Tagged Questions
0
votes
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 ...
1
vote
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 ...
3
votes
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 > ...
2
votes
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 ...
2
votes
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 ...
4
votes
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 ...
2
votes
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
...
0
votes
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 ...
1
vote
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 ...
0
votes
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)$, ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
3
votes
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 ...
0
votes
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 ...
1
vote
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 ...
6
votes
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 ...
0
votes
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 ...
4
votes
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 ...
1
vote
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 ...
3
votes
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? ...
0
votes
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 ...
1
vote
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 ...
2
votes
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 ...
-1
votes
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) ...
3
votes
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 + ...
0
votes
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 ...
1
vote
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 ...
1
vote
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 ...
2
votes
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 ...
3
votes
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$ ...
5
votes
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
$$
...
1
vote
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 ...
0
votes
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 ...
2
votes
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 ...
0
votes
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 ...
5
votes
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 ...
0
votes
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.
...
1
vote
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 ...
4
votes
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. ...
2
votes
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, ...
2
votes
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 ...
1
vote
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 $n$ ...
1
vote
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 ...
0
votes
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 ...
2
votes
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 ...
3
votes
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, ...
0
votes
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 ...
