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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8 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 ...
2
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1answer
7 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 ...
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0answers
21 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 ...
1
vote
1answer
9 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 ...
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0answers
6 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 ...
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0answers
15 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} ...
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0answers
13 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 ...
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1answer
14 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 ...
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0answers
6 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, ...
0
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1answer
14 views

Reason behind convergence in probability definition

A sequence ${X_n}$ of random variables converges in probability towards the random variable $X$ if for all $\epsilon > 0$ $$\lim_{n\to\infty}\Pr\big(|X_n-X| > \epsilon\big) = 0$$ But ...
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0answers
11 views

pdf for the sum of squared iid normal random variables

I am trying to find the distribution/pdf for the sum of squared $X_i$ iid observations from the normal distribution $X_1 ,..., X_n$ ~ $N(\alpha , \alpha)$, i.e. $X_1 ^2 + X_2 ^2 +...+ X_n ^2$. I was ...
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0answers
11 views

when is the maximum likelihood estimator measurable

For a random variable $X$, a class of probability measures $P_\theta$ for $\theta\in \Theta$ and their densities $f_{\theta}$ w.r.t. some common measure $\mu$, we can define the maximum likelihood ...
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0answers
15 views

$\lim\sum_{k=0}^{\lfloor\delta n\rfloor} \frac{n^k}{k!}e^{-n}$ and Poisson distribution

Problem: Let $X_1,X_2\ldots$ be some independent random variables with Poisson distribution with parameter 1. Show that for every $\epsilon > 0$ sequence $S_n-(1-\epsilon)n$ converges to ...
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0answers
23 views

Computing conditional expectation variable given variables X, Y

When it comes to conditional expectation, I can compute $\mathbb{E}(X|Y)$ when I know the distribution of $X, Y$ when they are continuous or discrete. But I don't know how to find $\mathbb{E}(X|Y)$ ...
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0answers
8 views

Interpretation of sufficient statistic in the continuous case

A statistic $S = S (X)$ is called sufficient for $\theta$ if there is a $P_{X \mid S} (\cdot \mid s)$ that doesn't depend on $\theta$. So if $S(X)$ is a discrete random variable and we know $S (X) = ...
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0answers
26 views

Expected value of conditional expectation, discrete variable

We are given a random variable $X$ on $\Omega_1$ and a discrete variable $Y: \Omega_2 \to \mathbb{N}$. We consider $\mathbb{E}(X|Y)$ as a random variable defined as follows: $$\mathbb{E}(X|Y)(\omega)= ...
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2answers
44 views

Conditional Probability Question. [on hold]

A letter is known to have come from either 'TATANAGAR' or 'CALCUTTA'. On the envelop just two letters 'TA' are visible. What is the probability that the letter has come from (i) TATANAGAR (ii) ...
3
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0answers
24 views

Conditional expectation, sigma algebra

Let $X$ be a random variable on $\Omega$ and $Y$ a discrete variable having values $y_1, y_2,...$. We define another random variable via conditional expectation $\mathbb{E}(X|Y)(\omega) = ...
2
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1answer
34 views

Sum of two normal numbers need not be a normal one

Using the translation invariance of Lebesgue measure how to show that sum and difference of two normal numbers need not be normal ? Normal number in $(0,1]$ is a number $\omega$ such that $\lim_{n ...
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0answers
2 views

Variance of Inhomogenous Poisson process in a given window

Consider some variable $X\sim \operatorname{Poi}(\lambda(t))$ to be Poisson-distributed with some parameter $\lambda$ dependent on time, where we know how the random variable $\lambda$ is distributed. ...
0
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1answer
19 views

$\Pr(X+Y\geq1)$

Two random variables X and Y have the following joint pdf: $$f_{X,Y}(x,y)\begin{cases}10x^{2}y & 0<x<1,0<y<x\\0 & \text{otherwise}\end{cases}$$ I am asked to find the marginal pdf ...
3
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2answers
44 views

A variation of Lévy's characterization of Brownian motion

It is shown here, without using stochastic calculus, that if $W_t$ is a standard Brownian motion, then $$ f(W_t)-\frac{1}{2}\int_0^t f''(W_s)ds $$ is a martingale, where $f\in C^2$ and compactly ...
2
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1answer
17 views

Conditional expectation - other formulation

Conditional expectation is defined as follows: We are given probability space $(\Omega, \Sigma, P)$ For $a \in \Sigma$ such that $P(A)>0$, random variable $X: \Omega \to \mathbb{R}$ we define: ...
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0answers
21 views

Levy process of argument in the complex plane

I am stuck on this question: Let $B$ be a Brownian motion in $\mathbb{C}$ started at $1$. Let $\theta_t$ be a continuous determination of the argument of $B_t$, i.e. $B_t = |B_t| e^{i \theta_t}.$ ...
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1answer
35 views

Show $\mathbb{E}(X \mid Y,Z) = \mathbb{E}(X \mid Y)$ if $Z$ is independent of $X$ and $Y$

Let $X,Y,Z$ be random variables, $X$ integrable, $Z$ independent of $X$ and $Y$. Then we have $E[X\mid Y,Z]=E[X\mid Y]$. Why is only assuming $Z$ independent of $Y$ not enough. I was able to ...
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2answers
37 views

Showing that the Lindeberg CLT Condition Holds

Suppose we have a sequence of random variables, $\{X_{n}\}_{n\geq 1}$ satisfying: $\mathbb{P}(X_{j} = 2^{j}) = \mathbb{P}(X_{j} = -2^{j}) = \frac{1}{2}$ Then is it true that the CLT holds? Or ...
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2answers
38 views

Help me find $P(A \cup B')$ under the given conditions

I was assigned the task to solve this problem by mathematics teacher which I can't solve because it doesn't make sense to me (I think that it is impossible to solve it). There was an error please try ...
-1
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0answers
14 views

Random variables set representation in the sample space [on hold]

Consider that I have two Random variables $ X : \Omega \rightarrow \mathbb{R} \space , Y : \Omega \rightarrow \mathbb{R}^d$ belonging to the same sample space and a measurable function $\space f : ...
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2answers
39 views

Inverse of a mean, exponential distribution, expected value

Could you help me find the expected value of this random variable? Let $X_1, X_2, ... $ be independent identically exponentially distributed with parameter $\lambda$ random variables. What is the ...
2
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2answers
28 views

integral of exponential of Brownian motion

I am currently reading a proof that uses the following fact without proof: If $B$ is a scalar standard Brownian motion, then $\int_0^\infty e^{B_s} \,ds = + \infty$ a.s.. How can we justify this ...
2
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2answers
52 views

Find the probability of solutions of an equation.

Let $x+y+z=20$. What is the probability that all the solutions are distinct? (No two variables have the same value). Assuming that the solutions are only positive integers or zero. I have tried- ...
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1answer
21 views

find distribution of $\max(x^2,x)$ and $\min (x,1)$

I have the following question. Find distribution of $Y=\max(X^2,X)$ and $Z=\min(X,1)$. My distribution function is $$ F_X(x)=\left\{\begin{array}{ll} 0 & \mathrm{if}\; x <0\\ 0.5x & ...
1
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1answer
12 views

FDD convergence of subsequences

First, sorry for this probably very stupid question. Let $(X_n)_{n\geq1}$ be a sequence of random variables (e.g. in $\mathbb{R})$ s.t. $X_n\stackrel{d}{\to} X$ in distribution. Now look at the ...
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1answer
23 views

Doubt on Caratheodory's extension theorem

This doubt is on the Caratheodory's extension in Billingsley. The main theorem says that a countably additive probability measure $P$ on a algebra can be extended to a countability additive ...
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0answers
10 views

Markov chain converges to boundary

I am learning martingale and related concepts recently and come across the following problem. Suppose $D$ is a bounded, connected, open subset of $\mathbb{R}^2$ with boundary $\partial D$. ...
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2answers
23 views

Mutual information expressed as Kullback-Leibler divergence

My lecturer defines the mutual information: $$ I(X;Y\mid Z) = D_{KL}\big(p(X,Y\mid Z)\parallel p(X\mid Z)\;p(Y\mid Z)\big)$$ Is this correct? I feel like it doesn't really make sense to say that; ...
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0answers
15 views

Question on proof of disintegration of measures

In a probabilistic setting: Let $\mu$ be a measure on the product space $S=S_1\times S_2$, both standard Borel, $\mu_1, \mu_2$ the marginal measures. Then there exists a Markov kernel $k$ such ...
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0answers
28 views

Teacher for Semi-Blind kid, Conditional expectation and bayes theorem.

I have an interesting question that I came across. I know that this uses Bayes Theorem, but I am stumped in terms of minimizing the expected squared error. This question is nothing I've ever seen ...
2
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1answer
47 views

Random variable with infinite expectation

I was trying to find $Y$,a random variable (non-negative, may be $E(|Y|)=\infty$), such that $$\sum_{n=0}^{\infty} E\Bigl(\frac{|Y|}{n^2 +|Y|}\Bigr)=\infty$$ I tried with Cauchy distribution but could ...
2
votes
1answer
41 views

Showing $E[X_{n+1}|X_1,…,X_n] = a_0+\Sigma_{k=1}^n a_kX_k$

$X_1,...,X_n,X_{n+1}$ are jointly distributed with a Gaussian distribution. We let $X^* = E[X_{n+1}|X_1,...,X_n]$. Show that there exists constants $a_1,...,a_n,a_{n+1}$ such that $X^* = ...
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0answers
20 views

Filtered Probability Space Understanding

Usually in my probability theory class, we define a filtered probability space in the background $\left(\Omega, F, \left\lbrace F_t \right\rbrace P\right)$ and do all of our work on that space. I'm ...
6
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2answers
80 views

Probability of getting A to K on single scan of shuffled deck

Let us say we have a regular 52-card well-shuffled deck. We scan through the deck (first to last) till we find an Ace. Then we continue (from that Ace) till we find a 2. Then we scan (from the 2) ...
2
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1answer
11 views

probability almost different and expectation

This relates to my question here, but for the difference case, i.e. is it true that $P(X \ne Y)=1 \Rightarrow E(X) \ne E(Y)$? I tried using the same proof technique as the answers to my other ...
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2answers
29 views

probability almost surely and expectation

It is true that $P(X= Y)=1 \Rightarrow E(X)= E(Y)$ and $P(X\ge Y)=1 \Rightarrow E(X)\ge E(Y)$, but is it true that $P(X > Y)=1 \Rightarrow E(X) > E(Y)$ ? The proofs for the first two don't ...
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0answers
20 views

probability theory question apex [on hold]

Probability theory predicts that there is a 64% chance of a team winning a particular match. If the team playing two matches is simulated 1,000 times in about how many of the simulations would you ...
1
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1answer
15 views

Metrics for weak convergence of probability measures

For a polish space $(S,d)$ I have see the metric: $$ \beta(P,Q) := \sup\left\{ \left| \int f dP - \int fdQ \right| \mid \|f\|_{BL} \leq 1 \right\} $$ where $f$ is taken to be Lipschitz and bounded and ...
2
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1answer
29 views

Checking if $B_t^3 $ and $3tB_t$ are martingales?

$$\mathbb{E}[ B_t^3 - 3tB_t + 3B_t | \mathcal{F}_s]$$ $$\mathbb{E}[B_t^3 | \mathcal{F}_s] - 3\mathbb{E}[t B_t | \mathcal{F}_s\}$$ $$\mathbb{E}[(B_t^3 - B_s^3 + B_s^3) | \mathcal{F}_s] + [ not \space ...
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0answers
27 views

Estimating a probability

I am interested in the probability of a random deal in bridge to be a par- zero-deal (a deal where no player can make any contract assuming perfect play with all hands visible) The events I need ...
0
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0answers
24 views

Random variables, indicator variables, conditional probability

Let $\{X_n \}$ be a sequence of independent identiacally distributed random variables. Let $A, B \in \mathcal{B}(\mathbb{R})$ be such that $P(X_1 \in B) \neq 0$. Let $S_n:= 1_{\{X_1 \in B \}} + ... + ...
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1answer
19 views

Proof of sufficient condition for $\mathbb{E}[|X|]<\infty$

If for all $\epsilon>0$ there exists a $\delta>0$ such that $\mathbb{E}[|X|1_A]<\epsilon$ for all $A\in \mathcal{A}$ with $P(A)<\delta$, then $\mathbb{E}[|X|]<\infty$. (X random ...