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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6 views

Does it hold $\sigma(X_1,\ldots,X_n)=\sigma(X_1,X_1-X_0,\ldots,X_n-X_{n-1})$?

Let $(\Omega,\mathcal{A})$ and $(\Omega',\mathcal{A}')$ be measurable spaces $X_1,\ldots,X_n$ be measurable with respect to $\mathcal{A}$-$\mathcal{A}'$ $Y_m:=X_m-X_{m-1}$ for $1< m\le n$ and ...
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1answer
4 views

Expectation of bernoulli trials

Could someone let me know if this looks correct? Let $X$ denote the number of successes in n bernoulli trials and let $Y$ denote the corresponding number of failures. Find an expression for ...
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0answers
8 views

Proving a.s. convergence for martingales

Let $ε_n, n > 1$, and $V_n, n > 0$, be independent random variables, with $P(ε_n = 1) = P(ε_n = −1) = 1/2$, $P(V_n = 1) = p_n, P(V_n = 0) = 1 − p_n$, for all n. Define $X_n$ inductively by $X_0 ...
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0answers
9 views

If two Brownian motion starts and end at the same points, can we say something about there difference?

Let $X$ and $Y$ be two standard Brownian motions with mean $0$ and variance $1$, both started at zero. If we know that \begin{align} X_n &= Y_n, \end{align} for some $n>0$, can we say ...
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0answers
6 views

Unbiased estimator geometric distribution of probability parameter

I am trying to identify an unbiased estimator of the probability parameter $p$ of the geometric distribution, but have only been able to find one - found here as Example 4. Can someone suggest another ...
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2answers
27 views

Isn't the statement of the Fatou's lemma somewhat problematic?

My lecture notes define $\int f := \int f^+ - \int f^-$ provided both $\int f^{\pm}$ are finite. And then the Fatou's lemma is stated in the following way: Let $f_n$ be a sequence of integrable ...
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1answer
8 views

If $B=(B_t,t\ge 0)$ is a Brownian motion and $(\mathcal{F}_t,t\ge 0$ is its generated filtration, then $X_t-X_s$ are independent of $\mathcal{A}_s$

A real-valued stochastic process $B=(B_t,t\ge 0)$ is called Brownian motion $:\Leftrightarrow$ $B_0=0$ $B$ has independent and stationary increments, i.e. ...
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1answer
18 views

If $X$ has density, when has $X\cdot I_A$ a density?

Let $(\Omega, \mathcal F, P)$ be a probability space, and $X$ be a random variable with some density function $f_X$. If $A \in \mathcal F$, then the indicator function $I_A$ has, as a discrete random ...
2
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1answer
16 views

Does it hold $\sigma(X_1,\ldots,X_n)=\sigma(X_n-X_1,\ldots,X_n-X_n)$?

Let $(\Omega,\mathcal{A})$ and $(\Omega',\mathcal{A}')$ be measurable spaces $X_1,\ldots,X_n$ be measurable with respect to $\mathcal{A}$-$\mathcal{A}'$ $Y_m:=X_n-X_m$ for $1\le m\le n$ I'm ...
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0answers
7 views

Transition rates and probabilities of a continuous markov chain

A certain type of component has two states: 0 = OFF and 1 = OPERATING. In state 0 , the process remains there an exponential amount of time with rate $ \alpha$, and then moves to state 1. The ...
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0answers
11 views

Mean Preserving Spread and concavity of the discrete function

Could someone help me with the understanding of the following thing? Consider a discrete distribution with pmf $p_k$ and its mean preserving spread (MPS) $p_k'$. Also let the set $a_1, \ldots, a_n$ be ...
1
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1answer
41 views

Probability confusing question

I saw this in my probability class past exam papers I saw the answer key but I still can't fully understand. I wish somebody can walk through this with me :) A company takes out an insurance policy ...
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0answers
9 views

Prove that an operator from $L^2(\Omega;C(s,T;\mathbb R^n ))$ into itself is well defined

I need an help proving the following estimate. First, we fix the notation. Let $L^2(\Omega;C(s,T;\mathbb R^n ))$ be the set of continuous and adapted processes $\{X_t:t\in [s,T]\}$ (valued in ...
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0answers
26 views

Joint pdf of N > 1 i.i.d. random variables isotropic if and only if variables are gaussian?

Is the Gaussian random variable with density given by $$f_X(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{x^2}{2 \sigma^2}}$$ the unique RV such that the joint pdf of $N > 1$ independent and ...
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0answers
16 views

Proving that a process is class DL

Let $(X_{t})$ be a stochastic process with $X_{t}\sim\mathcal{N}(\xi_{t},\sigma_{t}^{2})$ where $\xi_{t}\downarrow0$ and $\sigma_{t}^{2}\uparrow1/2$. What would be the most straightforward way ...
0
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0answers
14 views

Unbiased estimator for geometric distribution parameter p

I believe that the MLE of parameter $p$ in the geometric distribution, $\hat p = 1/(\bar x +1)$, is an unbiased estimator for $p$ and would like to prove it. So far, I have: $E[\bar x + 1] = E[\bar ...
0
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2answers
49 views

Probability of segment lying in circle

Given a circle of radius R: $x^2+y^2\le R$, find probability of horizontal segment with length $\frac{R}{2}$ lie whole inside this circle. Position of segment's center has uniform distribution in ...
0
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0answers
20 views

Almost sure closeness of random elements in a Hilbert space

Suppose I have a probability space $(\Omega,\mathscr{F},\mathbb{P})$ and a probability measure $\eta$ on a separable Hilbert space $H$ endowed with the Borel $\sigma$-algebra $\mathscr{B}$ arising ...
2
votes
2answers
40 views

Expected value with negative exponent

I am trying to solve identify the expected value of a statistic that involves a fraction. I have simplified the expression to: $E[\frac{1}{1+ \sum_i x_i}] = E[\frac{1}{1+ T}]$ However, I am not sure ...
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1answer
17 views

Order statistics when variables have different distributions

Let a,b and c be random variables, where a~U[0,8] and b~U[3,8]. Let c=max{a,b}, What is the mean of c? In general, let $(x_1, ..., x_n)$ be independently distributed in different supports. What is ...
0
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1answer
42 views

Approximating a joint pdf using normal density of two independent variables

I know that given these two random variables (which correspond to the $x$ and $y$ coordinates of a random walk after $n$ steps, their joint probability density function can be $approximated$ by a ...
0
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1answer
23 views

If $nE\|X_n-Y_n\|^2=o(1)$, why $\sqrt{n}(X_n-Y_n)=o_p(1)$ [on hold]

How to show that if we have $nE\|X_n-Y_n\|^2=o(1)$, then $\sqrt{n}(X_n-Y_n)=o_p(1)$? Here $X_n,Y_n$ are two random elements.
2
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1answer
22 views

Function of mean square continuous process

I have been asked to prove that, if $\{X_t\}$ is a ($n$-dimensional) mean square continuous process and $f:\mathbb{R}^n \rightarrow \mathbb{R}^d$ is a Lipschitz function, the process $\{f(X_t)\}$ is ...
2
votes
1answer
75 views

Is this chain irreducible and/or Aperiodic? What is its equilibrium mass function?

Consider a Markov chain with outcomes $\{0,…,n\}$ and transition probabilities $P_{i,i+1}=p$ $P_{i,i−1}=q$ for $1\le i\le n−1$ and $p+q=1$. Assume also that $P_{0,1} = P_{n,n−1} = 1$. Is this chain ...
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1answer
20 views

Application of martingale representation theorem

I am reading a proof that uses the following fact without proof (a bit strange): Let $W$ be a real Brownian motion generating the right-continuous, completed filtration $\{\mathcal{F}_t \}_{t \geq ...
0
votes
1answer
19 views

Binomial distribution tail inequality

Let $X \sim \mathrm{Bin}(n,p)$ does there exist $l$ ideally $l=f(n)$ such that $P(X<l)=o(1)$ in the limit $n\rightarrow \infty$? I'd be looking for the largest possible $l$.
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1answer
15 views

Application $\pi$-$\lambda$ lemma one-sided Markov shift

Let $(S_k^{\mathbb{N}},\Sigma_k^{\mathbb{N}},m,\tau)$ be the probability preserving transformation of the one-sided Markov shift, where $\Sigma_k^{\mathbb{N}}$ is the $\sigma$-algebra generated by the ...
2
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0answers
18 views

pythagorean theorem for conditional experience

Let G be a subsigma algebra and X is squareintegrable: => $ E[X²] = E[(X-E[X|G])²] + E[E[X|G]²]$ I know that this can be directly shown interpreting the conditional experience as a projection in ...
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0answers
28 views

Markov Chain States

Given a MC with five states $\{1,2,3,4,5\}$ and transition matrix \begin{bmatrix} 0.5& 0.5 & 0 & 0 & 0 \\ 0.75 & 0.25 & 0 & 0 & 0 \\ 0 & 0.25 & ...
3
votes
1answer
19 views

dependent “time change” of a.s. convergent random variables

Let $(X_n)$ be a sequence of random variables, s.t. $\frac{X_n}{n^p}\to X$ a.s. for some $p>0$. Now let $(Y_t)$ be a discrete stochastic process, s.t. $\frac{(Y_t)^p}{t}\to Y>0$ a.s. We only ...
1
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1answer
16 views

Augmentation of a Filtration

In class, we showed that Brownian Motion is a martingale with respect to the filtration $F_t = \sigma(B(s): 0\leq s \leq t) $. For a HW assignment, I need to show it's a martingale with respect to a ...
1
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1answer
25 views

Setting up the expected value for $x_t=\sin(2\pi U t)$.

We have the series $x_t=\sin(2\pi U t)$ where $t=1,2,3,\ldots$ and $U$ is uniform on the interval $(0,1)$. I have to find the expected value of $x_t$. I always thought that if $X$ is a continuous ...
1
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1answer
26 views

Show that $((N_t-t)^2-t)_{t \geq 0}$ is a martingale for a Poisson process $(N_t)_{t \geq 0}$

I am asked to show that if $N$ is a poisson process of intensity $1$, then: $X_t=N_t-t$ is a martingale. $X_t^2-t$ is a martingale. I have done the first part easily, using independence of ...
2
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0answers
14 views

Asymptotics of Laplace transform at minus infinity

I am interested in relating the asymptotic behavior of a function $f(t)$ for large values of $t$ with the asymptotic behavior of its Laplace transform $\hat{f}(s)$ for small values of $s$. In practice ...
0
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0answers
9 views

Calculating variance & expected value of a statistic with exponents

I am trying to calculate of a statistic: $Var(\frac{1}{1 + 1/n \sum_i x_i})$. Thus far, I have $=E[(1 + 1/n \sum_i x_i)^{-2}] - E[(1 + 1/n \sum_i x_i)^{-1}]^2$. How do you deal with exponents inside ...
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0answers
20 views

Mathematical statistics: Pollen dispersal directionality

What Information am I looking for? Think about a tree that is sending pollen all over the place. Because of wind, most pollen grain will go toward one direction. Imagine, we split the 2D area around ...
0
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1answer
27 views

Definition of an absolutely continuous random variable

Just what is the proper definition of an absolutely continuous random variable? It's supposed to be something like: $$\mathbf{P} (A) = \int_A f d \mu$$ for some Borel set $A$. But what is $\mu$? Is ...
1
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1answer
9 views

Proving that a process is a positive martingale

Let $X$ be the strong solution to the SDE $$ dX_t = \tanh X_t \,dt + dW_t, $$ where $W$ is a scalar Brownian motion defined on a probability space $(\Omega, \mathcal{F} ,\mathbb{P})$. (Such solution ...
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1answer
31 views

Geometric Brownian Motion [on hold]

I am new there. How can I calculate following expected value: $$E[X(s)\times X(t)]$$ where $X$ is Geometric Brownian Motion, i.e. $X(t) = exp[(\mu - 0.5\cdot \sigma^2)t + \sigma\cdot W(t)]$ ...
4
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0answers
26 views

Analysis of sorting Algorithm with probably wrong comparator?

It is an interesting question from an Interview, I failed it. An array has n different elements [A1 .. A2 .... An](random order). We have a comparator C, but it has a probability p to return correct ...
0
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0answers
24 views

What's the difference between a random variable and a measurable function?

I've tried to wrap my head around the measure theoretical definition of a random variable for a couple of days now. In his book Probability and Stochastics, Erhan Çinlar defines a measurable function ...
0
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0answers
19 views

expectation approximation

Note: You don't have to understand Approximation Algorithms to answer this Hello. I need to prove an algorithm approximation by using expectation. The algorithm takes $x_i \in {0,1,2}$ such that ...
3
votes
1answer
28 views

second Borel–Cantelli lemma (or converse result) application

I'm having issues with understanding how Borel-Cantelli lemma applies to the following exercise: If a coin is tossed infinite times, prove that the probability of getting 2 consecutive heads (or ...
0
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1answer
52 views

Compute Var(x=X1+X2+…+Xn)

Compute $Var(X_1+X_2+...+X_n)$ given $X_1,X_2...$ are iid.,$EX=\mu,Var(X)=\sigma ^2$,and $Var(N)=\sigma [n]^2$, N is a random variable of nonnegative integers independent with X, and my solution ...
3
votes
1answer
32 views

Pathwise solution of a stochastic integral equation, without stochastic calculus

Let $f$ be a Lipschitz continuous function from $\mathbb{R}$ to $\mathbb{R}$ and $W$ be a standard Brownian motion. I don't know any stochastic calculus (nothing about stochastic integrals, nothing ...
2
votes
1answer
16 views

Density of running supremum of Brownian motion until a stopping time

I am stuck on an exercise in my book: The question relies on the following fact: Let $M$ be a continuous, non-negative local martingale such that $M_0=1$ and $M_t \rightarrow 0$ almost surely as ...
2
votes
1answer
101 views

Probability of asymmetric random walk returning to the origin

Consider the random walk $S_n$ given by $ S_{n+1} = \left\{ \begin{array}{lr} S_n+2 & with & probability & p\\ S_n - 1 & with & probability & 1-p \end{array} ...
0
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0answers
28 views

Does the law of large numbers pin down the distribution of an infinite sample?

Imagine you draw (independently) an infinite amount of draws from a random variable with infinite support, and the strong law of large numbers applies. We know the average for sure will be equal to ...
2
votes
1answer
27 views

Expected value for sum of iid normal variables squared

Let $X_i$ be iid from a $N(\alpha, \alpha)$ distribution. I am trying to find $E[\sum_1^n X_i ^2]$ and thought that I would be able to transform the statistic $\sum_1^n X_i ^2$ into a chi-squared ...
0
votes
0answers
10 views

Lists of common sufficient statistics

Can someone suggest a source for common sufficient statistics for exponential families? For example, I'm looking for something more comprehensive than the Wikipedia page for sufficient statistics, ...