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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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) = ...
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1answer
28 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. ...
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1answer
17 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 ...
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1answer
19 views

Proving that $W$ is Brownian motion, without stochastic calculus

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
15 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
15 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
21 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
34 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
37 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 ...
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11 views

Random variables set representation in the sample space

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
29 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 ...
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2answers
47 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 & ...
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1answer
9 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
22 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
8 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
21 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
14 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
46 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
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1answer
39 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
19 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 ...
2
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1answer
40 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 ...
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1answer
13 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 ...
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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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26 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 ...
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0answers
23 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 ...
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12 views

A probability problem on outer measure

Let $P$ be a probability measure on a algebra $\mathcal{F}_0$ and for every subset $A$ of $\Omega$ define $P^*(A)$ by $$P^*(A) = \inf \sum P(A_n)$$ where $A_n \in \mathcal{F}_0$ and $A \subset \cup ...
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1answer
20 views

Finding $b$ such that $e^{5B_t - bt}$ is a martingale

I have $X_t = e^{5B_t}$ and Where $B_t$ is brownian motion at time $t$. $M_t = X_t \cdot e^{-bt}$ I need to find a value for $b$ such that $M_t$ is a martingale. I am encountering difficulty, ...
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1answer
26 views

Exchangeable Random Variable but not independent?

We flip a fair coin. If it is a head, we roll a die $n$ times, and if it is a tail, then we sample a number $n$ times with replacement from $\{1,2,3,4\}.$ The resulting random variable $X_1 X_2 \dots ...
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0answers
17 views

Size-biased distributions tight iff corresponding RRVs uniformly integrable

Let $\mathbf{P} \in \mathcal{M}_1\bigl([0,\infty)\bigr)$ with $m_\mathbf{P} := \int x \, \mathbf{P}(dx) \in (0,\infty)$, define a probability measure $\hat{\mathbf{P}}(A) \in ...
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1answer
7 views

Construct 95% lower CI for proportion of adults who voted poor. Of 1060 adults, 54% voted poor environment.

I know that the lower CI will be theta - Z(alpha/2)*sigma. The value for Z is 1.96 because it is a 95% CI, but I am not sure how to find theta and sigma. The only other piece of information was that ...
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0answers
29 views

Flip a coin, then repeat an experiment n times. Show exchangeable but not independent

We flip a fair coin. If it is heads then we roll a die n times, if it is tails we sample a number n times from the set {1, 2, 3, 4} with replacement. We denote the resulting n numbers by X1, ..., Xn. ...
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1answer
35 views

Proving that the characteristic function is uniformly continous.

I am trying to prove that the characteristic function is uniformly continuous. I understand how to get to this bound: And I would like to find the $\delta$ as a function of $\epsilon$ but I am ...
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0answers
29 views

Proving a sequence forms a martingale

Let $\Omega = \mathbb N = \{1,2,3,\cdots\}$ and $\mathscr F_n$ be the $\sigma$-field generated by the sets $\{1\},\{2\},\cdots,\{[n+1,\infty)\}$ Define a probability on $\mathbb N$ by setting ...
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1answer
48 views

There are 8 balls which appear identical. However, 1 is heavier than the rest. How do you find the ball with 2 weighings?

I understand there are similar problems but I am not sure how to go about constructing this problem with set of balls that are not exponents of 3^n. I know I need at least 2 weighings to find the ...
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1answer
18 views

Equivalent of random variable sequences in distribution?

Suppose that $X_n, Y_n$ are sequences of random variable on probability space $\Omega$. If $Xn,Yn$ converges to $X$ ( some random variable ) in distribution, then is $X_n=Y_n$ almost everywhere (a.s)? ...
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1answer
18 views

Characteristic function of $\sum_{t=1}^N a_t X_t$ given certain independence conditions

Let $S=\sum_{t=1}^N a_tX_t$ where each $a_t$ is Bernoulli with probability $\frac{1}{2}$ for $1$ and also $\frac{1}{2}$ for $0$. Moreover, it is also given that the vector $(a_1,\ldots,a_N)$ is ...
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3answers
55 views

Probability of dying

I have this scenario: 10 people with 100% of probability of dying in 1 month. So I assume that at the end of the month I'm gonna have: 10 dead people; 0 alive people. I also have this ...
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1answer
9 views

Convergence of expectation of product from truncation

Say $X$ and $Y$ are two random variables such that $(\mathbb{E}X)^2<\mathbb{E}X^2<\infty$ and $(\mathbb{E}Y)^2<\mathbb{E}Y^2<\infty.$ I want to argue that as $M\to\infty,$ (i) ...
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1answer
32 views

Prove $E[XE[Y\mid\mathcal{G}]] = E[YE[X\mid\mathcal{G}]]$

Show that for bounded $X$ and $Y$ that $E[XE[Y\mid\mathcal{G}]] = E[YE[X\mid\mathcal{G}]]$. Attempt: Suppose that $X = _{\mathcal{X}_F}$, where $F \in \mathcal{D}$. Then for every $B \in ...
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0answers
17 views

Variance of sample variance

We are given $\{X_i \} $ iid random variables with $\mathbb{E}X_i = \mu$ and $D^2X_i < \infty$. I'm trying to compute $D^2(\sigma^2_n)$ where $$\sigma^2_n= \frac{1}{n} \sum_{i=1}^n (X_i - \mu)^2$$ ...
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1answer
30 views

Distribution of an angle between a random and fixed unit-length $n$-vectors

Suppose I have a random unit-length $n$-element vector $\mathbf{x}$ that is uniformly distributed on an $n$-dimensional sphere, and let vector $\mathbf{a}$ be some other unit-length $n$-element vector ...
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1answer
36 views

An inequality regarding expectation of random variables

Let $X,Y$ be positive-valued, well-behaved random variables. Further, let $g(\cdot) \ge 0$ and $f(\cdot)\ge 0$ be two functions and $E(\cdot)$ denotes expectation operator. I am trying to prove the ...