, x_n \sim X$ have to be drawn such that the sample mean $\bar{x} = (x_1 + \dots + x_n)/n$ ...
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Is this a valid argument? Independence of sign and magnitude in a random walk

If $X_i$ are independent, symmetric random variables with mean $0$, and form the basis of a random walk $S_n =\sum_{i=1}^n X_i $, I am wondering if it's correct to say that the events: $$A=\{|X_i|\...
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1answer
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Finding the density of $X$ selected from $[Y,1]$, where $Y$ is first selected in $[0,1]$

Based on the problem statement: $f_{X|Y}(x|y) = \frac{1}{1-y}$ for $0<y<x<1$. $f_X(x) = \int_{-\infty}^{\infty} f_{X|Y}(x|y)f_Y(y)dy = \int_{-\infty}^{\infty}= \frac{1}{1-y}dy$ My ...
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Distribution function of a random variable (demonstration( [on hold]

Need some help solving this demonstration Prove pr(a
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1answer
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$\xi$ has modification $\eta$ with continuous paths $\iff \exists c \in \mathbb R$ s.t. $P(\xi_{0}=c)=1$

Let $\xi:=(\xi_{n})_{n\geq 0}$ be IID and $\eta:=(\eta_{n})_{n \geq 0}$: A path is defined as a map for fixed $\omega$ that $[0,\infty[\ni t\mapsto\xi_{t}(\omega)$ Show that: $\xi$ has modification ...
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Random Walk $\mathbb P(T_0>n $ and $S_n=a) = \mathbb P(T_a=n) =\frac{a}{n} \mathbb P(S_n=a)$

Consider the random Walk $S_n$ on $\mathbb Z$ starting in $x=0$. Let $a\in \mathbb Z$. Define $T_a(\omega)=\min\{n\in \mathbb N : S_n(\omega)=a\}$. Show for $a> 0$ $\mathbb P(T_0>n $ and $S_n=...
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1answer
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$X$ and $Y$ independent random variables then $E(XY) = E(X)E(Y)$ from the measure theory perspective.

Given $X:[0,1]² \to \mathbb{R}$ and $Y:[0,1]²\to \mathbb{R}$ random variables(i.e measurable functions), we say that $X$ and $Y$ are independent if $$m(w; X(w) \in A, Y(w) \in B) = m(w;X(w) \in A)m(w;...
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1answer
9 views

Bound on supremum of local martingale

Let $M$ be a continuous local martingale starting at $0$. How can I prove $$ P(\sup_{s\leq t}M_s>a,\langle M\rangle_t\le b)\leq 4\frac b{a^2} $$ for all $a,b>0$ and $t>0$?
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2answers
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Intuition: Portmanteau-Theorem

An important theorem in probability theory about weak convergence of measures is the Portmanteau-Theorem. Why should it be true - intuitively - though? EDIT: Our version of Portmanteau's Theorem is: ...
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0answers
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Do independent increments and continuous variance imply that random process is $L_2$ continuous?

Let $X_t$ ($t \in \mathbb R$) be a random process with independent increments such that its variance $DX_t$ is continuous as a function of $t$. Is $X_t$ $L_2$-continuous? It is well known that random ...
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Does Sørensen–Dice Coefficient only account for true positives?

I'm working in a project on medical image segmentation which uses the Dice Score as part of the loss function, but I got some doubts with the commonly adopted implementation. The definition of Dice ...
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2answers
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probability distribution proof $P(a < X \leq b) = F(b) - F(a)$

Let F be the distribution function of the probability $\mathbb{P}$ on $\mathbb{R}$ (induced by some random variable $X$). Prove: $\mathbb{P}((a,b]) =\mathbb{P}(a < X \leq b) = F(b) - F(a)$ This is ...
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0answers
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Tractability of Expectations

I'm working my way through a paper about bounds on the mutual information [1]. However, I have some issues in understanding claims they make about the tractability of the different bounds. Given: $ ...
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1answer
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show normality with non-linear transformation [duplicate]

This is one of the problem in the Allan Gut's Second Course for Probability. Let $X_1$, $X_2$ be independent standard gaussian e.g. N(0,1). Let $Y_1 = \frac{X_1^2 - X_2^2}{\sqrt{X_1^2 + X_2^2}}$, $...
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3answers
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Given the bivariate standard normal density, how to show that $X$ and $Y$ are standard normal densities? [on hold]

Given the standard bivariate normal density with correlation coefficient $\rho$ for $X$ and $Y$: $$f_{X,Y}(x,y)=\frac{1}{2\pi\sqrt{1-\rho^2}}e^{-(x^2-2\rho xy+y^2)/2(1-\rho^2)}$$ Is there a way to ...
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1answer
29 views

Conditional Expectation of XY When Individual Conditional Expectations Have a Certain Property

Imagine we have $X$ and $Y$ defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ such that $\mathbb{E}|X|, \mathbb{E}|Y|, \mathbb{E}|XY| < \infty$. Let $\mathcal{A}$ and $\mathcal{...
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1answer
43 views

Validity of Proof of Wald's identity

$\newcommand{\E}{\mathbb{E}}$ Theorem (Wald's identity): Suppose $\{X_i\}_{n \in \mathbb{N}}$ is a sequence of i.i.d random variables with $\E X_1 < \infty$. Let $\tau$ be a stopping time with ...
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1answer
24 views

Poisson process on nonintersecting sets

I try to show that for a homogeneous Poisson process $N$ with intensity $\lambda$ and nonintersecting sets $A,B$ the amount of events happening in the two sets are independent. My idea is to prove the ...
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1answer
34 views

Prove that stochastic prcess trajectories are continous.

Consider gaussian process $\{X_{t}, t \in [0,1]\}$ with zero mean and covariance function $R(X_{s}, X_{t}) = \min (s,t) -st$. We want to know does this process has continous trajectories, i.e. $\...
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1answer
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Confused about the expression relating the CDF and expected value of a random variable

Let X be a random variable that takes on nonnegative values and has distribution function $F(x)$. $E(X) = \int_{0}^{\infty}1-F(x)dx$ The proof my book gives is: This statement was included at the ...
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1answer
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Series Converging Almost Surely But Diverging in Mean

I am looking for an example of independent, non-negative random variables $X_1, X_2, \dots$ such that $$ \sum_{n=1}^{\infty} X_n \, \lt \, \infty $$ almost surely but $$ \sum_{n=1}^{\infty} \...
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Regression on trivariate data with one coefficient 0

Suppose {$(x_i,y_i,z_i):i=1,2,...,n$} is a set of trivariate observations on three variables:$X,Y,Z$, where $z_i=0$ for $i=1,2,...,n-1$ and $z_n=1$.Suppose the least squares linear regression equation ...
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1answer
42 views

$E(X_k|Y)$ for $X_k \sim Bern(\theta)$ and $Y := \sum_{k=1}^n X_k.$

Let $X_1, ..., X_n$ be i.i.d. random variables with $X_k \sim Bern(\theta)$ for $\theta \in (0,1)$. Furthermore, define $Y := \sum_{k=1}^n X_k.$ Determine $E(X_k|Y).$ Since $X_k$ and $Y$ are ...
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Error bound for Sparse Regression

My lecture notes state the following: We have, for all $t_0\geq 0$, $$ P\left(\|X(\beta-\beta^*)\|^2\geq 160k \cdot \log(2d/k)+8t_0\cdot k\right) \leq e^{-t_0\cdot k / 10}.$$ Then, using (for non-...
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1answer
15 views

A aperiodic state of a Markov chain has $N\geq 1$ such that $\forall n\geq N:p_{i,i}(n)>0$

The question I get asked is the following, I'm completely stuck on the problem: Let $i$ be an aperiodic state of a Markov Chain. Show that there exists $N\geq 1$ such that $p_{i,i}(n)>0$ for all ...
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Is the following always true: $\mbox{Var}[\mbox{Range}(X_1,\cdots,X_n)] = O(n^{-B})$ with $0\leq B \leq 2$?

Here $X_1,\cdots,X_n$ are i.i.d. The two extremes $B=0$ and $B=2$, and the standard case $B = 1$ are illustrated in the picture below. For the reference, see here.
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Constructing a Characteristic function

I'm working on an assignment and I am having difficulty with the following question: A particle is moving in one direction, covering each second 1 meter of distance starting at the the time $t = 0$ ...
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1answer
30 views

Understanding definition of random variable in textbook “Basic Stochastic Processes”

In Zdzislaw Brzezniak and Tomasz Zastawniak, "Basic Stochastic Processes" A random variable is defined as, How do I check the condition For every Borel set $B \in \mathcal{B}(\mathbb{R})$ ...
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26 views

Transformation of Bivariate Random variables

I am having trouble with this question and feel like my supports are not entirely correct: Given $f_u(u)= 1_{[1,0]}(u)$ and $f_v(v)= 2v*1_{[1,0]}$ with U and V being independent and $U,V \in \mathbb{...
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3answers
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How to calculate covariance with an minus like V(X-Y)?

my task is this: Be $ X $ and $ Z $ independent with the same distribution and $ Y :=X-Z . $ Calculate $ \operatorname{cov}(X, Y) $ and $ \operatorname{corr}(X, Y) . $ My Problem is the minus in $...
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1answer
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Yet another question on Expected number of flips to get 2 consecutive heads

So, there are quite a few solutions on the web to the problem of "what is the expected number of flips to get 2 consecutive heads for a fair coin". Many of theses solutions use the conditional ...
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1answer
65 views

Show that $X_n \to 0$ in probability under given condition.

Let $k > 0$. Suppose that $$\forall \epsilon > 0: \exists N: \forall n \geq N: P(|X_n| \geq \epsilon) \leq \epsilon k$$ Show that $X_n \xrightarrow{P}{ 0}$. Attempt: We have to show: $$\...
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Combinatorial inequality in Erdös-Kac proof.

I am reading a proof of Erdös-Kac theorem, in Durrett, "Probability: Theory and Examples", fourth edition. In some point, it is stated that $(\sum_{m=1}^nEZ_{n,m}^2)^k - \sum_{i_j} EZ_{n,i_1}^2 . EZ_{...
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1answer
33 views

Using Markov's inequality to find upper bound of tail probability for Gaussian random variable(Q-function).

Suppose $X$ be the standard normal distriution. Its tail probability will be $$Q(t)=\frac{1}{\sqrt{2\pi}}\int_{t}^{\infty}e^{-\frac {x^2}2}\,{\rm d}x$$ I need to find the upper bound of $Q$ function ...
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Existence of a weak solution to this SDE?

I am looking at an SDE of the form $d{X_t} = \left( {{1_A}({X_t}) - {1_{{A^c}}}({X_t})} \right)d{W_t}$ such that ${X_0} = 0$, $A \in \mathcal{B}(\mathbb{R})$ and ${A^c}$ has a lebesgue measure of zero....
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Lp Wasserstein Distance for two distributions which have equivalent partial marginals

Consider two distributions $p(x_1,x_2)$ and $q(x_1,x_2)$ which have equivalent partial marginals, say $p(x_1)=q(x_1)$. I am wondering if there is any relationship between two Wasserstein ditances $W_{...
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Some questions related to expected value of a random variable

My book states that if $\phi$ is a continuous function that maps from the range of $X$ to $R$, then $E(\phi(X)) = \int_{-\infty}^{\infty}\phi(x)f(x)dx$ (for the discrete case $\sum_{x\in\Omega}\phi(x)...
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24 views

Probability: Poisson distribution; unsorted elements after random review

The problem I am trying to solve is formulated as following: Random number $\mathit{N}$ of unsorted elements in a set is Poisson distributed with $\lambda>0$. These elements are found ...
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24 views

Relationship between tightness and weak convergence.

I am getting confused with the following set of notes http://www.stat.umn.edu/geyer/8112/notes/metric.pdf. On page 10, the author states Prokhorovs theorem as follows: Let $X_1,X_2,\dots$ be a ...
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1answer
23 views

Proving this sequence converges in $L^2(\mathbb{P})$

We have some IID sequence, $\left\{ {{X_n}} \right\}_{n = 1}^\infty $, of standard normal random variable on the probability space $\left( {\Omega ,\mathcal{F},\mathbb{P}} \right)$. Also $\left\{ {{\...
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3answers
61 views

Two dimensional induction

I have the following problem: I need to prove that given the following integral $\int_{0}^{1}{c(k,l)x^k(1-x)^l}dx = 1$, we the constant $c(k,l) = (k+l+1) {{k+l}\choose{k}} = \frac{(k+l+1)!}{k!l!}$, ...
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33 views

Using mirrored sample data to improve estimates

I will ask my question through an example game: Each round we are given a blue coin and a red coin. We are either given a pair of fair coins or a pair of biased coins (these four coins are the set of ...
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1answer
23 views

Probability Theory - Combinations

Please, I would like some help with the following problem. I tried to use combinations but I am wondering if I have to use also the Bayes formula, in the process of solving it. The problem is at it ...