# Tagged Questions

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 [tag:probability-...

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### A simple betting game

Consider the following betting game: Two players each have 100 cents to bet. If one player bets more than the other then that player gains a point and the other player loses a point. The goal of the ...
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### Trace of power of random matrix / sum of random variables with semicircle distribution

I want to calculate the expectation value for the trace of the $m$-th power of the $n\times n$ adjacency matrix $A$ of a large Erdos-Renyi random graph (without self-coupling, i.e., all diagonal ...
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### Why the moment-generating function, rather than the characteristic function?

I'm wondering why the moment-generating function is worth discussing (say, in basic probability courses, or in textbooks, rather than research), when the characteristic function appears to completely ...
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### Conditional expectation on Function space

This question is from a notation in section 13.4 of the book "Linear and Nonlinear Filtering for Scientists and Engineers" By N U Ahmed In this section, the author is deriving the Zakai ...
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### Skorokhod representation proof understanding

The conditions 1-3 were that $F$ is right continuous, $F$ is increasing and the limit of $F(x)$ as $x$ tends to $\infty$ is $1$ and $0$ if $x$ tends to $-\infty$. I don't follow the idea of this ...
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### If $\mathcal{M}_1(E)$ is compact, then $M \subset \mathcal{M}_1(E)$ is exponentially tight?

Let $E$ be a metric space, $\mathcal B$ the Borel $\sigma$-algebra on $E$ and $\mathcal M_1(E)$ the set of probability measures on $(E,\mathcal B)$. I'm interested in when and why the following ...
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### Conditional Expectation.

Let $X$ be a random variable in $L^2(\Omega, \Sigma, P)$ and $\mathcal G$ a sub-$\sigma$-algebra of $\Sigma$. Prove that $E[(X-E[X\mid\mathcal G])^2] \le E[(X-E[X])^2]$. As conditional expectation ...
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### Chernoff vs Berry-Esseen

Let $X_1,X_2,\ldots,X_n$ be iid random variables with mean $\mu > 0$, variance $\sigma^2$ and $X_i \in [-1,1]$ for all $i$. Let $S_n = \sum_{i=1}^n X_i$. Suppose $l < h < 0$. I am trying to ...
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### Tauberian theorems in queing theory

I'm trying to use Tauber's theorem below (Feller 1971, chapter XIII.5) "Let U be a measure with a Laplace transform $\omega(\lambda)$ defined $\forall \lambda >0$ and $t,\tau>0$ s.t. $t\tau=1$, ...
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### Norris exercise: Showing $P_0[\text{no return to}\ 0]=6/\pi^2$

Consider exercise 1.3.4 of Norris' Markov Chains. The question is as follows: Let $\{X_n\}_{n\geq 0}$ be a Markov Chain with state space $S=\{0,1,2,\dots\}$. Suppose the transition probabilities ...
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### Poisson distribution transformation method

Random independent variables $x_1, x_2 \sim \ \ \text{poisson(}\lambda )$ $y=x_1+x_2$ $z=x_1-x_2$ The possible density function $f(y,z)=?$ by using Inverse transformation method. Note that I ...
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### Random process involving CDF and PDF of standard normal.

Im studying old exams and found this one: Let $\Phi(x)=\int_{-\infty}^{x} \frac{1} { \sqrt{2\pi} } e^{-y^2 /2} dy$ and $\phi(x)=\Phi^\prime(x)=\frac{1} { \sqrt{2\pi} } e^{-x^2 /2}$ be the ...
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### Necessary and Sufficient Conditions for a CDF

This is an attempt to prove Theorem 1.5.3. in Casella and Berger. Note that the only things that have been proven are really basic set-theory with $\mathbb{P}$ (a probability measure) theorems (e.g., ...
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### Bayesian information criterion from measure theoretic point of view?

Bayesian information criterion (BIC) is well known and it is derived from the maximizing the posterior density function which is equivalent to solving the marginal likelihood integral. My question is: ...
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### Are two dot products of a random variable vector independent?

Let $w,v$ be two different vectors in the finite vector space $Z_p^m$ over $Z_p$ where $p$ is prime. Let $u$ be a vector chosen uniformly at random from $Z_p^m$. Are the random variables $u \cdot w$ ...
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### A sequence converging to 0 in probability times a sequence bounded in probability

I'm trying to prove the following from Lehman's "Elements of Large Sample Theory" Lemma 2.3.1: If the sequence $\{Y_n, n=1,2,\ldots\}$ is bounded in probability and if $\{C_n\}$ is a sequence of ...
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### Suppose $X$ and $Y$ have joint density $f(c,y)=6xy^2$ for $x,y\in(0,1)$. What is $P(X+Y<1)$?

My attempt: $$f(x,y)=6xy^2$$ $$P(X+Y<1)=\int_{0}^1\int_{0}^{1-y} 6xy^2dxdy$$ $$=\frac{1}{10}$$ However, the answer was supposed to be $\frac{3}{5}$, and the bounds on the integral were supposed ...
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### The boundedness of a certain sequence of expectations

In Bálint Tóth's paper, "No More Than Three Favourite Sites for Simple Random Walk", while proving one of the many technical lemmas in his theorem's proof, he makes the following claim: suppose for ...
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### Why the case of independence of random variables is more important than any other specific type of dependence?

Maybe a stupid question but why is the case of independence of, say, two random variables $X$ and $Y$ is in some ways considered to be more central'' or more important than any other type of fixed ...
Let $d_j$ be i.i.d. random variables, with$$P(d_j = 0) = P(d_j = 1) = {1\over2}.$$Define$$X = 0.d_1d_2 \dots = \sum_{j = 1}^\infty {{d_j}\over{2^{j}}}.$$I know how to show that $X$ is uniformly ...
Let $X_n \ge 0$ be a sequence of random variables (which are not necessarily independent) satisfying the following conditions: $E(X_n) = \mu_n$ where $l \le \mu_n \le u$ for some constants \$l, u <...