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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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 ...
2
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1answer
32 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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1answer
17 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 ...
0
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1answer
247 views

Probability - rolling a fair die 10 times, what is the probability you would match a separate set of 10 numbers?

Having some trouble with this problem... Say someone is rolling a fair die 10 times, and using that roll as an attempt to guess what number (1-6) someone else has written down on a piece of paper for ...
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2answers
24 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 ...
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0answers
41 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 ...
4
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1answer
102 views

Could you please help me understand the discrepancy metric?

I am trying to understand the discrepancy metric and its properties. It is defined as $$d_D(\mu,\nu):=\sup_{\small \mbox{ all closed balls}\,\, B}|\mu(B)-\nu(B)|$$ for probability measures $\mu$ and ...
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0answers
28 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 $$ ...
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1answer
25 views

Bayesian network and unknown probability

I'm trying to solve questions regarding bayesian network, and now I was wondering if it is possible to know the probability of an unknown variable in the tree. For instance, I have this tree, ...
5
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2answers
316 views

Existence of iid random variables

In probability theory we often used the existence of a sequence $(X_n)_n$ of independent and identically distributed random variables. This was already discussed here. One of the answers says: As ...
1
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1answer
25 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.
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12 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
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1answer
33 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 ...
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0answers
12 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
18 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 ...
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0answers
15 views

Measurability of the composition of sections and probability measures

Let $X,Y,Z$ be Polish spaces endowed with their respective Borel sigma algebras, and $\Delta(Y),\Delta(Z)$ be sets of Borel probability measures on $Y$ and $Z$, endowed with the weak*-topologies (and ...
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0answers
49 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
12 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 ...
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0answers
31 views

A probabilty of error calculation

Let's assume I have $N$ binary strings $\{T_1,T_2,\ldots,T_N\}$ of length $L$. All these strings satisfy a minimum hamming distance with respect to a reference binary string R with $\|R\|_1$ ones and ...
8
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1answer
829 views

What distinguishes the Measure Theory and Probability Theory?

It is clear that the Theory of Probability works primarily with limited measures on measurable spaces. On the other hand there is a folklore that says that what distinguishes Measure Theory and ...
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21 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 ...
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1answer
8 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 ...
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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} ...
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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}} ...
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1answer
19 views

Equivalent definition of random variables

I've come across the following two definitions of random variables and am trying to figure out if they are equivalent or not. Let $\Omega$ denote our sample space and $\mathscr{F}$ denote our ...
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2answers
21 views

What is the expected value of the mean of the highest $m$ numbers in a population of $N$ normally distributed random variables?

Suppose that I randomly generate $N$ numbers according to the standard normal distribution, $\mathcal{N}(0,1)$. Then suppose I pick the highest $m$ numbers, $x_1\leq x_2 \leq \cdots \leq x_m$. ...
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0answers
16 views

Please check this about Geometric brownian motion

Given $T>0$ and $L \in \mathbb{N}$, let $\delta t = T/L$, $t_i = i \delta t$. Consider the GBM given by $$S(t_{i+1})-S(t_{i}) = \mu S(t_i) \delta t + \sigma S(t_i) \sqrt{\delta t} Z_{i}$$ where ...
0
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1answer
10 views

tail limit of Laplace transform of a bounded random variable

Suppose that $X$ is a variable such that $0<X<m$. I would like to know some information on the behavior of the function $$\phi(p)=\frac{1}p \log E e^{pX} $$ when $p\to\infty$. Here are some ...
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1answer
10 views

Limiting random variable

My question is about limiting r.v. Suppose, we have a sequence of r.v.s. $\{X_n\}$. And we know that $\liminf X_n=-\infty$ and $\limsup X_n =\infty$ almost surely. What can we say about $\lim X_n$. ...
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1answer
54 views

$X_n$ are r.v.s, is it true that $E[\sum_{n=1}^{\infty} X_n] = \sum_{n=1}^{\infty} E[X_n] $? [duplicate]

$X_n$ are r.v.s, is it true that $E[\sum_{n=1}^{\infty} X_n] = \sum_{n=1}^{\infty} E[X_n] $? My feeling is that this is not necessarily true. But cannot come up with an example. Can someone provide ...
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1answer
264 views

finite additivity condition

The definition of a probability measure $ P $ requires countable aditivity: $ P \left( \bigcup_{n = 1}^\infty A_n\right) = \sum_{n = 1}^\infty P (A_n)$ whenever $ A_1, A_2, \ldots $ is a sequence of ...
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0answers
10 views

Finding the distribution of $\bar{Y}_{1+} - \bar{Y}_{2+}$ in randomized block design

Please help me, I've been stuck on this problem for so long! I'm trying to find the distribution of the following but I am having quite a hard time as Mathematics/Statistics has always been difficult ...
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1answer
25 views

Question in permutations

When we use this law? And in any case we use it? Thank you and I wish clarification.
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1answer
34 views

A question on independence of increments

How could I prove the following? Let $X=(X_t)_{t \in[0,1]}$ be a real-valued stochastic process on a probability space $(\Omega,F,P)$ with $X_0=0$ a.s Show that the following statement as are ...
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0answers
23 views

Probability of James Choosing Earlier Train Than Bill

James can choose to catch either train 1 or train 2. But Bill can choose to catch either Train 1, 2, 3 or 4. Both James and Bill choose their train at random. What is the probability that James ...
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1answer
254 views

Finding MLE of $f(x;\theta) =1$ if $\theta-1/2<x< \theta+1/2$

Let $X_1,...,X_n$ have density: $f(x;\theta) = \begin{cases} 1 &\mbox{if } \theta-1/2<x< \theta+1/2 \\ 0 & otherwise \end{cases}$ Let $Y_1=min \lbrace X_1,...,X_n \rbrace$ and ...
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16 views

Density function of $Y $given that $ Y = 2X, X \leq 2, Y = X^2, X > 2.$

So we have $X$ with density $$f_X(x) = 1/x^2, x \geq 1, f_X(x) = 0, x < 1.$$ And $$Y = 2X, X \leq 2, Y = X^2, x > 2.$$ So I drew my graphs of $f_X$ and $Y$, but where the functions change is ...
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36 views

Integral with respect to brownian motion

Let $f$ be a continuous function on $[0,∞)$ and $B_t$ be a standard Brownian motion. Define $Xt=\int_0^t f(s) dB(s).$ a) Show that $X_t$ is Gaussian and computer its covariance $C(X_s, X_t)$ b) ...
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2answers
34 views

Dice: Probability of rolling a number between two other dice throws

I was pretty suprised about this problem when I encountered it in one of my excercise sheets and would like to ask for an approach here because I have no idea how I'm supposed to get started here: ...
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2answers
67 views

“Back to square one” problem

There's a problem I've been stuck on in preparation for junior programming contest I'm going to participate in. It is as follows: The "back to square one" problem is played on a board that has ...
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0answers
25 views

Measure Preserving tranformation of the space of brownian paths

Let $O$ be an orthogonal transformation of $L2_{[0,\infty)}$. Let $1_{[0,1]}$ be the indicator function for $0 \leq s \leq t$. Also let $B(t)$ be a standard brownian motion. Define $W(t) = ...
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1answer
79 views
+50

Conditions on Poisson random variables to convergence in probability

Let $X_1,X_2,...$ denote iid random variables such that $X_j$ has a Poisson distribution with mean $\lambda t_j$ where $\lambda$ > 0 and $t_1, t_2,...$are known positive constants. a)Find conditions ...
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1answer
33 views

If X and Y are independent, then $\sigma(X)$ and $\sigma(Y)$ are

I want to show the following: If X and Y are independent, then their generated sigma-algebras $\sigma(X)$ and $\sigma(Y)$ are independent. Let $A \in \sigma(X)$ and $B\in\sigma(Y)$ be arbitrary. ...
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27 views

to be 99% certain of making a profit? central limit theorem?

Let $X_i$ be the profit card $i$ makes when its sold. I let $S_n = X_1 + ... + X_n$ so total profit. I found the mean of $X$ to be $0.1$. and $E[X^2] = 25$ so variance $= 24.99$ Are these correct? ...
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21 views

Do convergence in distribution along with uniform integrability imply convergence of mean?

Thorem about necessary and sufficient condition for $L_1$ convergence states that $X_n$ - non negative, $EX_n$ converges to $EX$ if and only if $X_n$ converges to $X$ in probability and $X_n$ is ...
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3answers
78 views

Prove that $\|x+y\|_{\infty} \leq \|x\|_{\infty} + \|y\|_{\infty}$.

Suppose $\left(X, \Sigma, \mu \right)$ is a measure space and $x,y \colon X \longrightarrow \mathbb{R}$ are random variables. We define $$\|x\|_{\infty} := \inf_{A \subseteq \Sigma, \mu(A)=0} ...
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1answer
30 views

The characteristic function of the random time $N$

The rv's $X_1,X_2,X_3,\ldots,X_n$ are I.I.D and have the following pmf's: $$p_x(-1)=1/4\quad p_x(0) = 1/2,\quad p_x(1) = 1/4$$ The random time $N$ is defined as: $$N = \min\{n \mid X_n = 0\}$$ ...
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2answers
35 views

Show that if X is a continuous random variable on $[b,\infty)$ then $\mathrm{E}[X]=b+\int_b^{\infty}(1- F(x))dx $

I have to use the definition $$\mathrm{E}[X]=\int_{-\infty}^{\infty}xf(x)dx $$ and integration by parts. I haven't done an improper integral in a while, so I'm pretty from the beginning that I made a ...
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2answers
27 views

Estimating P(X $\ge$ k) with Chebyshev's inequality

I have managed to derive non-rigorously that P(|X - E[X]| $\ge$ a) $\le$ $\frac{E[X - E[X]|^2}{a^2}$. for a random variable X. Now let X be a random variable with Poisson distribution, with mean ...
0
votes
1answer
34 views

Battery lifetime as normal distribution?

I want to model battery lifetime, which decrements continuously at every epoch (i.e., work-cycle) in the following way. So it takes values such as 100, 99.7, 99.3, 99.2, ... 0 (a continuous random ...