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 ...

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1answer
36 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 $D$ ...
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1answer
24 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 $D$ ...
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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
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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 ...
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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 ...
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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
33 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 ...
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1answer
66 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 ...
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1answer
46 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 ...
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1answer
39 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?
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1answer
54 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
41 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 ...
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0answers
19 views

What process does this SDE weakly converge to?

So my question is motivated by the following: Note that the ODE $$ dy_t = 2sgn(y_t)\sqrt{|y_t|}$$ $$y_0 = 0$$$$ has no unique solution. However, consider the SDE as follows: $$ dy_t = ...
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1answer
31 views

Let $X, Y, Z$ be iid. $N (0, 1)$. Find the joint MGF of $(X + 2Y, 3X + 4Z, 5Y + 6Z)$

Let $X, Y, Z$ be iid. $N (0, 1)$. Find the joint MGF of $(X + 2Y, 3X + 4Z, 5Y + 6Z)$ I ended up getting: $$M_X(s+3t)M_Y(2s+5p)M_Z(t+6p)$$ is this correct>?
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1answer
45 views

Conditional expectation w.r.t Lebesgue measure

Consider the probability space $(\Omega, \mathcal{F}, \mathbb{P})=((0,1)^{2},\mathcal{B}((0,1)^{2}),\lambda_{2})$, where $\lambda_{2}$ is the Lebesgue measure in $\Omega=(0,1)^{2}$. Then, for ...
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1answer
43 views

Is it true that $\sigma(X+Y) \subset \sigma(X,Y)$?

Suppose $X$ and $Y$ are two random variables. Is it true that $$\sigma(X+Y) \subset \sigma(X,Y)?$$ Intuitively I think above equation is correct, since if I know both $X$ and $Y$, I know $X+Y$. But ...
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1answer
73 views

Symmetric Random Walk / Prove $S = \inf\{n : X_n = 7\}$ and $T = 10^{12} \wedge S$ are $\{\mathscr F_n^Y\}$-stopping times.

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 = ...
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0answers
22 views

Poisson point process on a $\sigma$-finite space

Let $(X,H,\mu)$ be a $\sigma-$finite measure space. Show that the following construction gives a Poisson point process $V$ in $(X,H)$ with intensity $\mu$. Suppose that $0<\mu(X)<\infty$. Let ...
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1answer
62 views

Show that $E[X_T | T < \infty] \le E[X_0]$ and $cP(\sup X_n \ge c) \le E[X_0]$

From Probability with Martingales: I'm assuming the semi-colon means condition (Otherwise, why not say $T$ is a finite stopping time?). What I tried: $$X_T1_{T < \infty} = X_01_{T=0} + ...
3
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1answer
46 views

Balance equations looks the same in different cases for a problem

In a supermarket customers arrive at the cash desk with a Poisson process with an average of 30 customers per hour. There is one cash desk and the service time is exponential with an average of 2 ...
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1answer
42 views

ergodic theorem proof questions

I have questions concerning the proof of the ergodic theorem by Norris. http://www.statslab.cam.ac.uk/~james/Markov/s110.pdf Let $V_i(n)=\sum_{k=0}^{n-1}1_{\{X_k=i\}}$, $T_i^{(r+1)}:=\inf\{n\ge ...
2
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0answers
61 views

Integral equation: averaging

Let $X, Y, Z$ be Borel spaces, for simplicity we can assume that they are $\Bbb R$. Consider an equation $$ \int_{X\times Y}f(x,y,z) \kappa(x,\mathrm dy)\mu(\mathrm dx) = \int_{X\times Y}f(x,y,z) ...
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1answer
15 views

Probability function for group service time

I need help with the following problem: A batch of n items arrives at a service station where all items are being serviced simultaneously in parallel. The service time for each item is a stochastic ...
1
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1answer
33 views

How to calculate the convolution of constant function

Given that $f*g(x)=\int\limits_{-\infty}^\infty f(t)g(x-t)dt$. Calculate $f*f$ if $f(x)=\frac{1}{2}$ on $[-1,1]$ and $0$ elsewhere. So my initial thought is to calculate $\int\limits_{-1}^1 ...
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4answers
29 views

The problem of estimating a probability distribution from a closely timed sample?

I have a dataset, which is the hourly mean wind speed of each day(24 points each day), for 20 years. And I'm planning for using this dataset to estimate the ...
1
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1answer
44 views

Conditional expectation of joint probability conditional of sigma algebra

I was asked to proof the following property: Let X be $\mathcal{G}$ -measurable and let Y be independent of $\mathcal{G}$. Let $\mathcal{f}$ (x, y) be a bounded continuous function and defi ne ...
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1answer
42 views

Intuition of Probability Density Function.

So i know of many interpretations of the PDF of Random Variables. How does the notion of pdf's relate to the volume in the bivariate Random Variables case? Is the volume under the curve an ...
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2answers
82 views

Is $\{w:\lim_{n \rightarrow \infty}S_n/n \in A\}$ in tail $\sigma$-field?

Suppose $\{X_n\}$ are independent random variables on $(\Omega, \mathcal{F},P)$. Set $S_n:=\sum_{i=1}^{n}X_i$. Suppose $\frac{1}{n}S_n \rightarrow Y \;a.s.$ for some real-valued random variable $Y$. ...
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0answers
13 views

weak convergence of probability measures on a topological but non-metrizable space

Let $X$ be a topological space and let $\mathcal{B}$ be the Borel $\sigma$-algebra. Let $\Delta$ be the space of all (countably-additive) probability measures on $(X,\mathcal{B})$. Can I define on ...
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1answer
25 views

Convergence in distribution sequence max

Let $(X_n)_{n≥1}$ be a sequence of i.i.d. random variables with standard Cauchy distribution and let $M = \max\{X_1 ,...,X_n\}$. Prove that $(n M^{-1}_n)_{n \ge 1}$ converges in distribution and ...
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1answer
60 views

Expected number of rolls on two fair dice until sum is seven…?

They put me this question on a test: Two fair dice are rolled until the sum equals seven, or the dice are rolled twice. Let $X$ be the number of rolls in this experiment. a) Find the expectation of ...
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1answer
32 views

Total variation distance of two random vectors whose components are independent

Let $X^n=\left(X_1,\ldots,X_n \right)$ and $Y^n=\left(Y_1,\ldots,Y_n \right)$ be such that all $X_i$'s are independent and all $Y_i$'s are independent. I am trying to prove the following: $$ d_{TV} ...
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0answers
29 views

Question about confidence intervals

I have questions regarding the calculations and computations of confidence intervals basing upon the given accuracy. Here is the description from the textbook: The public health authority wants to ...
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1answer
37 views

Calculating $\mathbb{E}X^2$

When we could use the following equation: $$\mathbb{E}X^2=\int_0^\infty 2t \mathbb{P}(X>t)$$ I mean how is it possible to change $X^2$ to $2t$?
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2answers
78 views

$\max (X_t, a)$ is not a martingale, counterexample

It is quite easy to prove that if $[X_t \}_{t \in [0,T]}$ is a martingale, then for any number $a \in \mathbb{R}$ $ \{\max (X_t,a)\} _t$ is a submartingale and $ \{\min (X_t, a)\} _t$ but I cannot ...
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1answer
20 views

comparison of two probability measures

Let $P$ and $Q$ be two probability measures on some measurable space $(S,\Sigma)$. Assume that $\Sigma=\sigma(\Pi)$, where $\Pi$ is a pi-system. Then, I know that if $P=Q$ on $\Pi$, then $P=Q$ on ...
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3answers
34 views

Sigma algebra associated with sum of random variables

Let $Y_n$ be iid random varibles with $P(Y_n=-1)=p,$ $P(Y_n = 1) = q$, where $p,q>0$ and $p+q = 1$. Set $X_0 = 0$ and $X_n = \sum_{i=1}^n Y_i$. If I let $F_n = \sigma(Y_1,Y_2,\dots,Y_n)$, then ...
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0answers
21 views

Does there exist a closed form solution to this function, related to the moment generating function?

I'm wondering if similar to the multivariate moment generating function, there is a closed form solution to this function of normally distributed variables: ...
2
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0answers
73 views

Markov process and filtration

I would like to restate the question. I'm reading Revuz/Yor's definition of Markov process (P81), they started from transition function, and define the $P_t f(x)$ as usual (let's only consider the ...
2
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1answer
44 views

Probability exercise about SLLN

The question: Let {$X_n$} be i.i.d random variables. $EX_1=0$. Then $\sum_{i=1}^{n}X_i\over n$ converges almost surely to zero. I know that when the sequence $\{X_n\}$ satisfies ...
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2answers
27 views

Show that $Y\sim\Gamma(\frac{1}{2},\frac{1}{2})$

let $X\sim N(0,1)$ be a standard normally distributed stochastic variable and let $Y=X^2$. Show that $Y\sim\Gamma(\frac{1}{2},\frac{1}{2})$, ie. $Y\sim\chi^2(1)$.
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1answer
67 views

Use $\int_S f^{\pm} d\mu = \lim \int_S f_n^{\pm} d\mu$ to prove Scheffe's Lemma

Let $(S, \Sigma, \mu)$ be a measure/probability space. Scheffe's Lemma Part (ii): Suppose $\{f_n\}_{n \in \mathbb{N}}, f \in \mathscr{L}^1 (S, \Sigma, \mu)$ and $\lim_{n \to \infty} f_n(s) = f(s) ...
3
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1answer
92 views

How to apply law of large numbers for this problem

Let $X_1,...,X_n,...$ be independent variable satisfying $P(X_i=0)=P(X_i=1)=\frac{1}{2}$ for all i then denote $Z_i=X_iX_{i+1}$ for all i .I want to show that $\lim_{n\to ...
2
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1answer
107 views

Expected value for blackjack

In the game of blackjack, the odds of winning each hand are slightly less than 50 percent. As you play an infinite amount of hands, you would always lose money because you would win less than 50 ...
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0answers
39 views

The ratio of two asymptotically normal distribution

Let $(X,Y)$ be asymptotically normal with their means, variances, and a covariance. Then, I would like to show $X/Y$ is also asymptotically normal. I think there should be some references related to ...
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0answers
28 views

How to compute the possible outcome in a hypergeometric distribution

I have an urn that contains $n_0$ red balls and $n_1$ blues balls. Let $X$ be event randomly draw $r$ balls without replacement from that urn. What is the possible outcomes of event $X$, so that ...
2
votes
3answers
77 views

Probability of the Same Pair of Balls Drawn from Two Separate Urns

This morning, my friends and I discussed following problem. Problem: There are two persons named Mr. A and Mr. B. Each person has his own urn containing $N$ different balls. They uniformly randomly ...
1
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1answer
31 views

Computing random sum involving Poisson random variables

Let $N_t$ be a Poisson process. Let $Y_i$ be i.i.d. normal RV's with mean $m$ and variance $b^2$. Let $F_t$ be the sigma algebra of information acquired by observing $N_t$. Compute: ...