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Questions tagged [probability-theory]

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-distributions] for specific distribution functions, and consider [tag:stochastic-processes] when appropriate.

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8 views

CLT fro stochastic process.

Consider $\{X_i\}_{i=1}^{\infty}$ are i.r.v. with $\mathbb{E}(X_i) = \mu$ and $Var(X_i) = \sigma^2$. Then let $Z(t) = \sup \{n : \sum_{i=1}^{n} X_{i} \le t\}$. And we need to show that $\dfrac{Z(t) -...
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6 views

Expected value of a function of multiple random variables

I was wondering what is the definition of the expected value of a function of two or more random variables? And how does one show it is consistent? So if you have a random variable $z = g(x,y)$ ...
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Finding the limit $\lim_{n \to \infty} P_0^n$ for a European Cash-or-Nothing call option with $P=K^2\cdot \mathbf{1}_{\{S_T < K\}}$

Exercise : Let $K>0$. A European Cash-or-Nothing call option $P$ has the following pay-out profile : $$P=K^2\cdot \mathbf{1}_{\{S_T < K\}}$$ Let $P_0^n$ be the no-arbitrage value at time $...
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9 views

can a Markov chain have a periodic and transient state?

I want to say that in a Markov chain it is not possible for there to exist a state that is both transient and periodic. Here are the definitions I am working with. Let $P$ denote the transition ...
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12 views

Central Limit Theorem- Lapunow, Linderberg

I have a task: $(X_n)_{n>=1}$ are independent. $P(X_n=0)=1/n$ and $P(X_n=2n)=1-1/n$. Check the weak convergence $\frac{X_1+X_2+X_3+....+X_n}{n}-n$. I tried use the Lapunow theorem or Linderberg ...
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1answer
18 views

Question about an inequality in probability

Is the following inequality about probability true?$$\sum_{i=1}^{\infty}P(T\geq i)\leq \sum_{i=0}^{\infty}cP(T\geq ci)$$ Here $c>1$, an integer. $T$ is just a random variable taking non-negative ...
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1answer
16 views

Flip an unfair coin

An unfair coin has an probability of heads on a single flip $p=\frac 14$, the coin is flipped n times, and the probability of getting 2 heads is the same as the probability of getting 3 heads, what is ...
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6 views

If $\kappa$ is a transition kernel and $f$ is measurable, is $\int f(\;\cdot\;,\omega_2)\kappa(\;\cdot\;,{\rm d}\omega_2)$ measurable?

Let $(\Omega_i,\mathcal A_i)$ be a measurable space, $\kappa$ be a transition kernel with source $(\Omega_1,\mathcal A_1)$ and target $(\Omega_2,\mathcal A_2)$ and $f:\Omega_1\times\Omega_2\to[0,\...
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1answer
14 views

Proving $|\phi (t)|=1$ is sufficient for random variable to have lattice distribution

$\phi (t)=E[e^{itX}]$ is the characteristic function of a random variable $X$. $X$ has a lattice distribution if it's support is $\{a+nb,n\in Z, a,b>0\}$, for some $a,b>0$. ...
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Is it true that $\lim_{n \to \infty} {(P(\forall i,j\leq n \text{ } [X_i, X_j] = e))}^{\frac{1}{n}} = P(X_1 \in Z(G))$?

Suppose $G$ is a group. $\{X_n\}_{n = 1}^{\infty}$ is a sequence of i.i.d. random elements of $G$ satisfying the condition that $$\forall H \leq G \text{ } P(X_1 \in H) = \begin{cases} \frac{1}{...
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15 views

probability that a random variable is greater than a limit in given ordering of random variables

I am currently working on a modified version of the classic greedy algorithm for the 0/1 knapsack problem. Suppose that one has $N$ items with given weights and profits that are iid random variables ...
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1answer
25 views

Definition of ergodicity and ergodic process

I am confused by the definitions of ergodicity in wikipedia, see formal definition here which says that a measure-preserving transformation $T$ is ergodic if for every event $E$, $T^{-1}(E) = E$ ...
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1answer
34 views

$P(T_{1}<T_{2})=\frac{\omega_{1}}{\omega_{1}+\omega_{2}}$

Supposing that $ T_{1} $ and $T_{2}$ are independent and exponentially distributed, with parameters $\omega_{1} $ and $\omega_{2}$ respectively. Then, $$P(T_{1}<T_{2})=\frac{\omega_{1}}{\omega_{1}+...
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$\dfrac{1}{2d}\mathbb{E}N(x_{i}) \leq N(x^\prime_{i}) \leq \mathbb{E} N(x_{i})$.

1) Consider $n$ Points, $x_{1}, x_{2},...x_{n}$ distributed uniformly in $[0, 1]^d$. Term $d$ is the dimension. 2) Then, I construced a grid points $x^\prime_{1}, x^\prime_{2},...x^\prime_{n}$ that ...
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Gaussian concentration of measure, equivalent definitions

I need some help going between two equivalent definitions. First some notation : $\bullet$ For $A\subseteq \mathcal{X}$ and $r>0$ define what is called the r-blowup of $A$ as \begin{equation} ...
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14 views

Countable sum of point processes

Let $(\mathbb{X},\mathcal{X})$ be a measurable space. A point process is defined as a measurable mapping $$\eta : (\Omega, \mathcal{F}) \rightarrow (\textbf{N}, \mathcal{N}),$$ where $\textbf{N}$ ...
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1answer
23 views

Are $P(\min(X_iY_i)-\min( X_i) \min (Y_i) \leq x)$ and $P(\min (X_iZ_i)-\min (X_i) \min (Z_i)\leq x)$ equivalent?

Imagine I have three independent random variables $X, Y, Z$ and each with positive support; Now imagine I have an iid sample of each random variable, namely $X_1,...,X_n$, $Y_1,...,Y_n$ and $Z_1,...,...
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11 views

Proof of relation of Gamma and Beta distribution without using Jacobian

If X1 and X2 independent random variables follow Gamma Distribution, can we prove Y= X1/(X1+X2) is a Beta Distribution without using the Jacobian Change of Variable method? In our course, we haven't ...
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1answer
17 views

Bayesian updating with fair and unfair coins.

Given a list of coin tosses with 100,000 outcomes, suppose you know that they were generated by either a fair or a biased coin with a 51% chance of heads. How do you determine which coin it was ...
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17 views

Multi-dimensional integration by parts

Here's my problem: Given $X=(X_1,X_2)$ is a centered Gaussian random vector, i.e., $X\sim \mathcal{N}(0,C)$ and the density of $X$ is given by $$ p(x)=p(x_1,x_2)=\frac{1}{(2\pi)^{m/2}|C|^{1/2}}\exp\{...
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1answer
35 views

Bounds for the derivative of inverse Mills ratio of standard normal distribution

I'm trying to find bounds for the derivative of the inverse Mills ratio $\lambda(x)=\dfrac{\phi(x)}{\Phi(x)}$: $\lambda^{\prime}(x)=-\lambda(x) (x+\lambda(x))$ While my matlab numerical results ...
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13 views

Show that there exists a probability measure on $\mathbb{N}^{\mathbb{N}}$

For each $n\geq 1$ and each choice $u_1, \cdots, u_n \geq 1$, define nonnegative real numbers $p_n(u_1,\cdots, u_n)$ such that $\sum_{k} p_1(k)=1$ and for $n\geq 1$: $\sum_k p_{n+1}(u_1,\cdots, u_n, ...
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10 views

Minorization Condition for a Truncated Normal Distribution

Suppose $$t_1, \ldots, t_n | \beta, \sigma^2, y \sim \text{TN}(\lambda(y_i-x_i^T\beta),\sigma^2;0;\infty),$$ where $\lambda \in \mathbb{R},$ and $x_1,\ldots, x_n$ and $y_1,\ldots,y_n$ come from the ...
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1answer
29 views

Conditional Expected Value- maximum and minimum

I have to compute $E(max(X,Y)|min(X,Y)=t)$ where X and Y are independent and have uniform distribution on $[0,1]$. I made this task: $E(max(X,Y)|min(X,Y)=t)=f(t)$ $E(max(X,Y)\mathbf{1}_{min(X,Y)>...
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2answers
22 views

Interpretation of Expected value of Random Variable

When the sample space in question, is inherently quantifiable and the outcomes have quantifiable relations to each other the expected value is easy to understand, such as the expected value of 3.5 for ...
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0answers
9 views

Expected value of loss function

Problem Suppose $S=\{(\mathbf{x}_1,y_1),\cdots, (\mathbf{x}_n,y_n)\}$ is a set of i.i.d examples and we have a generic loss function $\ell(\cdot,\cdot)$, I am wondering if $\ell(h(\mathbf{x}_1),y_1),...
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15 views

Understanding the proof of inversion formula for density using characteristic function

The formula is: $f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-i\lambda x}\hat{f}(\lambda)d\lambda$ where $\hat{f}$ is the characteristic function, $f$ is continuous bounded on $R$ and both $f,...
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29 views

Prove that $P(\exists n\in N:S_n=n-2008)=1$

Let $X_n$, $n=1,2,...$ be independent random variables. $P(X_n=1)=\frac{1}{2}=P(X_n=-1)$ $S_n=\sum_1^nX_k$, $T=inf[n:S_n=n-2008]$ a) Prove that $P(\exists n\in N:S_n=n-2008)=1$ b) Prove $ET\...
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2answers
38 views

$X$ and $Y$ are independent if and only if $\textrm{Cov}(f(X), g(Y)) = 0$ for all $f,g$ measurable functions?

Let $X$ and $Y$ be real-valued random variables. Does it hold that $X$ and $Y$ are independent if and only if $\textrm{Cov}(f(X), g(Y)) = 0$ for all $f,g$ measurable functions? One direction is ...
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21 views

Confidence interval by Lagrange

Lagrange was the first mathematician that talked about confidence intervals? His contributions are relevant for the formulation of Maximum likelihood estimator?
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Probability of two first symbols in the string [on hold]

String is made from 33 symbols: A, B, C, D. We know that 30 symbols in the string are A. What's the probability that first two symbols will be A?
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2answers
28 views

link between cdf and expectation - learning the basics

Let $X$, $Y$ be random variables. Assume that the cumulative distribution function $F$ of $X$ and $Y$ satisfy the following relation: $$F_X(x) > F_Y(x) \hspace{2mm} \forall x$$ Is it possible to ...
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1answer
67 views

What is the largest possible expectation of difference between two i.i.d. random vectors in the separable Hilbert space?

Suppose $M$ is a compact subset of the separable Hilbert space $l_2$. Suppose, $X$ and $Y$ are i.i.d. random vectors with support in $M$. What is the largest possible $E \| X - Y \| $? Suppose $M$ is ...
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2answers
29 views

Show $B(t)/t\to0$ a.s. for $t\to\infty$ where $B(t)$ is SBM

Suppose $B(t),t\in[0,\infty)$ is Standard Brownian Motion. I know that $\exp(\alpha B_t-\dfrac{\alpha^2t}{2})$ is a martingale. I want to use this to show that almost surely, $B(t)/t\to0$ as $t\to\...
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2answers
92 views

Probability that $\vert x\vert +\vert y\vert +\vert z\vert +\vert x+y+z\vert=\vert x+y\vert +\vert x+z\vert +\vert y+z\vert$

Real numbers $x, y$, and $z$ are chosen from the interval $[−1, 1]$ independently and uniformly at random. What is the probability that $$\vert x\vert +\vert y\vert +\vert z\vert +\vert x+y+z\vert=\...
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14 views

Non-Uniqueness of the generative model of a stochastic process

To explain my question, I will start from a simple example. Suppose a binary random variable $X \in \{1, 0\}$ is generated by the following two steps. $P(Y = 1) = \frac{1}{2}$, $P(Y = 0) = \frac{1}{...
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1answer
24 views

Are the projections along orthogonal direction of multivariate normal distribution with diagonal covariance matrix independent?

I'm taking a probability class and my prof used the following theorem IIRC. Let $g\sim\mathcal{N}(\mu,\Sigma)$ where $\Sigma$ is diagonal( I don't know if this condition is necessary) and $\langle u,...
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33 views

MVU estimator for $e^{-\lambda}$ for a given Poisson Distribution

Give, $Y_1, Y_2, ... Y_n$ are IID random variables, each having a Poisson Distribution with parameter $\lambda$. Is there any unbiased estimator of either (i) $e^{-\lambda}$ or (ii) $e^{-2\lambda}$ ...
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1answer
27 views

How to show that the space of probability measures on $\mathbb{R}$ is separable under Lévy metric

The Lévy metric between distribution functions $F$ and $G$ is given by: $$\rho(F,G) = \inf\{\epsilon : F(x-\epsilon)-\epsilon\leq G(x)\leq F(x+\epsilon)+\epsilon\}.$$ Another way to write this is: ...
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0answers
26 views

How to derive the expectation of $\max(X,0)$ given the mgf function of $X$?

I got a moment-generating function for an r.v. $X$, $$M_X(t)=\frac{i\lambda\mu}{(\lambda +t)(\mu-t)},\forall t\in(-\lambda,\mu)$$ where $i$, $\lambda$, $\mu$ are non-negative constants. I know the ...
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0answers
25 views

Problem about weak convergence using continuous bounded functions

The question is from Billingsley. $X_n \in \{\gamma_n+k\delta_n; k\in N\}, \delta_n>0$. Suppose $\delta_n\rightarrow 0$ and $k_n$ is an integer varying with n s.t. $\gamma_n+k_n\delta_n\rightarrow ...
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2answers
51 views

Probability - Too simple but confusing - Why two different approaches for similar problems?

There are two questions and the approach seems to be different. a) Probability of choosing 3 queens from a deck of cards? Ans: 4/52 * 3/51 * 2/50 However, if the question is how many diamonds ...
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1answer
16 views

Varying definitions of a martingale

I frequently see that, over some filtration $\mathscr{F}_{n}, \{X_{n}\}$ is defined as a martingale if $E[X_{n+1}|\mathscr{F}_{n}]=X_{n}$. Sometimes, however, I see this extended to $E[X_{n+s}|\...
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3answers
42 views

Constructing a weird probability measure

Is it possible to construct a probability space $(\Omega,\mathcal{A},\mathbb{P})$ such that $\mathcal{A}$ is uncountable, there are uncountable events with probability $>0$ and there are also ...
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1answer
15 views

How many events can a space $\Omega$ contain in order for the elementary, discrete definition of expectation to still be valied

Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space and $X:\Omega\rightarrow \mathbb{R}$ a random variable on it. Now the definition of expected value would be $$\mathbb{E}[X]=\int _{\...
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1answer
17 views

Poisson Process Conditional Expectation

Given $X_t$ a Poisson process such that $\lambda = 1$ find $E[X_1\mid X_2]$ and $E[X_2\mid X_1]$. The first one is pretty straight forward since we have $E[X_2 - X_1] = E[X_1] = 1$ so then we get $E[...
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1answer
27 views

Property of median of probability distributiom

Suppose that a random variable $\mathbb{X}$ has density $f$ and a unique median $m$ . Suppose that $b$ is any real number. Show that $\mathbb{E(|X − b|) = E(|X − m|) + 2 \int_ b^ m (b − x)f(x)dx}$ , ...
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1answer
27 views

Calculate $EX_{\tau}$ where $\tau=[inf\space n: \space X_n=2 \space or\space X_n=3]$

I came up with this task myself so it might be blurry, actually I changed a bit another exercise which was easy, but I'd like to know the way of coming up to an answer if it was like that. Let $X_n$ ...
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1answer
29 views

Is Supremum of sequence of bounded random variables a random variable?

I was thinking about the following situation : Suppose we have ${\{X_n\}}$ is a sequence of bounded random variables . Is it true that $\mathbb{lim \ sup} X_n$ is also a random variable ? ( I get a ...
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0answers
23 views

Showing $f_n(f(e^{-C_{n+1}^{-1}})^z) = f_{n+1}(e^{-zC_{n+1}^{-1} })$ where $f_n$ is the pgf of a Galton-Watson process $Z_n$

I want to show $$f_n(f(e^{-C_{n+1}^{-1}})^z) = f_{n+1}(e^{-zC_{n+1}^{-1} })$$ In this case $Re(z)\ge 0$, $(C_n)$ is a sequence of constants and $f_n$ is the probability generating function of $Z_n$. ...