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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134 views

Deriving the transformation function of a random variable from the original and the final distributions

Consider a random variable $X$ and consider that this variable can be either real or integral (so I would like to cover both cases: continuos and discrete random variables). Consider to transform this ...
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3answers
229 views

Conditional expectation in the case of $\mathcal{A}=\{\emptyset,\Omega,A,A^c\}$

I have a small computation to do and I am not able to prove it: Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Let $X$ be an integrable random variable and $A\in\mathcal{F}$. Let ...
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1answer
154 views

How to find the odds of a horse wining a race?

The odds against a horse winning are 2 to 1. What is the probability of this horse winning?
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1answer
214 views

Is it possible to derive the CDF of $Z$?

Assume that $X_i$, $Y_k$, $i=0,\ldots,N$, $k=1,\ldots,K$ are non-negative independent non-identically distributed random variables. Let us define the random variable $Z$ as \begin{align} ...
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1answer
1k views

Questions about independence between random variable and $\sigma$-algebra

Given a probability measure space $(\Omega, \mathcal{F}, P)$, if a random variable X and a sub $\sigma$-algebra $\mathcal{A}$ are independent, I was wondering why: $$E (X|\mathcal{A}) = (EX)I_Ω;$$ ...
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2answers
2k views

Sigma algebra proof problem

Let $\mathcal{F}$ be a $\sigma$-algebra of subsets of $\Omega$ and $B \in \mathcal{F}$. Show that $E = \{A \cap B: A \in \mathcal{F}\}$ is a $\sigma$-algebra of subsets of $B$. Is it still true when ...
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3answers
2k views

Will this procedure generate random points uniformly distributed within a given circle? Proof?

Consider the task of generating random points uniformly distributed within a circle of a given radius $r$ that is centered at the origin. Assume that we are given a random number generator $R$ that ...
7
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1answer
102 views

If $A$ is the generator of $(P_t)$, then $A+f$ is the generator of $(P_t^f)$

Let $X=(X_t)_{t\geq0}$ be a Markov process on a state space $\Gamma$ (a Hausdorff topological vector space), let $A$ be the infinitesimal generator of $X$ and let $\mathcal C(\Gamma)$ the space of ...
7
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1answer
279 views

How can I prove this expression not to be a characteristic function

Let $\phi$ be a function of two real arguments defined as follows: $$ \phi\left(t_1, t_2\right) = \exp \left(-t_1^2-t_2^2 +i \frac{t_1}{3}\frac{ t_1^2-3 t_2^2 }{t_1^2+t_2^2} \right)$$ and whenever ...
6
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1answer
161 views

Characterizing a distribution by its projections

Consider the density $f(x,y)=\large\frac{1}{2\pi}\frac{1}{\sqrt{1-x^2-y^2}}$ on the unit disk centered at the origin. There is a particular characterization of this distribution: it is the unique ...
6
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1answer
273 views

Squeeze theorem for convergence in distribution

Let $(X_n)_{n\in\mathbb{N}}$, $(X'_n)_{n\in\mathbb{N}}$, $Y$ real random variables such that $$ X_n\leq Y\leq X'_n \quad\forall n\in\mathbb{N}\ .$$ Suppose that the sequences $X_n$ e $X'_n$ converge ...
6
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2answers
624 views

coupon collector and Markov chains

I need some help with my homework in probability. I need to prove that if $X(n) =$ the number of different coupons that the collector has in time $n$ then $X(n)$ represents a Markov chain. I proved ...
6
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2answers
386 views

Confidence band for Brownian Motion with uniformly distributed hitting position

Let $(B_t)$ denote the standard Brownian motion on the interval $[0,1]$. For a given confidence level $\alpha \in (0,1)$ a confidence band on $[0,1]$ is any function $u$ with the property that $$ ...
6
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1answer
915 views

What is the expected number of runs of same color in a standard deck of cards?

Standard deck has $52$ cards, $26$ Red and $26$ Black. A run is a maximum contiguous block of cards, which has the same color. Eg. $(R,B,R,B,...,R,B)$ has $52$ runs. $(R,R,R,...,R,B,B,B,...,B)$ has ...
5
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2answers
167 views

Sequence satisfies weak law of large numbers but doesn't satisfy strong law of large numbers

Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of independent random variables such that $$P(X_n=n+1)=P(X_n=-(n+1))=\frac{1}{2(n+1)\log(n+1)}$$ $$P(X_n=0)=1-\frac{1}{(n+1)\log(n+1)}$$ Prove that $X_n$ ...
5
votes
1answer
259 views

Intuition behind measurable random variables and $\sigma$-algebra

I've been trying to understand $\sigma$-algebras and how it encodes information in context of filtration. While certain parts seem clear and logical, I can't say I get the whole picture. I'll try to ...
5
votes
3answers
200 views

A recursive formula to approximate $e$. Prove or disprove.

Let the sequence $\{x_n, n=1,2,...\}$ be defined as follows: Let $x_2=x_1=1$ and for $n>2$ let $$x_{n+1}=x_n-\frac{1}{n}x_{n-1}.$$ This sequence, generated by the recursion above, tends to zero ...
5
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1answer
68 views

How to show that $p$th moment being finite is equivalent to a limit existing

Let $p \in (0,2)$ and let $\xi_n, n \geq 1$, be iid random variables. Show that the following two conditions are equivalent: With probability one, the limit $$ \lim_{n \rightarrow \infty} ...
5
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1answer
191 views

Interpretation of a tail event

I am currently reading about tail events wikipedia. And I was wondering: Where does the interpretation come from that events in this sigma algebra are independent from the behaviour of any finite set ...
5
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1answer
166 views

Probability convergence in distribution

$Y_1, Y_2,..., Y_n$ are i.i.d and uniformly distributed on the set $\{1, 2,..., n\}$. Define $X_n = \min\{k: Y_k = Y_j\; \text {for some}\; j < k\}$, and prove that $\frac {X_n}{\sqrt n}$ converges ...
5
votes
2answers
686 views

Is $\{\sin(\omega n), n \geq 1\}$ a strictly stationary process?

Let $X(t)=\sin(\omega t)$, where $\omega$ is is uniformly distributed R.V. on $[0,2π]$. Let $X_n=X(n)$, is $\{X_n,n \geq 1\}$ a strictly stationary process? I've calculated that the distribution ...
5
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1answer
178 views

An Integral Involving Brownian Motion

Let $B_t$ $(t \geq 0)$ be a Brownian motion on $\mathbb{R}^3$. That is, $B_t = (B_{t}^{(1)},B_{t}^{(2)},B_{t}^{(3)})$, where each $B_{t}^{(i)}$ is a Brownian motion on $\mathbb{R}$. Let $Y$ be a Borel ...
5
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3answers
9k views

Repeating something with (1/n)th chance of success n times

Is there anything that can be said about how many attempts it will take to correctly guess a random number out of 1000 numbers? If the number wouldn't change the probability would just increase every ...
5
votes
1answer
2k views

Dunford-Pettis Theorem

The Dunford-Pettis Theorem (see Uniform Integrability Wiki) states that: A class of random variables $X_n \in L^1(\mu)$ is Uniformly Integrable if and only if it is relatively weakly compact. Now ...
5
votes
2answers
2k views

Weak convergence of probability measure

I am working on a problem. Show that for each probability measure $\mu$, there exists probability measure $\mu_n$ with finite support such that $\mu_n$ converges weakly to $\mu$. I am thinking about ...
5
votes
1answer
590 views

Is Lindeberg's condition satisfied when variances of the sequence of r.v.'s are bounded from above and below?

This question is related to my previous question but I think it's sufficiently different to warrant a separate one. Suppose I have a sequence of independently distributed positive random variables: ...
5
votes
1answer
894 views

How does one prove that Lindeberg condition is satisfied?

I have a sequence of $n$ random variables, each drawn from a different distribution $X_1\sim A_1, X_2\sim A_2, \ldots, X_n\sim A_n$. The distributions $A_1, A_2, \ldots, A_n$ have nice properties: ...
5
votes
1answer
191 views

Existence of a completely supported probability measure

Given a compact Hausdorff space $X$, does there exist a probability $\mu$ on X such that the support of $\mu$ is $X$? This is equivalent to say, for any unital commutative C*-algebra, can we show the ...
5
votes
1answer
1k views

Inner product and norms for random vectors

From wikipedia inner product page: the expected value of product of two random variables is an inner product $\langle X,Y \rangle = \operatorname{E}(X Y)$. How it can be generalized in case of random ...
5
votes
1answer
857 views

Upper bound on the l1 norm of a multivariate normal random variable

Let $X \sim {\cal N}_d(0, \sigma^2I_d)$. I am interested in bounding the tail probability $P[||X||_1 > t]$ from above. A pointer to a known exponential or polynomial tail bound would be ...
4
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1answer
90 views

Question on part 3 of the Star Trek problem in Williams, Probability with Martingales

Consider this M.SE question, which is E12.3 in Williams. The answer of Robert Israel (and Xoff) seems to give an exponential bound on $R_n$ almost surely. Wouldn't this imply the convergence of ...
4
votes
1answer
45 views

Stopping time in Markov chains

A random variable $T : \Omega \rightarrow ${$1,2,3...$} $\cup$ {$ \infty$} is called a stopping time if the event {$T=n$} depends only on $X_0 , X_1 ,X_2 ,..., X_n$ for $n = 0,1,2,...$ I have trouble ...
4
votes
1answer
55 views

Does every smooth manifold carry a gaussian random field?

Let $M$ be an arbitrary smooth manifold of dimension $n$. For simplicity, let's assume that $M$ is boundary-less. Can we construct a gaussian random field on $M$? If the result is not true for ...
4
votes
1answer
113 views

Properties Least Mean Fourth Error

I am interested in whether a quantity \begin{align*} E[(X-E[X|Y])^4] \end{align*} has been studied in the literature before. I am not even sure if "least mean fourth error" is a correct name, since ...
4
votes
2answers
176 views

Is $\mathbb{E}\exp \left( k \int_0^T B_t^2 \, dt \right)<\infty$ for small $k>0$?

Suppose that $B$ is a Brownian motion. Does it hold that \begin{equation} \mathbb{E}\left[\exp\left(k\int_0^T[B(t)]^{2}\,dt\right)\right] <\infty\text{ ?} \end{equation} for some positive constant ...
4
votes
2answers
202 views

Sample path of Brownian Motion within epsilon distance of continuous function

Given a continuous function $f:[0,1]\rightarrow\mathbb{R}$, $f(0)=0$, how can one show that $P(\underset{0\leq t\leq1}{\sup}\left|B_{t}-f(t)\right|<\varepsilon)>0$, where $P$ is the probability ...
4
votes
1answer
261 views

Intuition behind Strassen's theorem

Currently I am dissecting a proof of Strassen's theorem, which states the following: Suppose that $(X,d)$ is a separable metric space and that $\alpha,\beta>0$. If $\mathbb{P}$ and ...
4
votes
1answer
200 views

Slowly varying function without limit at infinity

A function $f:\mathbb R \to \mathbb R$ is slowly varying at infinity if for any $t>0$ $$ \lim_{x\to +\infty}\frac{f(xt)}{f(x)}=1. $$ Is there a bounded function slowly varying at infinity whose ...
4
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1answer
99 views
+150

Representation of a linear functional Lipschitz in total variation

Let $\Omega$ be a Borel space and let $\mathcal P(\Omega)$ be the space of all Borel probability measures on $\Omega$ endowed with the topology of weak convergence. Define the total variation metric ...
4
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6answers
321 views

Logical issues with the weak law of large numbers and its interpretation

In several probability textbooks I have found what amounts to the following argument: Let A be an event in some probabilistic experiment. Let p=P(A) be the probability of this event occurring in ...
4
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1answer
127 views

Filtrations and Sigma-Algebras

I have been practising a question set by my lecturer and try to verify the answer, unfortunately I am unable to understand the following question and answer. $\textbf{Question:}$ Let ...
4
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3answers
883 views

Expected Value of Local Maxima and Local Minima

Recently I came across this question: Given a random permutation of integers 1, 2, 3, …, n with a discrete, uniform distribution, find the expected number of local maxima. (A number is a local maxima ...
4
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0answers
55 views

Is it reasonable to think of the expectation of an infinite-dimensional vector?

Given a probability space $(\Omega, \mathcal{F}, P)$, a random vector is an $\mathcal{F}$-measurable mapping $X: \Omega \rightarrow \mathbb{R}^{k}: X(\omega) = (X_{1}(\omega), ...
4
votes
1answer
202 views

Triangular arrays and almost sure convergence of row averages

Suppose we have the triangular array $\{\{X_{in},i=1,\ldots,n\},n=1,2,\ldots\}$: $$\begin{array}{ccccc} X_{11}&&&&\\ X_{12}&X_{22}&&&\\ ...
4
votes
1answer
695 views

A sequence of random variables that does not converge in probability.

I was doing a problem about the converge of the sum of random variables which has two parts: Let $X_1, X_2 ,\dots$ be independent and identically distributed random variables with $E X_i = 0$, $ 0 ...
4
votes
1answer
168 views

Measurable structure on the space of probability measures

My advisor only half-jokingly mentioned that sometimes people like to consider the measurable structure on $P(X)$ where X is a locally compact polish space and $P(.)$ denotes the probability measures ...
4
votes
1answer
163 views

What is the right invariant $\sigma$-algebra for the Birkhoff ergodic theorem?

I have been reading stuff about ergodic theory, and I have encountered two versions of the involved "invariant sigma field". let the underlying probability space be $(\Omega,\mathcal{F},P)$, and let's ...
4
votes
1answer
490 views

Prove that it is a random variable iff it is constant on each partition

Let $\mathcal{G} = \{A_1, \ldots, A_n\}$ be a partition of a set $\Omega$, $\mathcal{F} = \sigma(\mathcal{G})$. Prove that $X : \Omega\to\mathbb{R}$ is a random variable if and only if it is constant ...
4
votes
2answers
547 views

Convergence of random variables (Durrett: Probability Theory and Examples)

I was working out some problems from Rick Durrett's Probability theory and Examples (2010 edition), when I came across a very unusual question(reproduced here ad-verbatim): If $X_n$ is ANY sequence ...
4
votes
2answers
977 views

Covariance of two random variables with monotone transformation

Suppose I know that, for two random variables $X,Y$, we have $$Cov(X,Y)\neq 0.$$ What happens if we take a monotone transformation of $X$; will the inequality persist? That is, say $f(.)$ is a ...