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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22 views

What conditional independence theorem is being used here

In stanford's machine learning lecture 1, linear regression is defined on page 11, section 3 as: For $i = 1, \ldots, m$, $y^{(i)} = \theta^T x^{(i)} + \epsilon^{(i)}$, where $\epsilon^{(i)}$ are IID ...
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1answer
19 views

The probability that two randomly selected $2$ year old male feral cats will live to be $ 3$ years old is? [on hold]

The probability that a randomly selected $2$ year old male feral will live to be $3$ years old is $0.82666$. (a) what is the probability that two randomly selected $2$ year old male feral cats will ...
2
votes
4answers
50 views

Uncountable increasing family of $\sigma$-algebras

Could someone give an example of what an uncountable increasing family of $\sigma$-algebras $\{\mathcal{F}_t\}_{t\geq 0}$, $(\mathcal{F}_s \subset \mathcal{F}_t$ for $s<t)$ might look like? For ...
0
votes
1answer
29 views

If independent r.v. converge in probability to a constant, do they converge almost surely?

I've seen several examples when a sequence of r.v. converge in probability but not almost surely, yet none of them had the sequence to be independent. Would additional conditions of independence and ...
0
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1answer
27 views

How to change the measure to make a non standard normal random variable standard normal

Given a probability space $(\Omega, \mathcal{F}, P)$ consider a standard normal random variable $X$. Let $\tilde{X} = X + c$, $c \in \Bbb R$. Now consider the following probability measure ...
1
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0answers
27 views

Expectation of a continuous function

Can someone help with the following? I have a continuous function $g: A_i \times A_{-i} \to \mathbb{R}^k$, and a probability measure $\mu \in \Delta(A_{-i})$. We can let $A_i=\mathbb{R}^n$ and ...
-1
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0answers
24 views

Distribution Problem based on unknown function [on hold]

I got struck at this problems as Function is not given. Any help will be appreciated
2
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0answers
44 views

What is the General Central Limit Theorem?

General Central Limit Theorem says: Let $ \{(X_{n,j} , 1 ≤ j ≤ n), n ≥ 1\} $ be a triangular array of rowwise independent random variables, set $ S_n = \sum_{j=1}^n X_{n,j}, s_n^2 = \sum_{j=1}^n ...
0
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0answers
20 views

Conditional Expectation with respect to two Random Variables

Consider the quantity $$ \mathrm E[U \mid S,T]. $$ Is this shorthand for $$ \mathrm E[U \mid \sigma(S) \otimes \sigma(T)]? $$ If so, the defining characteristics are that $\mathrm E[U \mid S,T]$ is ...
3
votes
1answer
23 views

Almost surely on a subset

I often meet in the literature on probability theory statements like "$\phi$ almost surely on $S$", where $\phi$ is a property and $S$ a subset of the underlying complete probability space $(\Omega, ...
0
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1answer
33 views

Probability of Union of Events in a Probability Product Space By Counting Event Size

This question is about probability of union of events in a probability product space. Let’s say a fair die is thrown twice and we’re interested to find out probability of getting face value one in 1st ...
0
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0answers
27 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
votes
2answers
33 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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votes
2answers
92 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 ...
7
votes
1answer
124 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 = ...
1
vote
1answer
22 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 ...
1
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1answer
24 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 ...
2
votes
1answer
22 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
votes
0answers
18 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 ...
2
votes
0answers
52 views
+50

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
votes
1answer
35 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 ...
0
votes
1answer
35 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.
2
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1answer
33 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
votes
3answers
36 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 ...
2
votes
1answer
30 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 $x_1 x_2 \ldots x_n$ ...
2
votes
2answers
44 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 ...
-1
votes
1answer
59 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 $ ...
3
votes
1answer
57 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 ...
3
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0answers
29 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
votes
0answers
25 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 ...
2
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0answers
37 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 ...
2
votes
1answer
41 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 ...
1
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0answers
34 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 ...
3
votes
1answer
64 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 ...
1
vote
3answers
108 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] = ...
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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votes
1answer
25 views

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

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*} ...
0
votes
1answer
37 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 ...
1
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0answers
20 views

differentiating $\phi(u,t)=e^{(iu^T\hat{x_t}-\frac{1}{2}u^TP_tu)} w.r.t$ t?

Here $\phi(u,t)=E\{e^{iut}\} $ is a characteristic function, $x_t$ is Gaussian. Differentiating $\phi(u,t)=e^{(iu^T\hat{x_t}-\frac{1}{2}u^TP_tu)}$ w.r.t t the result is $\phi_u=\phi[i\hat{x}_t-P_tu]$
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votes
0answers
39 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
vote
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 ...
2
votes
1answer
42 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 ...
0
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0answers
36 views

expected value of expected value

I want to quantify the error of phase noise in terms of its normalized mean squared error. I define the error measure as (x is the error free function, y the distorted): $$ \rm NMSE = \frac{\int ...
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 ...
0
votes
1answer
30 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 ...
0
votes
1answer
31 views

Defining the set of pre-images of a product of random variables in terms of the sets of pre-image of the original random variables

Say I have two random variables $X$ and $Y$. Their respective $\sigma$-algebras are $$\sigma(X) = \{ X^{-1}(B) \mid B \in \mathscr{B} \}$$ and $$\sigma(Y) = \{ Y^{-1}(B) \mid B \in \mathscr{B} \}.$$ ...
0
votes
2answers
58 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, ...
1
vote
1answer
20 views

Indicator Functions with Random Variables

Let $E$ be an event and $Y$ a random variable. What exactly is meant by $\mathrm E[\mathbf 1_E \mathbf 1_{Y\in B}]$? I have two guesses, the first is that $\mathbf 1_{Y\in B}$ is an indicator random ...
2
votes
1answer
37 views

Independence, conditioning, and correlation part 2 [closed]

Suppose $X$ and $Y$ are independent random variables. I now want to consider conditioning on some event $C$. Under what conditions will $X\mid C$ and $Y\mid C$ be correlated?
3
votes
0answers
26 views

Ito's formula and Infinitesmal generator

Consider an Ito process $$ dX_t = \sigma_t dB_t $$ where $\sigma_t$ is a two-state continuous-time Markov chain with state space $\{ \sigma_1, \sigma_2 \}$ that switches state with Poisson ...