# 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 ...

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### If $P(X<Y)=P(X<g(Y))$ then what could be the form of $g$?

Let $X$ and $Y$ are two continuous random variable and $$P(X<Y)=P(X<g(Y)),$$ for some convex function $g$. Is it true that $g$ will always be a linear function?
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### limit of average of independent, but not identically distributed r.v.

Let $\{X_i\}$ be a collection of independent r.v., but with distribution dependent on index $i$, such that $P(X_i=2^i)=2^{-i}$ and $P(X_i=0)=1-2^{-i}$ for $i \in \mathbb{N}$. What can I say about ...
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### CLT version for $ER_n(p)$ graphs

We defined the Erdôs- Rényi graph as follows: $ER_n(P)$ is the random graph with vertex set $[n]$ where each pair $\{u,v\}$ of vertices is added to the edges set $E$ independently with probability ...
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### How to prove that $cov(f(X_1, X_2, \ldots ,X_n), g(X_1, X_2, \ldots ,X_n)) \geq 0$ for $X_1, \ldots, X_n$ independent and $f,g$ increasing?

I read in a talk that a consequence of the FKG inequality is that: $$cov(f(X_1, X_2, \ldots ,X_n), g(X_1, X_2, \ldots ,X_n)) \geq 0$$ for $X_1, X_2, \ldots ,X_n$ independent and $f,g$ increasing ...
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### Family of decompositions of a probability space and sigma algebra generated by a discrete random variable.

While reading the textbook "Martingale Methods in Financial Modelling" by Musiela and Rutkowski I am puzzled with a new definition (i.e. "the family of decompositions") that I never encountered and ...
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### Problem with change of variables (with regard to kernel density estimation)

I'm trying to understand why \begin{equation*} \begin{split} E(R(\hat{f}'')) & = \frac{1}{nh^6} \int \int K'' \left( \frac{x-y}{h} \right)^2 f(y) dy dx \\ & \quad + \frac{n(n-1)}{n^2 h^6} ...
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### Does this self-conjured RV converge almost surely?

I thought of this example in hopes of helping me understand almost sure convergence a little better. So, if you could add any additional (relevant) details in your response I would greatly appreciate ...
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### Showing that $E[X|G_1,G_2]=E[X|G_1]$, where $\sigma(X,G_1)$ and $G_2$ are independent

Suppose that $X$ is an integrable r.v. on probability measure space $(\Omega,F,P)$. Show that $E[X|G_1,G_2]=E[X|G_1]$, where $G_1,G_2$ are sub $\sigma$-algebras and $\sigma(X,G_1)$ and $G_2$ are ...
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### High school Math: confusion about the basic probability

I am confused about the following two scenarios: Out of a bag of 3 apples and 3 oranges, you pick 2 items. 1) What is the probability that you will have 2 apples? 2) What is the probability that ...
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### Expectation and the Survival Function: Measure Theory

I have an example from class notes that I do not understand and would appreciate some clarification. Particularly, I haven't found a direct explanation online or in my text with regards to the limits ...
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### Monotone convergence implies $\mathbb{E}\sum X_n = \sum \mathbb{E}X_n$?

My professor stated the following implication made from the Monotone Convergence Theorem: \begin{align}\mathbb{E}[\sum X_n] = \sum \mathbb{E}[X_n] \end{align} Up till now I have been assuming the ...
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### If $P(X = x) = 0$ for a continuous RV $X$, then isn't that it is impossible to observe any data at all?

For example, if I let $X$ be the weight of a dog, then I weigh my dog and his weight is 10 lb. Theorem says the probability of observing this 10 lb is zero, i.e $P(X=10) = 0$. However, I DO observe ...
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### $X_T = \lim_{n \to \infty} X_{T \wedge n}$ if X is a supermartingale and T is a finite a.s. stopping time?

Given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}, \mathbb{P})$, let $X = (X_n)_{n \geq 0}$ be a $(\{\mathscr{F_n}\}, \mathbb{P})$-supermartingale and $T$ be a finite ...
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### Exercise 1.2.4 (From Grimmett and Stirzaker)

This is yet another problem, where I have run into trouble. Let ${F}$ be a ${\sigma}$-field of the subsets of $\Omega$ and suppose that $B\in{F}$. Show that $G=\{A\cap{B}:A\in{F}\}$ is a ...
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### Compute $E(Y\mid X=m)$ [closed]

The random vector $(X,Y)$ has the following joint distribution $$P(X=m, Y=n) = {m\choose n} \frac{1}{2^m}\frac{m}{15}$$ where $m=1,\ldots,5$ and $n=0,\ldots,m$. Compute $E(Y\mid X=m)$
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### Odds of winning a lottery

My question will be similar to the one asked here Odds of Winning the Lottery Using the Same Numbers Repeatedly Better/Worse? I understand that each lottery is an independent event so selecting any ...
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### Probability of a set that has infinite Lebesgue measure

Forgive, for the title didn't know how to name this questions. Please change to something better. Let $B_1(n)$ denote a unit ball around $n\in \mathbb{Z}^{+}$. Suppose that for every $n$ there exists ...
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### Where is my error in finding the edgeworth expansion of the binomial distribution?

Let $B_n$ be a standardized binomial distribution. To illustrate the Edgeworth Expansion I made a plot showing $f(x)=P(B_n \le x)-\Phi(x)$ and $g(x)=\frac{p-q}6 (x^2-1) \phi(x) \frac{1}{\sqrt{npq}}$ ...
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### Show that if $a_n(X_n-X) \overset{\mathcal{D}}\to Z$ then $X_n \overset{P}\to X$.

Let $(a_n)\subseteq \Bbb{R}$ be a sequence such that $a_n \to \infty$. Let $(X_n)$ be a sequence of random variables such that $a_n(X_n-X) \overset{\mathcal{D}}\to Z$ fore some random variables $X$ ...
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### Expected value, E(X), where does the “x” in the formula come from?

So if we have the formulas $$Eg(X) = \int \cdots \int g(x_1,\dots,x_k)f_X(x_1,\dots,x_k) \, dx_1\dots dx_k$$ or, for discrete, $$Eg(X) = \sum_{x\in M} g(x)f_X(x)$$ with $X:\Omega\to\mathbb{R}^k$ a ...
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### Renewal theory distribution of residual lifetime in delayed case with delay distribution of $F_0$

Show that $H(t,x)= F_0(t+x)-F_0(t) +F_0*H(t,x)$ where $H(t,x)$ is the probability that at least one renewal epoch between $t$ and $t+x$.Here $F_0(x)=\frac{1}{\mu}\int_0^x(1-F(x))dx$ where F is the ...
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### Questioning convergences in respect to $0$: almost surely, in probability, in distribution of random variable with $\varphi$ density given.

Questioning convergences in respect to $0$: almost surely, in probability, in distribution of random variables with $\varphi_{n}(x)$, density given. \varphi_n(x)=\frac{n}{\pi(1+n^2x^2)}, x\in ...
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### Asymptotic variance for Markov chain

Let $\hat{\mu_t}(f)=\frac{1}{t} \sum_{i=1}^{t} f(X_i)$ is an estimator for a finite, irreducible Markov chain $\{X_t\}$ with its stationary distribution $\pi$. In addition, assume the estimator is ...
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### Proving $P(\bigcap A_i)\geq\sum P(A_i)-(n-1)$

I need to prove $P(\bigcap A_i)\geq\sum P(A_i)-(n-1)$. I tried playing with $\sum P(A_i)\geq P(\bigcup A_i)\geq\sum P(A_i)-\sum_{1\leq i\lt j\leq n}P(A_i\cap A_j)$, but didn't get anywhere. A hint ...
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### Writing an event as a disjoint union of events - $\bigcup A\cap B_i$ or $\bigcup A|B_i$

I am having trouble with two ways of writing an event as a disjoint union of events - as the union of $A\cap B_i$ or $A|B_i$. For example: Say we have a bag with $a$ blue balls, $b$ red balls and ...
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### Limit of expectation of L2 norm unbounded implies function ill defined

Given a random process $A(t,x) := \lim_{N\rightarrow\infty} A^N(t,x)$ where $x \in (0,1)$ and $t \geq 0$ and given that \mathbb{E}\left[ \int_0^1 A^N(t,x)^2 dx \right] = tN ...