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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10 views

Interpretation of integral as ratio of joint and conditional densities?

A common exercise in Bayesian statistics is specifying a prior $p(\theta)$ on some parameter $\theta$. We then observe a collection of data $D=(X_1,\dots,X_N)$, the distribution of which is ...
2
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2answers
24 views

Is there accepted notation for the pushforward measure that doesn't mention $\mathbf{P}$?

Let $(\Omega,\mathcal{F},\mathbf{P})$ denote a probability space, $(S,\mathcal{M})$ denote a measurable space, and $X : (\Omega,\mathcal{F},\mathbf{P}) \rightarrow (S,\mathcal{M})$ denote a measurable ...
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2answers
84 views

What do you call this thing in probability theory? [on hold]

I have studied it before but I forgot the name. It is like when the possiblity of something happens is so small, but you created the experience so so many times, then the probability of that thing to ...
5
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0answers
57 views

When does $\sum_{i=1}^{\infty} X_i$ exist for random sequences $\{X_i\}_{i=1}^{\infty}$?

Suppose $\{X_1, X_2, X_3, \ldots\}$ is an infinite sequence of random variables such that $E[X_i]=0$ for all $i$, and $E[X_iX_j]=0$ whenever $i \neq j$. Further suppose the variances $\sigma_i^2 = ...
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0answers
24 views

Kolmogorov's sufficient and necessary conditon for SLLN - What about pairwise uncorrelated RV?

Kolmogorov proved, that, as one considers independent (not necessary equally distributed) Random Variables: $\{X_n\}_{n\ge0}\subseteq \mathcal L^2$ With $\mathrm{Var} (X_n)=\sigma^2_n$ and without ...
2
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1answer
33 views

Independent increments of a Poisson process

In the following $\{X_t\}$ is a Poisson process. Assume that I've proved that $P(X_s=i,X_t-X_s=k)=P(X_s=i)P(X_t-X_s=k)$ so that the two events are independent, does it follow that ...
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0answers
13 views

Proof of a classical Theorem of Martin-Löf on complexity dips for Kolmogorov complexity,

I have a question on the first Theorem from the article Complexity of Oscillations in Infinite Binary Sequences by P. Martin-Löf, which could be downloaded from the publisher or from here. Theorem ...
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1answer
19 views

What is the countable product sigma algebra of powersets of a countable set $E$? The powerset of the space of all sequences in $E$ or not?

Let $E$ be a countable set with power set $\mathcal{P}(E)$. $(E,\mathcal{P}(E))$ is a measurable space. Let $E^{\mathbb{N}}$ be the space of sequences in $E$ and $\mathcal{P}(E)^{\mathbb{N}}$ the ...
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1answer
55 views

Prove the series converges a.s in Probability

I have an article as follows Why are they enough to prove that $ \sum_{n=1}^\infty \dfrac{X_n \textbf{1}_{\{|b_n|< |X_n|\}}}{b_n} $ converges almost surely? I want to know why must prove $ ...
2
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4answers
187 views

Probability interview question

Suppose we have three positive integers $A, B, C$. We randomly choose an integer $a$ in the range $[0,A]$ and an integer $b$ in the range $[0,B]$. Find the probability that $a + b\leq C$. I am unable ...
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0answers
34 views

Find a probability density

I am going through a paper trying to understand all the single steps, but I got stuck. I need to calculate $$p(x+\delta t) \mid x(t), t)= \int p(x(t+\delta t) \mid \mu , x(t), t)p(\mu\mid x(t), t) ...
2
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1answer
19 views

Probability in knockout games.

Suppose in a knockout tournament 32 players p1 , p2 .....p32 participate. In each round players are divided into pairs at random and winner goes to the next round. If p5 reaches semifinal what is ...
0
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2answers
193 views

If X,Y and Z are independent, are X and YZ independent?

If yes: I know that f(X) and g(Y) are independent if X and Y are independent and f and g are "measurable".* If that is to be used, is g(Y) = YZ measurable? If not, how else to approach this? If ...
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0answers
16 views

Calculating Variance of payment in patterns of balls.

We have five different bags labeled from 1 to 5 and several colored balls. There are 9 different possible colors. We know how many balls of each color there are in each bag. We have a grid of 5x3 ...
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1answer
32 views

Show that for any random variable $X$, and any $a > 0$, $P(|X| > a) \leq {EX^4 \over a^4}$.

Show that for any random variable $X$, and any $a > 0$, $$P(|X| > a) \leq {EX^4 \over a^4}.$$ Maybe I need to use Markov's Inequality, but I don't know how.
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3answers
33 views

Understanding the definition of $P(Y = y \mid X = x)$

Let $X: \Omega \rightarrow E_X$ and $Y: \Omega \rightarrow E_Y$ be random variables. By definition, we have that $P(Y = y \mid X = x)$ is defined as follows: $$ P(Y = y \mid X = x) = \frac{P(X = x ...
3
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1answer
56 views

sign of the conditional expectation

I'm working on the following problem: Let $X$ be a random variable defined on $(\Omega,F,P)$ and $G$ a $\sigma$-algebra contained in $F$. Show that, if $E(|X|)<\infty$ and $E(X\mid G)$ has the ...
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0answers
24 views

Does convergence in probability preserve the weak inequality?

Suppose I have two sequences of random variables $\{x_n\}$ and $\{y_n\}$ such that $x_n\leq y_n$ and $\text{plim}\;x_n=L_x$ and $\text{plim}\;y_n=L_y$, can I say $L_x\leq L_y$ (almost surely)? Does ...
2
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2answers
38 views

What exactly is $\cap$-stable here?

From my lecture notes: Theorem: Let $(\Omega, \mathcal A, P)$ be a probability space, $A \in \mathcal A, \mathcal M := \{ M_1, \ldots, M_n \} \subset \mathcal A$. The following statements are ...
2
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1answer
23 views

Intuition on Martin-Löf-Test for finite strings

The followng example is from An Introduction to Kolmogorov Complexity and Its Applications, Example 2.4.1. and is concerned with Martin-Löf-Tests for finite strings: A string $x_1 x_2 \ldots x_n$ ...
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2answers
57 views

Minimizing $\mathbb E((X-m)^2)$

Let $X$ be a real random variable such that $X^2$ is integrable. I have to find $m$ minimizing $\mathbb E((X-m)^2)$. How I want to do this is by saying: $$ \begin{align} \mathbb E((X-m)^2) &= ...
3
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1answer
63 views

How to prove that this process is always positive?

I would like to ask is there any way to prove that following process $$ \mathrm dY_t=\left(a+\frac{b}{Y_t}\right)\mathrm dt +\mathrm dW_t, \ \ Y_0=y_0>0, $$ where $a\neq 0$ and $b\geq 1/2$, is ...
0
votes
1answer
14 views

MCMC( Markov Chain Monte-Carlo Simulation )

Suppose $Q=[q(i,j)]$ is a transitional probability matrix for an irreducible Markov chain. Suppose also $\lbrace X_{n}, n \geq 0 \rbrace$ is Markov Chain such that if $ X_{n}=i$, generate $Y=j$ such ...
2
votes
0answers
22 views

What is the meaning of the cumulant generating function itself?

If we define the characteristic function for a random variable X as $\Phi(t)=<e^{itX}>$ then it seems like we can think of it as essentially a spectral decomposition that measures the ...
1
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4answers
86 views

Difference between $E[X^2]$ and $E[X^3]$

Hope to ask a dumb question. $Y = aX$,with $a \in N_+$. Here, we know the correlation coefficient is 1. Now, suppose $X \sim N(0,1)$. Here, we know $X, Y$ are not independent. Cov($X,Y$) = ...
2
votes
1answer
314 views

Why does the Doob-Dynkin lemma show that $\sigma$-algebras are the carriers of probabilistic Information

The Doob-Dynkin lemma states that for two functions $X, Y \rightarrow \Omega$ the following two statements are equivalent: There is a Borel-measurable function $h:{R}^n\rightarrow R^n, f(X)=Y$. Y is ...
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1answer
35 views

Why is $f(X_t)-\int_0^t Af(X_s) \, ds$ a martingale for a Markov process $(X_t)_{t \geq 0}$?

I think if $A$ is the usual generator for the Markov process $(X_t)_t$ $$A f (x) = \lim_{t \downarrow 0} \frac{\mathbf{E}^{x} [f(X_{t})] - f(x)}{t}$$ then we get that for any "nice" $f$ the process ...
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0answers
32 views

Conditional expectation with Cauchy-Schwarz Inequality

Consider real-valued random variables $X$, $Y$, and $Z$; and a scalar, positive constant $k$. I want to prove the following \begin{equation} E[1|X+Y<Z<X+Y+k]E[X^2|X+Y<Z<X+Y+k]\ge ...
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1answer
30 views

Cauchy Schwarz inequality for random vectors: need help with the proof

I have a question related to the proof of Cauchy Schwarz inequality (for discrete random variables). We have two random discrete variables $X,Y$ and we know that $\mathbb E[X^2] \text{ and } \mathbb ...
2
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1answer
41 views

Expected value of a mean when previous values determine stopping point

I recently came across this brain teaser: There's an island and every family on the island wants to have a boy. So each family continues having kids until they have a boy, then they stop ...
2
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0answers
35 views

Can Monotone Class Theorem be easier to check than $\pi$-$\lambda$ Theorem?

I've been working on problem 14.4 in Billingsley's "Probability and Measure", which says: "Let $C$ be the set of continuity points of $F$. Show that for every Borel set $A$, $P(F(X) \in A, X \in ...
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1answer
18 views

How to derive formula for marginal probability of choosing nest in nested logit model?

I am trying to understand all the details of the nested logit and what confuses me is the formula for marginal probability of choosing the nest. In more details: the joint probability of individual n ...
2
votes
1answer
621 views

expectation and sign of the Radon-Nikodym derivative

I am new here. I have some questions about the Radon-Nikodym derivative. I hope someone is willing to help me with these. The questions are stated below. Also I added my attempts to the problem to ...
1
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1answer
44 views

Prove that risk function is analytic?

I'm considering the statistical minimax estimation problem of the bounded normal mean: Specifically, the problem is to find the minimax estimator of $X \sim N(\theta,1)$ where $\theta \in ...
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0answers
34 views

Intuition for probability density function as a Radon-Nikodym derivative

If someone asked me what it meant for $X$ to be standard normally distributed, I would tell them it means $X$ has probability density function $f(x) = \frac{1}{\sqrt{2\pi}}\mathrm e^{-x^2/2}$ for all ...
6
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1answer
428 views

Girsanov: Change of drift, that depends on the process

Known: If I am looking at an SDE like: $dX_t = b(t,\omega) dt + dW_t$ with $W_t$ a Brownian motion under a measure $P$. I know that I can change the drift by using Girsanov to $dX_t = ...
2
votes
1answer
52 views

$\mathrm E [f(X,Y) \mid Y=y] = \mathrm E [f(X,y)]$?

Let $X,Y$ be independent integrable random variables and let $f :\mathbb{R}^2 \to \mathbb{R}$ be integrable. It makes intuitive sense that $$ \mathrm E [f(X,Y) \mid Y=y] = \mathrm E [f(X,y)], $$ but ...
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0answers
29 views

Conditional expectation and conditional variance decomposition

I am preparing an exam and I have found in my lecture notes the two following formulas that the professor uses again and again, but I have no clue where they come from: ...
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3answers
106 views

$\mathrm E [X \mid X=x] = x$?

I've gotten so caught up in measure-theoretic probability that I'm actually having trouble showing this simple result. Let $X$ be an integrable random variable. Then $$ \mathrm E[X \mid X=x] = ...
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1answer
26 views

Prove that limsup and liminf of an independent sequence are independent of finite number of terms

Let $X_1, X_2, ...$ be an independent sequence of random variables on $(\Omega, \mathscr{F}, \mathbb{P})$. What I'm trying to prove is: Prove that $X_1, X_2, ..., X_k$ is independent of $\liminf ...
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0answers
38 views

Finding distribution from PGF not in closed from.

$X_1,X_2,\ldots,X_N$ are independent and identically distributed random variables. We have $X = e^{-Y}$, where $Y\sim\mathrm{Poisson}(\lambda_u)$, and $$Z =X_1+X_2+\cdots+X_N ,$$ where $N \sim ...
1
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1answer
46 views

Understanding the Markov property of Brownian motion

I'm trying to understand the Markov property for Brownian motions in full generality. The textbook I'm following states it like this: Recall that we have a family of measures $P_x, x \in ...
1
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1answer
40 views

Computing $E[ {\rm Tr}\{(ZZ^T)^2 \}]$ for $Z$ Gaussian.

Let $Z \in \mathbb{R}^n$ be a Gaussian random vector with zero mean and $Cov(Z)=I$ where $I$ is identity matrix. How to compute \begin{align*} E[ {\rm Tr}\{(ZZ^T)^2 \}] \end{align*} I know that ...
0
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2answers
57 views

Are random variables independent of their tail sigma-algebra?

Let $X_1, X_2, ...$ be independent random variables. Define $$\mathscr{T}_n = \sigma(X_{n+1}, X_{n+2}, \ldots)$$ and $$\mathscr{T} = \bigcap_{n} \mathscr{T}_n,$$ the tail σ-algebra of $(X_1, X_2, ...
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1answer
24 views

Does conditioning reduces conditional variance i.e. $Var(W|Y) \le Var(W|Y,Z)$ [on hold]

Let $W,Y,Z$ be are be some random variables. My question is does conditioning reduce variance on in other words is the following inequality true? \begin{align*} Var(W|Y) \le Var(W|Y,Z) \end{align*} ...
1
vote
1answer
443 views

Simulation of a Gaussian process on $R^2$ with a stationary kernel using the Karhunen-Loève expansion

Assume $X(\omega, t) \sim \mathcal{N}(0, K(\cdot, \cdot))$ is a real-valued, centered Gaussian process on $R^2$, i.e., $X: \Omega \times R^2 \to R$. Let the covariance function of the process be ...
1
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1answer
40 views

Equivalence of Conditional Expectations w.r.t. Discrete Random Variable

Let $X$ and $Y$ be integrable random variables such that $P(Y=y) > 0$ for all $y \in Y(\Omega)$. Then the conditional expectation of $X$ given $Y=y$ is defined as $$ \mathrm E[X \mid Y=y] : = ...
6
votes
3answers
2k views

Weak Law of Large Numbers for Dependent Random Variables with Bounded Covariance

I'm currently stuck on the following problem which involves proving the weak law of large numbers for a sequence of dependent but identically distributed random variables. Here's the full statement: ...
1
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0answers
21 views

How to bound $E \left[\left(E[Z^2\mid Y] \right)^2\right]- 2E \left[ |E[Z\mid Y]| \sqrt{E[Z^2\mid Y]} \right]$

I am looking for an upper bound on the following quantity \begin{align*} A=E \left[\left(E[Z^2\mid Y] \right)^2\right]- 2E \left[ |E[Z\mid Y]| \sqrt{E[Z^2\mid Y]} \right] \end{align*} where $Z$ is ...
1
vote
2answers
68 views

Proving Cantelli's inequality

I'm assuming that the random variable $X$ has mean $0$ and finite variance ${\sigma}^2$. It is immediate from Chebyshev's inequality that $$P(X\geq x)\leq \frac{{\sigma}^2}{x^2},$$ but I'm still ...