Tagged Questions

1answer
21 views

Conditional expectation is square-integrable

I am given the following definition: Let $(G_i:i\in I )$ be a countable family of disjoint events, whose union is the probability space $\Omega$. Let the $\sigma$-algebra generated by these events ...
1answer
24 views

Showing that $\mathbb{P}[X\geq a]\leq \exp[-ta]\mathbb{E}[\exp[tX]]$

The problem is to show that $\mathbb{P}[X\geq a]\leq \exp[-ta]\mathbb{E}(\exp[tX])$ given $\exp(tX)<\infty$ for $t\in \mathbb{R}$ where $X$ is a random variable. Then to show that ...
3answers
64 views

$E[X]$ finite iff $\sum\limits_{n} P(X>an)$ converges

Show that: \sum\limits_{n \in N } P(X>an) < \infty\ \text{for some}\ a > 0 \Rightarrow E[X] < \infty \Rightarrow \sum\limits_{n \in N } P(X>an) < \infty\ \text{for every}\ a > ...
4answers
90 views

What exactly is a probability measure in simple words?

Can someone explain probability measure in simple words? This term has been hunting me for my life. Today I came across Kullback-Leibler divergence. The KL divergence between probability measure P ...
1answer
36 views

Proof of conditional expectation

Suppose that we have three integrable random variables $x,y,z$ on a probability space $(X,\Sigma, \mathbb{P})$ such that $x$ and $z$ are independent, and $y$ and $z$ are also independent. Show that ...
0answers
48 views

Conditional expectation as a random variable

We have three random variables $x,y,z$. Is the condition "$y$ and $z$ are independent" enough to guarantee that "$\mathbb{E}(x\,|\,y)$ and $z$ are independent"? Would anyone give me a brief proof or ...
1answer
34 views

About independence and conditional expectation

Can anyone give me a little hint on a the following question? Many thanks!! The question is: If we know that $x$ and $z$ are independent, and $y$ and $z$ are independent, is it true that ...
0answers
31 views

Tail $\sigma$-algebra

With a simple symmetric random walk such that $S_n=\sum\limits_{k=1}^n X_k$ and $\mathbb{P}[X_i=\pm1]=1/2$ with $S_0=0$ like in this post: Tail events and exchangeable events where Did answered some ...
2answers
51 views

About conditional expectation

Can someone give me some hints on the following problem? Many thanks!! Let $x$, $y$, and $z$ be integrable random variables on a probability space $(X,\Sigma, \mathbb{P})$. Show that if both $x$ and ...
1answer
18 views

If $P$ is a statistically complete set of distributions, the only sufficient subfield is the trivial one

In this thread i solved a claim stated without proof by Bahadur that if $P=\left\{p\right\}$ is the set of all probability measures on the measurable space $\left(\Omega,\mathcal{A}\right)$, ...
2answers
36 views

densities being absolutely continuous wrt Lebesgue measure

I'm reading an article with an assumption similar to: "The density $f(.)$ exists and is absolutely continuous with respect to Lebesgue measure". I don't understand this assumption because $f$ is not ...
1answer
56 views

Probability of two events are indepedent

Given a probability space $(\Omega, \mathscr {B}, P)$, then $\sigma : \mathscr{B}\times \mathscr{B} \to [0,1]^2$ is defined as, for any $A, B \in \mathscr{B}$ $$(A,B) \mapsto (P(A),P(B))$$ Now take ...
1answer
55 views

Counterexamples for Borel-Cantelli

Our teacher mentioned to construct two counterexmaples for Borel-Cantelli using the following ways. (a) Construct an exmaple with $\sum_{i=1}^{\infty}\mathbb P(A_i)=\infty$ where \$\mathbb ...
0answers
26 views

Exponential Order Statistics Independence

Are the order statistics from the $n$-sample with $X_i\sim \text{Exp}(\lambda)$ (taking, without loss of generality, $\lambda=1$) $\Delta_{(k)}X=X_{(k)}-X_{(k-1)}$ independent? Can show that for an ...
2answers
60 views

Conditional Expectation of Exponential Order Statistic $\text{E}(X_{(2)} \mid X_{(1)}=r_1)$

Having already worked out the distributions of $\Delta_{(2)}X=X_{(2)}-X_{(1)}\sim\text{Exp}(\lambda)$ and of $\Delta_{(1)}X=X_{(1)}\sim\text{Exp}(2\lambda)$ where $X_{(i)}$ are the $i$th order ...
1answer
79 views

Exponential Distribution Function

If $X\sim \text{Exp}(X)$ then for all positive $a$ and $b$, $P(X>a+b\mid X>a)=P(X>b).$ So given independent random variables $X \sim \text{Exp}(\lambda)$, $Y \sim\text{Exp}(\mu)$ we would ...
0answers
28 views

Gap distribution independence proof

I have a question bout the proof of the independence of gap RVs. Given the independent exponentially distributed random variables $\xi_1$, $\xi_2$ ~ $\text{Exp}(\lambda)$, and a corresponding order ...
0answers
28 views

Cylindrical sigma algebra answers countable questions only.

I got a missing link in some in the following (standard) textbook question: Show that the cylindrical sigma algebra $\mathcal{F}_T$ on $\mathbb{R}^T$ (equals \$\bigotimes_{t\in ...
1answer
42 views

How can a $\sigma$-algebra be “treated” or computed? Example

My question is: I have a random variable $X:\Omega \rightarrow \mathbb{R}$, the $\sigma$-algebra generated by $X$ is: $\sigma(X) := \{X^{-1}(B), B\in \mathcal{B}(\mathbb{R})\}$. But, imagine now that ...
1answer
64 views

Motivation for Measure Theory example

I was taking a look at this book while trying to pick a book for learning some rigorous probability theory. I have been totally stumped by the motivating eg. on the first page. Specifically, I am ...
1answer
63 views

Is Cesaro convergence still weaker in measure?

I've encountered a question I couldn't answer, and I would appreciate any help: Is it true that $f_n \xrightarrow{m}0$ $\Rightarrow$ $\frac{1}{n} \sum_{k=1}^{n}f_k \xrightarrow{m}0$? Where ...
1answer
18 views

If a function is $L^p$ small, is its expectation with respect to a $\sigma$-algebra $L^p$ small?

This came up in my homework, but isn't strictly my homework. I've just gotten very curious, and I keep going in circles trying to prove it. Consider a probability measure space $(X,\Sigma,\mu)$ and ...
1answer
67 views

Random Walk on Z

Let $S_n$ be the symmetric random walk on $\mathbb{Z}$. How do i calculate $P(\limsup_{n\rightarrow\infty} S_n=\infty)$? I already know that the probability is 1 but I don't really know how to start? ...
1answer
28 views

Scheffe's lemma with dominated convergence

Suppose that $f_n, f$ are non-negative measurable functions with $\mu(f_n)$ and $\mu (f)$ finite and such that $f_n\to f \text{ a.e.}$, Then $\mu(|f_n-f|)\to 0 \iff \mu(f_n)\to\mu(f)$. For ...
1answer
31 views

$\mathbb{P}(\{X>a\}) = 1 \Rightarrow \mathbb{E}(X)>a$

The implication $$\mathbb{P}(\{X>a\}) = 1 \Rightarrow \mathbb{E}(X)>a$$ seems obviously true to me, but I can't nail a half-way rigorous proof of it. (Coming up with a counterexample seems to ...
2answers
98 views

Proof on Fubini's Theorem

The Fubini's Theorem states that for any two $\sigma$-finite measure spaces $(S,\mathcal{S},\mu)$ and $(T,\mathcal{T},\upsilon)$, there exists a unique measure \$(\mu \otimes \upsilon)(A\times B)=\mu A ...
1answer
30 views

The conditional expectation of a random variable

The conditional expectation of a random variable $\xi$ given $B$ is defined as its expectation with respect to the conditional probability measure given $B$: \$\Bbb{E}[\xi|B]=\int\xi(\omega) ...
1answer
61 views

Stopping time computations via martingales

I'm studying probability, and having trouble with the following problem (from this exam). Suppose $X_j$ are i.i.d. random variables with $P(X_i=1) = P(X_i = -1) = 1/2$. Let $S_0=0$ and \$S_n = X_1 + ...
1answer
41 views

Find the distribution of a random variable

Let $\Omega=[0,1]$, $\mathcal{F}=\mathcal{B} \cap [0,1]$, and $P$ be the Lebesgue measure restricted to $[0,1]$. Let $\Phi_{\mu,\sigma^2}(x)=\mathcal{N}_{\mu,\sigma^2}((-\infty,x])$. Then it is clear ...
0answers
30 views

Total variation norm against uniform metric in $\mathbb{R}^n$

Let's consider probability functions $G(\mathbf{x}, a)$ and $F(\mathbf{x}, a)$ of 2 continuous random vectors. $a$ is a parameter. Let the convergence ...
2answers
50 views

Definition of atomic $\sigma$-field.

Reading an article in probability theory I faced with phrase atomic $\sigma$-field. I tried to search for the definition, but google doesn't give any meaningful result. As a result I'm looking for the ...
1answer
42 views

Equivalence of measures and $L^1$ functions

Suppose we have two probability measures $\mu$ and $\delta$ on $(X, \mathcal{B})$ such that $\delta <<\mu << \delta$. How can I prove that $f \in L^1(X,\mathcal{B}, \mu)$ iff \$f \in ...
1answer
54 views

Total variation norm in $\mathbb{R}^n$ [duplicate]

Let's consider total variation norm ρ( , ) on $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n)),$ where $\mathcal{B}(\mathbb{R}^n)$ is a Borel $\sigma$-algebra. Is it true that for probabiblity measures $P$ ...
1answer
82 views

Measure on a separable Hilbert space

Let $H$ be a real separable Hilbert space. Is it true that there exist a probability space $(\Omega, \mu)$ and a measurable function $\pi\colon \Omega \to H$ such that for any $h \in H$ we have  ...
2answers
52 views

Show $\displaystyle \mathbb{E}\left[\frac{X^a}{Y^a}\right] \geq \frac{\mathbb{E}[X^a]}{\mathbb{E}[Y^a]}$

Given independent RVs $X$ and $Y$, with $Y>0$, $\mathbb{E}[Y^a]< \infty$ and $\mathbb{E}[X^a]< \infty$ for some real $a\geq 0$. I need to show that \$\displaystyle ...
0answers
28 views

Question about probability measures on the real line [closed]

http://www2.imperial.ac.uk/~boz/M34P6/P11_2.pdf Dear comrades. I am struggling with Ex 1.4(i) on here. I think that I have proved that the measures $\mu$, $\nu$, $\lambda$ are all equivalent, in the ...
1answer
54 views

sigma algebra problem [duplicate]

Let $f$ be any function from $\Omega$ to $X$ and $\mathcal{C}$ an arbitrary nonempty collection of subsets of $X$. Show $$\sigma(f^{-1}(\mathcal{C}))=f^{-1}(\sigma(\mathcal{C}))$$ I already know how ...
2answers
30 views

What is meant by closed under complementation?

I was going through the probability and measure chapter of testing of hypothesis book by L.H. Lehman, where I found this "A class of sets that contains Z and is closed under complementation and ...
4answers
209 views

What is the probability you guess the number I am thinking of?

Probability is defined as the likely number of outcomes over all total outcomes. In this case, 1 over infinity; which would equate to zero. But, there is a chance you can guess the number I am ...
1answer
29 views

Almost surely constant [duplicate]

Let $x$ and $y$ be two independent random variables on a probability space $(X,\Sigma,\mathbb{P})$ such that $x+y$ is almost surely constant. Show that both $x$ and $y$ must be almost surely constant. ...
2answers
97 views

Almost sure convergence of random variable

I see a lot of examples of limit theorems in terms of functions, and sequences of functions. But I think the transition from the general measure space to the probability space ...
1answer
40 views

Double Jumps of a Poisson Process

If $N_t$ be a Poisson Process with rate $\lambda>0$, surely for any prescribed $t>0$, the probability that $N_t$ "jumps (at least) twice" at $t$ is zero, i.e. ...
1answer
59 views

Measurable Maps

Let $X$, $M$ be two metric spaces and $\nu:X\rightarrow \mathcal{M}_1(M)$, $x\mapsto \nu_x$ a map, where $\mathcal{M}_1(M)$ is the space of all probabilities over $M$ with the Boral $\sigma$-algebra, ...
1answer
88 views

Probability Curiosity

Let $X_{1},X_{2},\ldots$ be i.i.d. random variables with $E\left[X_{1}\right]=0$ and $0<Var\left(X_{1}\right)=\sigma^{2}<\infty.$. Let $S_{n}=\sum_{j=1}^{n}X_{n}$. Consider now ...
1answer
168 views

Show that if $f_n \leq g$ for all $n$ and $g$ is integrable, then $\{f_n\}$ is uniformly integrable

A sequence {$f_n$} of measurable functions is called uniformly integrable if $$\lim_{M \to \infty} \sup_{n} \int_{[|f| >= M]} |f_n|\ \mathrm{d}\mu = 0$$ Show that if $|f_n| \leq g$ for all $n$ ...
1answer
77 views

Does the sum of sequence of measurable functions converge outside a set of measure zero?

Let {$f_n$} be a sequence of measurable functions defined on a probability space, such that: $$P(f_n = 1/n) = 1 - P(f_n = 0) = 1/(n^2)$$ Does $\sum_{n=1}^{\infty}{f_n}$ converge outside a set of ...
0answers
43 views

If one finite measure is less than another on a field, then the same holds for the generated sigma-field??

Suppose $μ_1$ and $μ_2$ are finite measures on $F = \sigma(F_0)$, where $F_0$ is a field. If $μ_1 \leq μ_2$ on $F_0$, then show that $μ_1 \leq μ_2$ on $F$. This is an exam review question and the ...
1answer
45 views

Behavior of the tail of a cdf.

If X is an integrable real random variable it is true that $$\lim_{x \to \infty} x P(X > x) = 0 \, ?$$ I know it is true for the $L^2$ case since it can be derived easily from Chebyshev ...
0answers
54 views

Is there a canonical probability measure on smooth curves?

For continuous curves, we have Brownian motion giving the most natural probability measure. However, the sample paths of Brownian motion are almost surely terribly behaved (not of bounded variation, ...
1answer
89 views

Uniform measure on the rationals between 0 and 1

I am trying to think of a probability measure on the set of rationals between 0 and 1 ($X:=\mathbb{Q}\cap[0,1]$). I want to achieve something like a uniform measure, i.e. every number should have the ...
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