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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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8 views

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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12 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
21 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
11 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
47 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
20 views

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

Let $X$ , $Y$ be two discrete random variables. What is the probability mass function of $Z=X Y$?
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17 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
22 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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20 views

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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13 views

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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26 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
10 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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16 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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0answers
9 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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16 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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45 views

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
30 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 $E[X_0] = \mu > 0$. ...
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0answers
18 views

justification of $\sum_{j\neq i}\mathbb{E}[Y_i Y_j] = (n - 1)\mu^2$

I am learning the justification of Sample variance $${\displaystyle {\begin{aligned} \operatorname {E} [\sigma _{Y}^{2}]&=\operatorname {E} \left[{\frac {1}{n}}\sum _{i=1}^{n}\left(Y_{i}-{\frac {...
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2answers
25 views

which rule or definition apply ${\displaystyle {\begin{aligned} \operatorname {E} [Y_{i}^{2}] = (\sigma ^{2}+\mu ^{2}) \quad (3.1) \end{aligned}}}$

I am learning the justification of Sample variance $${\displaystyle {\begin{aligned} \operatorname {E} [\sigma _{Y}^{2}]&=\operatorname {E} \left[{\frac {1}{n}}\sum _{i=1}^{n}\left(Y_{i}-{\frac {...
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0answers
35 views

Is every probability measure a Radon measure? [on hold]

Let $\mu$ be a probability measure defined on a compact convex subset $K$ of a locally convex Hausdorff space $X$. Is $\mu$ a Radon measure?
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24 views

Probability of being below a threshold given the sample mean and variance

I have been thinking about the following apparently simple problem. Given a sample $X_1,...,X_n$ from a $N(\mu,\sigma^2)$ population. What is the probability that the first observation is below a ...
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53 views

Showing $E(X) = \sum_{i}E(X\mid A_i)P(A_i)$

Following is my proof. Suppose $X$ is a discrete-type random variable ranging in the set $S$ and $\{A_i : i=1,2,3,\dots\}$ is a finite or countably infinite partition of a sample space $\Omega$. We ...
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1answer
23 views

Uniform distributed success probability for a coin

$n\in \Bbb N$. Let $X_1 \sim \text{Uni}_{(0,1)}$ and $X_2 \sim \text{Bin}_{n, X_1}$ conditional on $X_1$. I want to find the distribution function of the law of $X_1$ given $X_2 = k$, i.e. $\Bbb P (...
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0answers
12 views

Upper bound on number of cliques in a Vietoris-Rips complex

Does there exist an upper bound on the number of cliques of order $k$ in a Vietoris-Rips complex? I found this work --> https://arxiv.org/pdf/1104.0914.pdf I understand it makes the assumption of ...
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1answer
25 views

Odd moment and the characteristic function of a random variable

Let $X$ be a random variable and $\phi_X(t)$ be its characteristic function. Let $n$ be a positive even integer. If $\phi_X(t)$ is $n$-times differentiable, then the $n$-th moment of $X$ exists and ...
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2answers
32 views

What will be the probablilty in these cases?

So we have a fair and unbiased dice, which is rolled thrice in a row. 1)What is the probability to get the sequence [1,2,3] in the three continuous trials? 2)What is the probability of getting the ...
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2answers
18 views

Finding the expected number of a certain colored ball drawn from an urn in k draws

Suppose we have an urn containing c yellow balls and d green balls. We draw k balls, without replacement, from the urn. Find the expected number of yellow balls drawn. Hint: Write the number of ...
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2answers
22 views

How is P(B) derived and why is $P(D_i)$ equal to 55/72 and not $(55/72)^i$

So this is a question with its solution below to which I don't understand 2 things. How is P(B) derived? And, why is $P(D_i)$=55/72 and not $(55/72)^i$. Since, for example, obtaining heads in the n ...
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1answer
19 views

$P(T_2=T_{-3}), P(T_1<T_4<T_{-1})$ and $P(T_3<2)-P(T_{-3}<2)$

Let $W(t)$ be a Brownian motion and $T_x=\inf\{t:W(t)=x\}$. I need to calculate $P(T_2=T_{-3}), P(T_1<T_4<T_{-1})$ and $P(T_3<2)-P(T_{-3}<2)$. I'm not sure if I understand these ...
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31 views

Is this application of a law of large numbers rigorous in this not identically distributed case?

Let $ \{ X_t \}_{t \in \mathbb{N} }$ be a sequence of indipendent random variables such that $X_t \sim N(u_t, 1)$ for all $t$ where the mean $u_t$ is given by the equation $$u_t = \theta u_{t-1} + \...
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0answers
15 views

Variance of linear combination

This is a follow up question to this. Let $(X_1,\ldots, X_n)$ be non-independent random variables such that $$\sum_{i=1}^{n} X_i\sim\sum_{i=1}^{n} \alpha (\mathcal{N}(0,1))^2$$ where $\mathcal{N}(0,1)...
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1answer
35 views

$m$ minimizes $E(|X-a|)$ over $a\in R$ if and only if $m$ is a median for $X$.

I'm trying to show that $E(|X-a|)$ attains its minimum value if and only if $a=m$ where $m$ is the median. I know this particular problem has been discussed before, but I want to prove it in a ...
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0answers
23 views

Quadratic martingale bound

I know that if $a_1,a_2,\dots$ are random variables and {$\mathcal{F}_{t}$ } is a filtration such that $$\mathbb{E}[a_i \mid \mathcal{F}_{i-1}] \leq K$$ for all $i$, then for any stopping time $\...
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38 views

Understanding i.i.d. random variables from product measure space perspective

I have a very weak background in measure theory, and I am having some troubles understanding i.i.d. random variables from a measure theoretic perspective. Let $X$ be a random variable defined on a ...
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21 views

probabilility point closer to distance circle

the probability that the point is closer to the distance to the center of the circle than to the circumference is $\frac{1}{4}$ find probability: (A) When several points are selected sequentially ...
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2answers
27 views

What is p(Evidence) exactly in a bayesian model?

I'm having a hard time intuitively understanding what this means in a machine learning context. When using the variables $A$ or $B$ or some trivial example, it all makes sense, but when looking at ...
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1answer
33 views

Deriving the mean of the binomial distribution

According to textbooks, the mean of a random variable $X\sim\displaystyle\operatorname{Bin}\left(n,\theta\right)$ that follows a binomial distribution is given by $\mathbb{E}\left[X\right]=\theta$. ...
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1answer
37 views

Compute the distribution function when $n=2$ and $n=3$

Two players $A$ and $B$ play a series of games that ends when one of them has won $n$ games. Suppose that each game played is, independently, won by player $A$ with probability $p$. Let $X$ be the ...
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1answer
17 views

a condition for a tightness of measure

$f: X\to \mathbb R$ be nontrivial continuous, given the fact that $\sup \int\limits_df\mu_n<\infty\forall n$, then could anyone tell me whether $\{\mu_n\}$ is a tight sequence of a probability ...
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0answers
32 views

Taylor's series in the binomial approximation to Poisson random variable.

The Poisson random variable has a probability distribution $$P_X(k) = e^{-(\lambda T)} \frac{{(\lambda T)}^k}{k!}$$ We can relate this expression to the binomial distribution by dividing the temporal ...
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0answers
23 views

A geometric interpretation of the conditional Variance

We know that we can see conditional expectation as a projection. To be more specific, let be $<X,Y> = E[XY]$ the usual inner product in $L^{2}$. We know that: $$E(Y|X) = \hbox{argmin}_{Z \in \...
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1answer
36 views

Find a consistent estimator for $E[X^2]$ when $X \sim \text{Exp}(\beta)$

I am working on this problem. Find a consistent estimator for $E[X^2]$ when $X \sim \text{Exp}(\beta)$ . So far I am thinking of using the invariant property of MLEs, so I let $$\hat{\theta} = \...
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0answers
22 views

How can we show that the $k$th order statistic is measurable?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(E,\le)$ be a partially ordered set and $\mathcal E$ be a $\sigma$-algebra on $E$ $n\in\mathbb N$ $X_1,\ldots,X_n:\Omega\to E$ be $(\...
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0answers
30 views

Can be a transition matrix of homogeneous Markov's Chain to be disjoint in t?

Can be a transition matrix of homogeneous Markov's Chain to be discontonuous of $t$, if $P(X_τ = j∣X_{t_n} = i_n, X_{t_n−1} = i_{n−1}, . . . , X_{t_1} = i_1) = P(X_τ = j∣X_{t_n} = i_n)$ $\forall t_1&...
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0answers
34 views

How to build normal probability table in excel

Question: We've looked at some other data and know that the time taken to complete a transaction (in minutes) follows a normal distribution. Can you help management a little bit by providing basic ...
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0answers
24 views

Is stochastic integral the same as Lebesgue integral of Banach-valued function?

I'm reading a book about stochastic processes and I've come to "stochastic integration." In this section stochastic processes are basically functions $$X:[0,\infty)\rightarrow L^2(\Omega, \mathbb{C})....
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1answer
30 views

Is a measurable space a set or a tuple?

According to Wiki: Consider a nonempty set $X$ and a $ \sigma$-algebra $ F$ on $ X$, then the tuple $(X,A)$ is called a measurable space. Also according to Wiki: A random variable $X$ is a ...
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1answer
31 views

Does $|X_n|\le\Delta_n+\delta$, with $\Delta_n\overset{p}\to0$ imply $|X_n|\overset{p}\to0$?

Suppose $\Delta_n\overset{p}\to0$, and for any $\delta>0$, we have $$|X_n|\le\Delta_n+\delta.$$ Can we conclude that $$|X_n|\overset{p}\to0?$$ Here $\Delta_n\overset{p}\to0$ means that for any $\...
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0answers
15 views

What does Markov Property Have to do with $P^{x}(X_{n}=x)P^{x}(X_{i}\neq x \operatorname{ for all} i \geq 1)$

I am aware of the Markov Property, i.e. that: $P(X_{n+1}=x_{n+1}|X_{n}=x_{n},...,X_{1}=x_{1})=P(X_{n+1}=x_{n+1}|X_{n}=x_{n})$ But I cannot seem to understand the following: Let $\sigma:=\{k \in \...
2
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1answer
23 views

Dominating function for derivative of moment generating function

Let $X$ be a random variable and the moment generating function $$\psi_X:(-\varepsilon,\varepsilon)\rightarrow \mathbb{R}_+,\quad \psi_X(t):=E[e^{tX}]$$ be defined, such that $\psi_X(t)<\infty$ ...