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-...

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1answer
36 views

Prove the uniqueness of the Markov distribution

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. I want to prove the uniqueness of a probability distribution under which a process $(X_n)_{n\geq0}$ is a Markov chain with transition ...
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1answer
29 views

Does convergence everywhere imply convergence in mean square?

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of real-valued square integrable random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ satisfying $\forall \omega \in \Omega, X_n(\...
3
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0answers
31 views

Convergence in distribution, in $L^p$ and convergence of first and second moments

For some application, we have the following three assumptions about a sequence of Random Variables $X_n$ (with values in $\mathbb{R}^+$, $n \geq 1$: There exists a $X \geq 0$ such that a) $X_n \...
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2answers
44 views

How to write $E(Y|X)$ as a function of $X$?

How to write $E(Y|X)$ as a function of $X$ ? If for example $X$ and $Y$ are discrete valued as below then $E(Y|X)$ should be equal to $-1+2X^2$ but I don't get it. If I write for example $\...
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0answers
31 views

Invariant measure and invariant distribution for markov chain

I am so lost between the measure and distribution! I do know the definition for each but I couldn't apply it for this question : Comment on the existence and uniqueness of invariant measure and of ...
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1answer
32 views

An example s.t. $E(E(X|\mathcal F_1)|\mathcal F_2)\neq E(E(X|\mathcal F_2)|\mathcal F_1)$

If $\Omega=\{a,b,c\}$ give an example that $E(E(X|\mathcal F_1)|\mathcal F_2)\neq E(E(X|\mathcal F_2)|\mathcal F_1)$ If I choose the $\sigma$-algebras disjoint (or better said: one is not fully ...
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0answers
17 views

Moment Generating Function of a Multivariate Random Variable

Let $X,Y$ be two i.i.d. random variables with a distribution given by $N(0,1)$. Set $U=X+Y,V=X^{2}+Y^{2}$. Find the moment generating function of $(U,V)$ and $\rho_{(U,V)}$ (the correlation factor) ...
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1answer
32 views

Almost sure convergence of sum implies bounded sumands a.s./Proof of Kolmogorov's continuity theorem

Yesterday I asked a question about Kallenberg's proof of Kolmogorov's continuity theorem for stochastic processes. I understood that part of the proof, but now i face another problem: After showing ...
2
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0answers
30 views

Convergence of Sample Mean Via WLLN

I am trying to show that the sample variance converges to the population variance in using the Weak Law of Large Numbers $$\begin{align} \\ \Rightarrow S_n= \frac{1}{n} \sum_{i=1}^n (X_i-\bar{X})^2 &...
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0answers
34 views

Variance Bound for a Random Variable

Let $X$ b a random variable such that $P(a \leq x \leq b)=1$ for $-\infty<a<b<\infty$. Show that: $Var(X) \leq \displaystyle\frac{(b-a)^2}{4}$ I'm not sure on how to get started with the ...
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2answers
53 views
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1answer
24 views

Prove that if $X$ and $Y$ are i.i.d. then $\phi_{X-Y}(t)=|\phi_{X}(t)|^2$.

Prove that if $X$ and $Y$ are i.i.d. then $\phi_{X-Y}(t)=|\phi_{X}(t)|^2$. So as $X$ and $Y$ are i.i.d. this implies that $\phi_X(t)=\phi_Y(t)$. Also, $$\phi_{X-Y}(t)=E[e^{it(X-Y)}]=E[e^{itX}e^{it(-Y)}...
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0answers
53 views

Suppose that $Y_{\lambda}=^{d}P(\lambda)$. Prove that $[Y_{\lambda}-\lambda]/{\sqrt{\lambda}}\to^{d}N(0,1)$

Suppose that $Y_{\lambda}=^{d}P(\lambda)$. Prove that $[Y_{\lambda}-\lambda]/{\sqrt{\lambda}}\to^{d}N(0,1)$ when $\lambda \to \infty$ using characteristic functions. So $$\phi(t)=\sum_{k=0}^{\infty} \...
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0answers
42 views

Law of large numbers for continuous periodic functions

Suppose $f$ is a continuous function on $\mathbb{R},$ with period 1. Prove that $$ \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N f\left(n \theta\right)=\int_{0}^1 f(t) \ dt $$ for every ...
1
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1answer
16 views

Transformation of a random variable bijectivity

How does one determine the density of a transformed random variable? I understand the theorem, but I have a really hard time applying it. For example, if the transformation is $t(X) = X(1-X) = X - X^...
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0answers
20 views

How to show this set $\mathfrak{B}=\left \{ A \in \mathfrak{A}: \int f \mathbb{1}_Ad\mu=\int g \mathbb{1}_Ad\mu \right \}$ is Dynkin-System?

Let $\Omega$ be a set, $\mathfrak{A}$ a $\sigma$-Algebra and $\mu$ a measure of ($\Omega$,$\mathfrak{A}$). Then $f,g : \Omega \to \overline{\mathbb{R}}$ and $\int f d\mu=\int g d\mu$. $$\mathfrak{B}=\...
0
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1answer
32 views

Uniform integrability of reciprocal of random variables

Let $\{X_n, n\geq 0 \}$ be a sequence of positive random variables that are uniformly integrable. Assume that $\frac{1}{X_n}$ is integrable. Then, is it true that $\left\{\frac{1}{X_n}, n \geq 0\...
2
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2answers
83 views

Maximum of martingales

I need to show whether or not the maximum of two martingales is also a martingale. Originally, I thought yes. But supposedly the answer is no. So as a counter-example, let $U_i$ be $iid$ $unif(0,1)$, $...
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0answers
28 views

Prove M\M\1 queue is in fact CTMC

As i understand the number of customers in the M\M\1 queue is expressed as $N(t)=A(t)-\int_0^tI_{\{N(s-1)\geq1\}}dD(s)$ Where A(t), D(t) are independent Poisson processes with rates $\lambda$ and $\...
2
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1answer
68 views

Verify that $L = \lambda W$, $M/M/2$

I have a $M/M/2$ system, with traffic intensity $\rho = \frac{\lambda}{2\mu}$. I will call $\boldsymbol \pi$ the stationary distribution. Then, I have formulas for $\pi_0$ and $\pi_k$, and I also have ...
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1answer
26 views

Random Sampling and Measurability.

In a probability space $\left(\Omega,(\mathcal{F}_t)_{t=0,..,T},\mathcal{F},\mathbb{P}\right)$ let $\tau$ be a stopping time. Consider the definition of "stopped" filtration as $$ \mathcal{F}_{\tau} ...
2
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1answer
48 views

Convergence in distribution and characteristic function

Convergence in distribution ( Two equivalent definitions) I am referring to the above problem. There was one thing in the proof that I was not clear. Once we have the $\textbf{existence of a}$ ...
2
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1answer
23 views

Let $X$ ~ Geometric with $p=\frac{1} {4}$ and $Y$ ~ Uniform on $(1,2,3,4)$.

Given X and Y are independent, I need to find $VAR(2X + 2Y)$. Does independence mean that $VAR(2X + 2Y) = VAR(2X) + VAR(2Y)$? If so, then $VAR(X)=\dfrac{1-p} {p^2} = \dfrac{\dfrac {3} {4}} {\dfrac{1} {...
2
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1answer
26 views

If $\{A_n,n\geq 1\}$ are independent events show that $\frac{1}{n} \sum_{i=1}^{n} 1_{A_i}-\frac{1}{n}\sum_{i=1}^{n}P(A_i) \rightarrow^{P} 0$

If $\{A_n,n\geq 1\}$ are independent events show that $$\frac{1}{n} \sum_{i=1}^{n} 1_{A_i}-\frac{1}{n}\sum_{i=1}^{n}P(A_i) \rightarrow^{P} 0$$. Proof so far $P(|\frac{1}{n} \sum_{i=1}^{n} 1_{A_i}-\...
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1answer
90 views

Is it obvious that this integral converges given the following assumptions?

The integral is $\int\limits_{p(x) > 0}p^{-\lambda + 1}(x) \, \left| \ln p(x) \right|^k \, dx$. Assumptions: $\lambda > 0, k > 0$ $\int\limits_{p(x) > 0}p^{-\lambda + 1}(x) \, dx < \...
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0answers
40 views

Martingale Strong Law of Large Numbers

Consider a Probability Space $(\Omega,\mathcal{F},P)$ Let $\{X_n\}$ be a Sequence of Random variables such that $X_n \in L^2 \; \; \forall n$ and $$\sum_{j=1}^{\infty}\frac{E[X_j^2]}{j^2} < \infty ...
0
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0answers
14 views

replicating a probability model for classification

I'm trying to replicate a probability model for a binary classification based on some data. Until now I been able to replicate the left panel, the histogram and the gaussians curves, but I dont know ...
1
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1answer
27 views

Suppose $\{X_n, n > 1\}$ are independent non-negative random variables satis­fying $E(X_n)= \mu_n \> Var(X_n)=\sigma^2$ [closed]

Suppose $\{X_n, n > 1\}$ are independent non-negative random variables satis­fying $$E(X_n)= \mu_n \> Var(X_n)=\sigma_n^2$$ Define for $n \geq 1$, $S_n = \sum_{i=1}^{n} X_i$ and suppose $\sum_{n=...
2
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1answer
54 views

True or false: The process $\{ X(t), t \geq 0 \}$ at a $M/M/s$ queue is a reversible Markov process.

Let $X(t)$ denote the number of customers in a system at time $t$. The process $\{ X(t), t \geq 0 \}$ at a $M/M/s$ queue is a reversible Markov process. Is this statement true or false for: (a) $...
1
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1answer
34 views

Sequence of iid $U(0,\theta)$ random variables

$(X_n)_{n=1}^\infty$ iid $U(0,\theta)$, $X_{(1)} = \min{\{X_1, \dots, X_n}\}$. Consider the sequence $Y_n = nX_{(1)}$. Does $Y_n$ converge in distribution to some r.v. $Y$? This is not homework, just ...
1
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1answer
132 views

Asymmetric Random Walk / Prove that $E[T:= \inf\{n: X_n = b\}] < \infty$

Given random variables $Y_1, Y_2, \ldots \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \...
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1answer
67 views

Asymmetric Random Walk / Prove that $T:= \inf\{n: X_n = b\}$ is a $\{\mathscr F_n\}_{n \in \mathbb N}$-stopping time

Given random variables $Y_1, Y_2, ... \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \...
1
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1answer
24 views

product of Bernoulli and Categorical distribution

I have random variable, which is the product of two random variables, derived such that. $Z = X_i*Y$, where $X_i\sim Ber(p_i)$ and $Y \sim Categorical(i,\frac{1}{n}) $, here $n$ is the number ...
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0answers
20 views

Conditional expectation, show two random variables are equal almost surely. [duplicate]

$ \newcommand{ \F }{ \mathcal{F} } \newcommand{ \prob }{ \mathbb{P} } \newcommand{ \G }{ \mathcal{G} } \newcommand{ \L }{ \mathcal{L} } \newcommand{ \exp }[1]{ \mathbb{E}\left[ #1 \right] } \...
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2answers
137 views

What is wrong with this answer to: expected time of return to origin in random walk on edges of a cube

(Quant Job interviews Questions and Answers Q3.22) Suppose we have an ant travelling on edges of a cube going from one vertex to the other. The ant never stops and it takes it one minute to go along ...
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0answers
112 views

Proof of strong Markov Property of double sided Levy Process

Let $(X_t)_{t\in\mathbb{R}}$ be a Levy Process, i.e. $X_0 = 0$ a.s., $X$ has independent and stationary increments, and almost all paths $t\mapsto X_t(\omega)$ are right continuous with left hand ...
1
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1answer
37 views

If I have that $\sqrt{n}(X_n-\mu) \overset{D}\to \mathcal{N}(0,\sigma^2)$, how to show $X_n \overset{D}\to \mathcal{N}(\mu, \frac{\sigma^2}{n})$?

If I have that $\sqrt{n}(X_n-\mu) \overset{D}\to \mathcal{N}(0,\sigma^2)$, how can I show via a transformation that $X_n \overset{D}\to \mathcal{N}(\mu, \frac{\sigma^2}{n})$? Here I have that the $D$ ...
0
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1answer
25 views

Symmetric Square Root of a Matrix

I have to prove the following Theorem: $X_1$, $X_2$, ... $X_n$ are Random Variables with the density function: $f_X(x) = k \times exp\{-\frac{1}{2}(x-\mu)^TD(x-\mu)\}$ where $\mu \in R^n$ and $D$ ...
1
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2answers
51 views

Conditional Probabilities and Independence

I have a simple question about conditional probabilities and independence, suppose that $X, Y$ are independent random variables while as well $N$ is a random variable (which both $X$ and $Y$ are ...
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0answers
32 views

What is $E[X|3X+4Y]$ for $X$ and $Y$ i.i.d $\mathcal{N}(0,1)$?

$X$ and $Y$ are two random variables drawn independently from standard normal distribution $\mathcal{N}(0,1)$. Set $Z:=3X+4Y$. What is the conditional expectation of $X$ given $Z$, i.e., $E[X|Z]?$ By ...
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0answers
36 views

Most recent jump probability

Assume I have two independent Poisson processes with respective parameters$$ \sim\text{Poisson}(\alpha_1),\sim \text{Poisson}(\alpha_2)$$ that I observe over a time interval $[0,t].$ What is the ...
2
votes
2answers
51 views

Hidden Markov Model and Viterbi algorithm: Understanding the Casino Problem?

I am deeply struggling with understanding how to apply the Viterbi algorithm. From my course notes, I have the following simple(I'm told) example: If the sequence ...
0
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1answer
17 views

expectation of distribution where arguments are distributions too

I want to find an expectation and variance of c, where $ c | a, b \sim Poiss(0.5 a + 0.05 b) $ $ a \sim R(15, 30) $ $ b \sim R(250, 350) $ Where ...
0
votes
1answer
17 views

Measurability of a function related to Skorohod space

Let $(E,r)$ be a metric space, $D_E[0,\infty)$ be the Skorohod space on $[0,\infty)$ takes value in $E$. Consider the function $$(D_E[0,\infty)\times[0,\infty),\mathcal B(D_E[0,\infty)\times \mathcal ...
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0answers
37 views

Is exponential of GUE random matrix Haar random?

Consider the matrix exponential map $H \mapsto e^{i H t}$ acting on the Gaussian unitary ensemble (GUE) of Hermitian matrices. I would expect that for large $t$, the resulting measure on the unitary ...
0
votes
1answer
70 views

Asymmetric Random Walk / Prove $E[T] = \frac{b}{p-q}$ / How do I use hint?

Given random variables $Y_1, Y_2, \ldots \stackrel{\mathrm{iid}}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}...
0
votes
1answer
50 views

Asymmetric Random Walk / Prove $E[X_{T \wedge n}] = (p-q)E[T \wedge n]$

Given random variables $Y_1, Y_2, ... \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \...
2
votes
1answer
44 views

Characteristic function using conjugate property

To prove that $e^{−i|x|}$ is not a characteristic function: $$e^{−i|x|} =\cos|x|-i \sin|x|.$$ Its conjugate will be $\cos|x|+i \sin|x|$ which is not equal to $\phi(-x)$. Is my solution correct?
0
votes
1answer
56 views

Symmetric Random Walk / Find $E[X_S]$ and $E[X_T]$

Given a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $\mathscr F_n = \mathscr F_n^Y$, let $Y_1, Y_2, ...$ be iid random variables w/ $P(Y_n = ...
1
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1answer
42 views

Limiting distribution of a sequence of random variables

$X_1$,...$X_n$,... are iid random variables with mean being $1$ and variance being $1$. Let $S_n=\sum_{i=1}^{n}X_i$. Let $\Phi(\cdot)$ be the cdf of standard normal distribution. What is the limiting ...