Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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Convergence in Probability and in Quadratic Mean for a sequence of random variables

I have been trying to determine whether a sequence of random variables, $X_1,X_2,\ldots,X_n$, such that $$P\left(X_n= \frac{1}{n}\right)=1-\frac{1}{n^2}\\ \text{and}\\ ...
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40 views

One basic probability theory problem

Let $\beta_{n,i} = \sum_{j=1, j\neq i}^{N} \frac{\delta_j(n)}{\delta_i(n)}\nabla_j{J(X_n)}$ where $X_{n+1} = X_n + f(X_n, \delta_i(n), 1 \leq i \leq n)$ and $\{\delta_i(n): 1 \leq i \leq n, n \geq ...
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47 views

Brownian motion motivation of construction

I have read Stochastic Differential Equations by Bernt Oksendal It constructs Brownian motion by Kolmogorov extension theorem by consider $p(t,x,y)=(2\pi t)^{-n/2} e^{- \frac{|x-y|^{2}}{2t}}$ Why ...
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88 views

Large deviation for gaussian distribution

Given a random variable $X$ that is $N(0,1)$ distributed and a sequence $(X_i)$ of iid distributed $N(0,1)$ random variables(copies of $X$) and I am supposed to calculate $P(X \ge 5)$ by means of ...
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84 views

The expected number of visits before hitting zero in simple random walk

I am learning Markov chains and encounter the following problem: Suppose in simple random walk, we start from state k. What's the expected number of visits to k before we hit 0? The book does not ...
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57 views

If X is an uncountable set, will countably many random draws be dense in X?

I'm interested in how large samples would approximate a distribution. Suppose $X$ is a set of finite dimension of uncountable cardinality. If I take countably many i.i.d. random draws from the set ...
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42 views

Sum of bounded in probability random variables

I'm self-studying probabilistic order notation, and I wanted to show some properties to get used to it. But now I'm having trouble showing that the sum of two random variables that are bounded in ...
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54 views

Representation of Stochastic Integrals as Lebesgue/Bochner Integrals

Just as the Riemann–Stieltjes integral can be equivalently defined as a Lebesgue integral with the corresponding Lebesgue–Stieltjes measure, I am looking for the corresponding results for the ...
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55 views

Poisson/ jump process distribution for process $z(t)=2t+B(t)+\sum_{k=0}^{X(t)} J_k$

For the process: $z(t)=2t+B(t)+\sum_{k=0}^{X(t)} J_k$, where $X(t)$ is a poisson process with paramater $\lambda$, and: $J_k$ are i.i.d . random variables (jumps). $B(t)$=brownian motion. I want to ...
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18 views

Distribution functions of a probability measure on a probability space $(\mathbb{R},\mathcal{B})$

Let $F$ denote a distribution function of a probability measure $P$ on a probability space $(\mathbb{R},\mathcal{B})$, where $\mathcal{B}$ denotes the Borel $\sigma$-algebra on $\mathbb{R}$. Given ...
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15 views

action of transition operator on function

Let $P$ be the transition operator of a markov chain with discrete time and discrete state space $X$. The action of the transition operator on a function $X \to \mathbb{R}$ is defined by $Pf(x) = ...
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94 views

Derivation of the negative hypergeometric distribution

Suppose we've given an urn which contains $R$ red and $W$ white balls. These balls are drawn randomly from the urn and are not placed back. Let $X:=$ number of attempts, before we've drawn at least ...
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93 views

Girsanov theorem conditions

If we have an adapted function $f(t)$ such that $\int_0^t f(s)ds\,<\infty$, then the Girsanov exponent can be defined: $$ Z(t):=\exp\left( \int_0^t f(s)dW(s) - \frac{1}{2} \int_0^t ...
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1answer
87 views

Conditional Probability and the Complement Rule

Does this identity hold for all events? $$ P(A|B) = 1-P(A'|B) $$ Logically speaking, if the probability of $A$ given $B$ occurred is $X$, shouldn't the probability that $A$ does not occur, $A'$, ...
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101 views

Almost sure convergence of a sequence of random variables

Once again I've encountered a problem, which might not be difficult: I'm given a sequence of random variables $ (X_n) $, each with density function $g_n(x) = nx^{n-1} \textbf{1}_{(0,1]} $. I am to ...
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1answer
41 views

Show that a stochastic process is a martingale

Use Ito's formula to prove that the following stochastic process is a $\{\mathcal{F_t}\}$- martingale. a) $X_t = e^{\frac{1}{2}t}cosB_t \ \ \ \ (B_t \in \mathbb{R})$ So ...
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126 views

Convegence of regularized sequence in $L^2$

Let $(\rho_n)_{n \geq 0}$ be a standard regularizing sequence on $\mathbb R$. Let $P$ be a probability measure on $\mathbb R$ such that the sequence $(P*\rho_n)_{n \geq 0}$ is bounded in $L^2$. Then, ...
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1answer
74 views

Convergence almost sure pointless?

A very common type of convergence in probability theory is 'almost sure convergence'. I don't understand why this type is used at all. In principle, we should always be able to substitute it by a ...
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194 views

PDF of sum of two random variables

Assume an $n$ dimensional random variable $U$ that is uniformly distributed in the volume of an $n$-sphere with radius $R$. Assume another $n$ dimensional random variable $N$ that is distributed ...
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1answer
117 views

Convergence in probability of iid normal random variables

Let $X_1, X_2,\ldots$ be a sequence of iid normal random variables with zero mean and unit variance. I read the following as a trivial example: (1) $X_n \to X_1$ in law, (2) $X_n \not\to X_1$ in ...
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12 views

Weighted variance of a small sample

I am trying to calculate the variance of a small sample. I have the data: ...
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1answer
37 views

Types of Convergence (Random Variables)

Suppose that for every $n\ge 1$, the law of $X_n$ is given by $P[X_n=n^2]=\beta_n$ and $P[X_n=0]=1-\beta_n$, determine if $(X_n)_{n\ge 1}$ converges in probability, in $L^1$ or almost sure to zero, ...
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41 views

How many ways to represent a probability density function?

I have read accidentally in a book this sentence: " ... consider a random sample $X_1, X_2, \ldots, X_n$, each $X_i$ having probability distribution $f(x)dx$. Thus, we have $$\mathbb{P}(X_1\in ...
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20 views

The distribution of minmax and maxmin deviations of a Random variable

Let $X_1,X_2,X_3,......,X_n$ be $n$ independently and uniformly distributed random variables in the interval $[a,b]$. Further let $P=\min \{X_i,i=1,2,3..,n\}$ and $Q=\max\{X_i,i=1,2,3..,n\}$. ...
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38 views

What is the optimal strategy?

There are $m+n+1$ cards numbered $1,2,\ldots m+n+1$. Participants A and B respectively get $m$ and $n$ cards. Meanwhile, they only know what they get. The remaining card is face down on the desk. ...
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1answer
70 views

(Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$

The purpose of this question is to complete my personal exposition on the rigorous proof of Ito's lemma. I have consulted more than half a dozen mathematical finance texts and not a single one, for ...
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1answer
42 views

Random variables $x_i$ with $\lim_{k\to\infty}\frac1k\sum_{i=1}^kx_i=0$

I am looking for a sequence $(x_n)_{n\in\mathbb N}$ of random variables such that the sequence hasn't any expected value and $\lim_{k\to\infty}\frac1k\sum_{i=1}^kx_i=0$. I thought about using a ...
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53 views

Definition of conditional expectation

We have the following definition for a conditional expectation: Let $X: \Omega \rightarrow \mathbb{R}$ be a randomn variable on ($\Omega, \Sigma, \mathbb{P})$. Let $F \subset \Sigma$ be a sub ...
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14 views

Dividing the student's t-distribution into independent and identically distributed random variables

It is well known (see [1], [2], etc.) that the student's $t$-distribution is infinitely divisible, that is, it can be expressed as the probability distribution of the sum of an arbitrary number of ...
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27 views

Weighted random walk in 1-dimension

Suppose we have random walker on a line, he can only stay on sites which are, say, a distance $a$ from each other. At each step he can go left or right. Every time he steps on a site, makes the ...
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39 views

$\sup_nX_n<\infty$ almost surely iff $\sum_nP(X_n>A)<\infty$

Let $(X_n)_{n\in\mathbb N}$ be a sequence of independent random variables. Show that $\sup_nX_n<\infty$ almost surely iff there exists $A>0$ such that, $\sum_nP(X_n>A)<\infty$ By ...
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16 views

Probability question on mutual exclusiveness

Okay, it's like this: Let A, B and C be events such that P[A|C] = 0.05 and P[B|C] = 0.05. I was just wondering, should the events A and B be looked at as mutually exclusive, as in only one or the ...
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43 views

How do I prove the special case of the central limit theorem?

Let $(X_n)$ be an i.i.d. sequence such that $\mathbb P(X_1=1)=\frac{1}{2}+\varepsilon$ and $\mathbb P(X_1=-1)=\frac{1}{2}-\varepsilon$ for some $\varepsilon\in(0, \frac{1}{2})$. I'd like to show that ...
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36 views

Inequality on bounded integer random variable

I am trying to solve the following problem. Let $X$ be a non-negative integer-valued random variable such that $X \leq m$ and $E[X] = 2 m^{1-td/2}$. Proof that $$Pr[X \geq m^{1-td/2}] \ge ...
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Rolling $n$ times with an $m$-sided dice. Closed, finite formula for the distribution of the sum? [duplicate]

My current idea is the following: practically we want to get the distribution of the sum of $n$-times of a discrete uniform distribution between $1,...,m$ . It is practically the discrete convolution ...
3
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1answer
195 views

Convergence of marginal distribtution

Here I have a question which looks a little bit weird: $(q_n)_n$ is sequence of probability density functions of the couple $(x,y) \in \mathbb R^2$, $p_n$ is the marginal density of $q_n$, i.e. ...
3
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1answer
79 views

Conditional probability independent of one variable

Let $X,Y$ and $Z$ be Borel spaces (that is, Borel subsets of Polish spaces) and let $\mathcal P(X)$ denote the Borel space of all Borel probability measures on $X$. For a product measure $P\in ...
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1answer
40 views

Prove $\{Z=0\}\subset\limsup\limits_{n}\{X_n<\epsilon\}$

Let $(X_n)_{n\in\mathbb N}$ be independent real random variables, with values in $(0,\infty)$. Consider the random variable $Z(w):=\inf\limits_{n\in\mathbb N}X_n(w)$. Prove that for every fixed ...
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1answer
28 views

Question about derivation of normal distribution properties

The normal distribution is given by the density $$g_{\alpha,\sigma^2}(x) = (2 \pi \sigma^2)^{-1/2} \exp\left(-\frac{(x-\alpha)^2}{2\sigma^2}\right) $$ in the probability book of bauer it is claimed ...
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38 views

Finding the distribution of $M$

Let $(X_1,X_2)$ be uniformly distributed in $[0,1]^2$ and define $Y_1=\max(X_1,X_2)$, $Y_2=\min(X_1,X_2)$. What is then the distribution of $M:=Y_1-Y_2$ ? To find the joint distribution we take a ...
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111 views

martingale and stochastic Integral

Let ${W_t}$ be 1 dimension Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why it become ...
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46 views

Markov Chains and Return Times

Let $(X_n)_{n≥0}$ be a Markov chain with transition kernel $p$ on a countable state space $S$, starting at $x∈S$ $T^{(1)}=\inf\{n≥1:X_n=x\} \quad \quad$ first return time to $x$ ...
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1answer
31 views

Convergence of random variable

I've been facing the following problem: Let $(X_k, Y_k)_k$ be a sequence of $2$-dimensional, independent random variables, each with uniform distribution over $ B(0,k) $ Verify if the following ...
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25 views

Determine the Law of $F^{-1}(U)$

If, $F^{-1}(u)=\inf\{x\in\mathbb R:F(x)>u\}$ and $U$ is uniformly distributed in $[0,1]$, what is the law of $F^{-1}(U)$ ? ($F$ is a distribution function of some random variable $X$) How can ...
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1answer
66 views

SDE and Stochastic calculus

$W_t$ is 1 dimension Brownian morion. $X_t=(cosW_t,sinW_t)$ Write SDE about $X_t$ I thought that $f(t,x)=(cosx,sinx)$, but I can't how "$t$" is expressed. I heard that the hint of this question is ...
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59 views

Question on Gauss map - application of Birkhoff's ergodic theorem

Take a Gauss map $G: [0,1] \longrightarrow [0,1]$ which is $$G(x) = \frac{1}{x} \mod 1, 0<x<1$$ and $0$ if $x=0$. Let $\mu$ be the Gauss measure. For $x \in [0,1]$ let $[a_{1}(x), ...
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50 views

expectation by integral

I am trying to compute the expectation of $\mathbb{E}[XY]$ where $X$ and $Y$ are dependent on a third non-negative random variable $Z$. I can now compute the expectation as follows ...
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1answer
55 views

How to find the expectation value?

Suppose that an insurer has an exponential utility function $u(x)=−2e^{-2x}$. What is the minimum premium $P^{-}$ to be asked for a risk X? After solving this we reached the following, So,only ...
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38 views

Reconstruction of state covariance from output covariance

Let us be given an LTI system $$ \frac{d}{dt} x (t) = A x(t), \;\; x(0)=x_0 \\ y(t) = Cx(t) $$ where $x_0$ is a random vector (e.g. uncertainty). Then it is known that the expectation $\mathbb ...
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--Should I bother to vote?? — so thats toughest one Ive ever faced… :/ any help to solve that?

So thats the question Im facing, Im still stack at Bayesian equilibriums, any help would be very much appreciated Thanks very much for your time guys :)