Tagged Questions

Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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1answer
13 views

Probability of non repeated value in a set of vectors (with integer values) for any number in the same vector position.

Suppose a set with $m$ vectors ($m$ finite) defined by $V_{i} = (x_{vi1},x_{vi2},\dots,x_{vin})$, with $i \in \left\{1, 2, \dots, m \right\}$ and $2 \leq n \leq p$, for a given $p \in \mathbb{Z}$ ...
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2answers
33 views

Optimal Number of White Balls

There are C containers, B black balls and infinite number of white balls. Each container should have at least one ball. Each of the containers may contain any number of black and white balls. Action ...
1
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1answer
30 views

Paley Wiener stochastic integral

Sorry for the stupid question, no answers necessary anymore! let $(B_t)_{t\in [0,1]}$ be a standard Brownian motion and $F\in C[0,1]$ differentiable. Then the sequence (which is an easy version of ...
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1answer
24 views

Bounding the difference of random variables by coupling.

Suppose we have two probability densities differing by atmost $\delta$. Is it possible to use coupling to have two random variables with the above two densities differing by less than $\delta$? I ...
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0answers
37 views

$\tau$ is stopping time. Check if $\tau + 1$, $\tau - 1$, $\tau^2$ also are stopping time.

Suppose that $\tau$ is stopping time. Is is true that a) $\tau + 1$ b) $ \tau - 1 $ c) $ \tau^2 $ also are stopping time? My prove: a) Yes, because forall t we have $$\{ t: \quad \tau+1 \le t \} ...
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2answers
19 views

Inner dependence of Independent random vectors

If $X = (X_1,X_2)$ and $Y = (Y_1,Y_2)$ , $X$ and $Y$ are stochastically independent can $X_1$ and $Y_1$ be dependent?
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0answers
22 views

Find $E[Z_1 | aZ_1 + bZ_2]$

Let's $Z_1,Z_2$ be a random variable such that $EZ_1^2 < \infty$ and $EZ_2^2 < \infty$. Find $E[Z_1 | aZ_1 + bZ_2]$ where $a,b \in \mathbb{R}$. We don't know what is distribution of $Z_1$ and ...
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0answers
16 views

expectation approximation error

Let $X$ be a random variable with no mass taking values in $\mathbb{R}$, and $f:\mathbb{R}\mapsto\mathbb{R}$ be a "smooth" function. I want to approximate $\mathbb{E}[f(X)]$ with $\mathbb{E}[g(X)]$ ...
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1answer
22 views

A Question on the Scaling Invariance of Brownian Motion

I read the following paragraph. Let $B_t, \ t \in [0, \infty)$ be a standard linear Brownian motion. For each $q > 4$, define the following sequence of sets. $$ \Omega_k := \left\{\omega \in ...
2
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0answers
30 views

how to related a weakly convergent random variable with its k-th moment

Let $\{X_n\}$ be independent random sequence with zero mean and unit variance. Suppose $$S_n:=\sum_{m=1}^n \frac{X_m}{\sqrt{n}} \Rightarrow X\sim \mathcal{N}(0,1)$$ holds. (Here "$\Rightarrow$" ...
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1answer
25 views

Simple bounding question for an expectation with truncating function

Let $\{X_m\}$ be independent random sequence. I want to show the following result Given $E[X_m^2]:=\sigma^2 < \infty$ and $$0 = \mathop {\sup }\limits_m P\left( {\left| {{X_m}} \right| > ...
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1answer
36 views

How to show this inequality in the law of iterated logarithm

Let $\psi(t) = \sqrt{2t\log\log t}$ and $q>4$. Then we have for large $k$, $$ \psi\left(q^k - q^{k-1}\right) \geq \psi\left(q^k\right) \left(1-\frac{1}{q}\right). $$ To prove this, I can tell by ...
1
vote
1answer
62 views

Requesting deeper understanding of binomial coefficient

I noticed that $\binom {52} 4$ * $\binom {48} 1$ is $5$ times that of $\binom {52} 5$. So for example, if we were to draw $4$ cards from a standard deck then draw $1$ more, we cannot just say there ...
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0answers
17 views

Probability distribtuions [closed]

A 10 metre by 10 metre plot of land is divided into 100 equally sized squares. Suppose that 300 seeds are randomly scattered on the plot of land. Use a suitable approximation to find the probability ...
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1answer
43 views

$X$ normally distributed in $\mathbb R^n$ iff components $x_i$ normally distributed?

We've had the normal distribution today in class and I was thinking about the following: Let $X$ be normally distributed, $X\sim N(a,\Sigma)$ with a symmetric positive definite matrix $\Sigma$ and ...
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0answers
25 views

Cameron Martin Theorem

I am struggling with two versions of the Cameron Martin Theorem. 1) We define the measure spaces $(\Omega,\mathcal{F},P)$ and $(C[0,1],\mathcal{C},\mathbb{L}_0)$, where $\mathcal{C}:=\sigma(f\mapsto ...
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0answers
17 views

Function of Nakagami Distribution

Does anyone know what the distribution of the sum of squared Nakagami is? $$\sum_i^n X_i^2$$ $$X_i\sim \text{Complex Nakagami-m }$$ Is the distribution Erlang? Is the distribution the same as ...
1
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1answer
9 views

How to interpret this variance

If I have a probability measure defined by $P( \Omega ) = \int_{\Omega} (1-a) \delta(x) + a \delta(x-a^2) dx,$ then I noticed that the variance is given by $a^5(1-a)$. This is somewhat strange, cause ...
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0answers
19 views

What is the magnitude of Complex random variable Gaussian Case?

Let $X_1$ and $X_2$ be independent complex Gaussian random variables, $$X_1 \sim \mathcal{CN}(0,\sigma)$$ $$X_2 \sim \mathcal{CN}(0,\sigma)$$ If $X= aX_1 + bX_2$ where $a,b$ are constants then the ...
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2answers
13 views

Conditional probability: $P(B'|A) = 1-P(B|A)$

Suppose that $A$ and $B$ are events with $P(A) > 0$. Show that $$P(B'|A) = 1-P(B|A),$$ where $B'$ is the complement of $B$. I get stuck after I go from $P(B'|A)$ to $P(AB')/P(A)$. I would greatly ...
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0answers
39 views

Law of iterated logarithm proof

I am trying to master this proof of iterated logarithm. However, I get stuck at the last part. Here is a link In the last two line at fourth page. We calculate the probability that: $$ ...
2
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0answers
33 views

Analyzing a coin tossing game with cheating

Consider a game where you toss $N$ coins, and let $H$ denote the number of heads. Let's say you win the game if $|H - N/2| \geq K$, i.e. if the number of heads deviates east least $K$ from what you ...
3
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1answer
132 views
+50

A dyadic decomposition of a random variable

Let $X$ be a real-valued random variable with mean equal to zero. We consider $n$ identical copies $X_1,\ldots,X_n$ of $X$ and denote with $S_n=X_1+\cdots+X_n$ the sum of them. We decompose the ...
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0answers
13 views

average distance between vectors of n dimenstions

In a recent group of experiments I did, I performed clustering on vectors with about 10k dimensions, where the individual values were drawn from a standard normal distribution. Then, for each ...
0
votes
2answers
35 views

For a non-negative absolutely continuous random variable $X$, with distribution $F$. Why is $\lim_{t\rightarrow \infty}t(1-F(t))=0$?

So I am given a non-negative absolutely continuous random variable $X$ with distribution $F$, and density $p_X$. I am given the definition of expectation using simple functions and the survival ...
1
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1answer
31 views

How to calculate $\mathbb{P}[Y\in F|X]_{\omega}$

Here I have an exercise of book: Probability and Measure of PATRICK BILLINGSLEY of conditional probability in the page 442, exercice 33.4 (b): Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability ...
1
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2answers
37 views

Why is the expected value of $|X|^p$ equal to $p\int_{0}^{\infty}y^{p-1}\mathbb{P}(|X|>y) dy$?

I'm trying to understand a passage from the book: A Basic Course in Probability Theory, Rabi Bhattacharya Edward C. Waymire, in the page 21. The calculation is the following: If $X$ is a random ...
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0answers
27 views

Sigma algebra generated by a random vector

I understand this question is very basic, but I found this confusing while I am learning measure theory myself.. Suppose we toss a coin twice (once afeter once), and denote by each $X$ and $Y$ the ...
2
votes
1answer
68 views
+50

Proving (a part of) Hoeffding's lemma

Hoeffding lemma goes like this: *Let $X$ be a scalar variable taking values in an interval $[a,b]$. Then for any $t>0$ $$\mathbb{E} e^{tX}\leq e^{t\mathbb{E} ...
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0answers
22 views

Weak convergence, measurability, uniform integrability

I've been facing the following problems: a) Let $ X_n \rightarrow X $ and $f $ be a measurable, bounded function. Prove that $ \mathbb{E}f(X_n) \rightarrow \mathbb{E}f(X) $ (we also assume that the ...
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0answers
18 views

Proof that, if $X_n\rightarrow X $ weakly and $\mathcal u_x(D)=0$, then $\mathbb Ef(X_n) \rightarrow \mathbb Ef(X)$

Proof that, if $X_n\rightarrow X $ weakly and $\mathcal u_x(D)=0$, then $\mathbb Ef(X_n) \rightarrow \mathbb Ef(X)$ $D$ is a set of discontinuous points X and $f$ is bounded, measurable. We can ...
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0answers
15 views

Hitting time of two dimensional continuous martingale

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...
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2answers
31 views

theory of probability question [closed]

There is a lottery. From 10,000 people only 100 win, so the probability to win is 1%. Question: what is the probabily to win if you join/buy ticket to the same lottery 100 times? I am sure you can ...
3
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2answers
39 views

Probability Assignment to Intervals in $\mathbb{R}^{n}$.

Given a random variable $\bf{X}$ distributed on $\mathbb{R}^{n}$, let $F_{X}(t)$ be its distribution function. Suppose we want to find $P\left(\textbf{X} \in (\textbf{a}, \textbf{b}]\right)$. I was ...
1
vote
1answer
21 views

Is an infinite product probability space compatible with an almost surely statement?

My question pertains to the following Theorem which can be found here. Theorem (Existence of product measures): Let $A$ be an arbitrary set and for each $\alpha \in A$ let ...
2
votes
2answers
36 views

Relationship “finite mean” <-> "absolutely integrable

What is the relationship between the property of a random variable (i.e. a measurable function defined on some probability space) being absolutely integrable, i.e. $$\mathbb{E}|X|<\infty$$and ...
0
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0answers
44 views

If $X_n$ are independent random variables, then does $\sum_n \mathbb{E}(X_n)=\mathbb{E}(\sum_n X_n)$? [duplicate]

If $X_n$ are independent random variables, then does $\sum_n \mathbb{E}(X_n)=\mathbb{E}(\sum_n X_n)$? This is not a homework problem but rather a question I had. If it is not true, what are the ...
1
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0answers
33 views

Application of Law of Iterated Expectations

Could you please explain something from a text I am reading? We're given that $E(\epsilon_i)=0$ and $E(\epsilon_i x_i)=0$ for $i\geq 1$. Write $g_i=\epsilon_i x_i$ and we further know that $$ ...
2
votes
1answer
46 views

Simple Question about Almost Sure Convergence

If I can show for some event $A_n$, $$ \sum_n P(A_n) < \infty. $$ Then by First Borel-Cantelli Lemma, I get $P(A_n \; i.o.) = 0$; But I still confused about how this infinitely often connects to ...
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0answers
17 views

Conditional expectation, conditioning on a random variable [duplicate]

I am asked to show that if $X,Y$ are integrable, and $E[X| Y]=Y$ and $E[Y|X]=X$ almost surely, then $X=Y$ almost surely. Moreover, is the first equality sufficient for $X=Y$ almost surely? My attempt ...
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0answers
13 views

Predictive analysis based on history

Let me first say that I am a CS person and my knowledge about statistics is quite basic. I am trying to see what predictive analysis to use for a problem I am trying to solve. I will try to make my ...
1
vote
1answer
37 views

Weak convergence of random variables implies $\mathbb E | X| \le \liminf_n \mathbb E|X_n|$

Proof that, if $X_n \rightarrow X$ weakly, then $\mathbb E | X| \le \liminf_n \mathbb E|X_n|$. I know, that I should use Fatou's lemma but I don't know what can I do first.
1
vote
1answer
38 views

Convergence in probability realated question

Consider $X_n$ and $Y_n$ be two real-valued random sequences, if $$P(X_n \neq Y_n) \rightarrow 0 \text{ as $n \rightarrow \infty$}$$ is it equivalent to say that $X_n$ converges to $Y_n$ in ...
1
vote
0answers
22 views

Easy question: Multiple random variables vs. product of probability spaces

I never had a course in probability theory and the definitions we work with are quite informal, so I am a little bit confused about the difference between "multiple" random variables and the notion of ...
2
votes
0answers
62 views

Measure-preserving maps on probability spaces

A professor posed me a problem a few days ago, and I have not been able to find an answer to it. Let $(X_1,\Sigma_1,\mu_1)$ and $(X_2,\Sigma_2,\mu_2)$ be probability spaces. Suppose the following ...
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0answers
18 views

Expected Probability of a Random Agent and a Probabilistic Agent

I'm running simulations on two agents: random agent and probabilistic agent. The world they are running in is the Wumpus World where the agent is dropped in a 4x4 grid where each cell has a 20% chance ...
0
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0answers
26 views

Kolmogorov zero-one law in continuous time?

Let $(X_t : t \geq 0)$ be a stochastic process. Is it necessarily the case that $$P (\limsup_{t \geq 0} X_t \leq a) \in \{ 0,1\}$$ as it is in discrete time? If some conditions are needed on the ...
0
votes
1answer
9 views

Simple linear regression for predictive purposes

Relationship between X and Y, the first and second year batting averages of a random baseball player is expressed as simple linear regression Y=0.159 + 0.4X + e with e ~ N(0,variance) If a player's ...
0
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0answers
15 views

Hermite polynomials with non standard variance (not equal to one)

It is known for probabilists that if $p(x)=\frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$ then the derivatives of $p$ can be expressed in terms of the Hermite polynomials as follows: $$\frac{d^n}{dx^n} ...
-1
votes
1answer
26 views

How to derive this inequality

I learnt that for a standard normal random varialbe $Z$ and positive $x$, we have $$\mathbb P (Z > x) \geq \frac{x}{x^2+1} \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} = \frac{1}{x+\frac{1}{x}} ...