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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22 views

How to know the result of entropy function using uniform distribution set

In the entropy function here $H(s) = -\sum P(class=i|S)log_2{P(class=i|S)}$ I am trying to understand what is the domain of it's output for any input. I know that given a set where the frequency of ...
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3answers
516 views

What does actually probability mean?

I am a beginner in quantum information. Reading about it has made me question the definition of probability. If the probability of an outcome $m$ in an experiment is $p(m)$ then it means that if I ...
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1answer
70 views

Intro to probability chapter 4 ex 31

A group of 50 people are comparing their birthdays (as usual, assume their birthdays are independent, are not February 29, etc.). Find the expected number of pairs of people with the same birthday, ...
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62 views

Exercise in measure theory/probability

This is an exercise in chapter 2 of Probability with Martingales by David Williams. Question: Let $\mathcal{A}$ be the set of all maps $\alpha : \mathbb{N} \rightarrow \mathbb{N}$ such that ...
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49 views

number of moves in the “memory”-game

Suppose, a player with a perfect memory starts with $2n$ cards in the game memory. A move consists of choosing two cards, the game ends if all pairs are found. Let $X$ be the number of moves the ...
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131 views

Sigma algebra generated by a homeomorphic random variable

Let $\Omega = [0,1]$ be our probability space with sigma algebra of borel sets on $[0,1]$ and Lebesgue measure on $[0,1]$. Let Y be a random variable such that $Y(\omega) = Y(1-\omega)$ for every ...
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1answer
23 views

Given the distribution of $X$, whats the distribution of $cX$

Let's say $X \sim \chi_k^2(\lambda)$ with pdf $f_x(x)$ (i.e. noncentral chi-squared distribution). What can we say about the distribution of $Y = cX$ ? where $ c \in \mathbb{R}^+$ I know that $f_y(y) ...
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1answer
51 views

An equality involving the Wiener process

The equality below appears as a step in a proof in a chapter titled "Itô Stochastic Calculus" in Brzeźniak and Zatawniak's textbook "Basic Stochastic Processes", Springer 2005 (in a solution to ...
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0answers
54 views

Moments bounds VS Chernoff bounds

I have to prove that, when bounding tail probabilities of a nonnegative random variable, the moments method is always better than the classical Chernoff method. In mathematical language, I have to ...
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1answer
62 views

No arbitrage iff there EMM $P^*$ theorem [closed]

The definition of an arbitrage I was given: "An arbitrage strategy is an admissible strategy with zero initial value and positive probability of a positive final value." I think that an initial ...
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0answers
56 views

Conditional density along a curve?

Let $X,Y$ be two random variables with $f_{X,Y}$ as their joint pdf and with $f_X(x)$ and $f_Y(y)$ as their marginal pdf's. The conditional pdf of, say $X$ with respect to $Y$ is $$f_{X\mid ...
4
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1answer
223 views

Convergence to the Dirac Delta Function

Let $h\colon[0,1]\to \mathbb{R}^+$ be any bounded measurable non-negative function with a unique maximum at $a$ and $h$ is continuous at $a$. For $\lambda>0$ define $h_\lambda(x)=C_\lambda ...
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1answer
29 views

Formula for average time in Markov chain

I have a model like: A B C A 0.80 0.10 0.10 B 0.20 0.75 0.05 C 0.10 0.10 0.80 How do I get the average time from B to A? I understand that ...
3
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1answer
126 views

Almost sure convergence of the Poisson process

Let $N = \{N(t) \}_{t\geq 0 }$ be a Poisson process. I already know that $N(t)- \lambda t$ is a martingale where $\mathbb{E} [ N(t) ] = \lambda t$. I want to prove that $$ \frac{N(t)}{t} \rightarrow ...
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1answer
26 views

conditional expectation conjugate exponents

If I know that $X\in L^p(\Omega,F,P)$ and $Y\in L^q(\Omega,G,P), \ G\subset F$, $\frac{1}{p}+\frac{1}{q}=1$, $F$ is $\sigma$ algebra on probability space $\Omega$, $G$ is sub $\sigma$ algebra. How ...
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0answers
234 views

Conditional expectation, quadratic function, absolute value

We are given two random variables defined on $[0,1]$: $$X(\omega) = 2 \omega -1 + |2 \omega -1|$$ $$Y(\omega) = 1-|2 \omega^2 -1|$$ I am supposed to find $\mathbb{E}(X|Y)$ which by definition is a ...
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1answer
42 views

Binomial Distribution with probability $P$ such that $P$ is Uniformly distributed

A number $P$ is random chosen from the uniform distribution from [0,1]. Then a coin with probability $P$ of getting a head is flipped $n$ times. Let $X$ be the number of heads showing and compute ...
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0answers
66 views

Discrete measures converging weakly

Let $a_1, a_2, \ldots$ be any sequence of non-negative real numbers with $\sum_i a_i = 1$. Define the discrete measure $\mu$ by $\mu(\cdot) = \sum_{i\in\mathbb{N}} a_i \delta_i(\cdot)$, where ...
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1answer
23 views

Weak convergence equivalent to convergence of distribution functions

I'm trying to understand the following proof: I don't see where the Lipschitz property of the $f$ is needed. Why isn't bounded and continuous enough? Can somebody explain it to me? Thanks.
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0answers
51 views

Limit of an integral (Arrow theorem)

Im not sure about a limit of an integral. I would like to prove that there is a solution for this integral for d, and this solution is unique. The integral is: $$\beta = \int_d^{\infty}(x-d)f(x) dx$$ ...
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0answers
41 views

Conditional density boundary problems

The definition of $Y$, given $X$, is uniform on the interval $[0,X]$. The marginal density of $X$ is $ f(x)= \begin{cases} 2x, & \text{for }0<x<1 \\ 0, & \text{otherwise} \end{cases} $ ...
2
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1answer
170 views

Interchange of expectation and summation

Assume $(\Omega,\mathscr{F},P)$ is a probability space and $\{X_n\}_{n\geq 1}$ is a sequence of random variables. Let $\{A_n\}_{n\geq 1}$ be a measurable partition of $\Omega$. My question is when the ...
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2answers
42 views

joint probability and conditional probability question

The number of workplace injuries, $N$, occuring in a factory on any given day is Poisson distributed with mean $\lambda$ . The parameter $\lambda$ is a random variable that is determined by the level ...
4
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1answer
42 views

A question about projections of product measure space

I am considering the space $\mathbb{R}^{\mathbb{N}}$ of real-valued sequences with the sigma-algebra $\mathcal{F}$ generated by sets of the form $$\{\omega \in \mathbb{R}^{\mathbb{N}} : \omega_k \in ...
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1answer
271 views

Intuition behind measurable random variables and $\sigma$-algebra

I've been trying to understand $\sigma$-algebras and how it encodes information in context of filtration. While certain parts seem clear and logical, I can't say I get the whole picture. I'll try to ...
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0answers
47 views

conditional density of $X$ given $XY^2$

I was asked the following problem. Given that $X$ and $Y$ are random variables with joint density $f(x,y)$, find the conditional density of $X$ given $XY^2$. My thought was to first change variables ...
4
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2answers
258 views

Are there any differences between distributions (generalized functions) and probability distributions?

A distribution/generalized function is an element of the dual space of $$S=\{f\in C^{\infty}(\mathbb{R})\colon \|f\|_{\alpha,\beta}<\infty \text{ for all } \alpha ,\beta\}$$ Where ...
2
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0answers
93 views

Transition function for absorbed Brownian motion

I need an help with the following exercise. I've already seen this question Prove that Brownian Motion absorbed at the origin is Markov but I don't understand the answer. Also I would like to prove ...
4
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1answer
64 views

Computing Conditional Variance

I have been tasked with trying to solve a conditional variance. I have red and black pens with respective exponential probability parameters 2 and 4. I have 70% red pens and 30% black pens. What is ...
2
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1answer
43 views

$X \sim Rice(\nu,\sigma)$, what is the distirbution of $X^2$?

Let $X = |\nu e^{j\theta}+W|$, where $W \sim \mathcal{CN}(0,2\sigma^2)$, i.e. $X\sim Rice(\nu,\sigma)$, what is the distirbution of $X^2$? Note that X also can be writen in terms of real and imaginary ...
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3answers
199 views

Coin toss game - Probability of winning

Question: Two players A and B, alternatively toss a fair coin (A tosses the coin first, then B, than A again, etc.). The sequence of heads and tails is recorded and if there is head followed by a tail ...
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1answer
39 views

Dice roll, estimator, epsilon

We roll a non-symmetric die. Let $X_n$ be the reulst of $n$-th roll. $$P(X_n = 6)= \frac{1}{6} + \varepsilon, \ P(X_n = 1) = \frac{1}{6} - \varepsilon, \ P(X_n=2) = ... = P(X_n = 5) = \frac{1}{6} $$ ...
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0answers
25 views

$X + Y \overset{\mathcal{D}}{=} X \Longrightarrow \mathbf{P}[Y = 0] =1$

Let $X$ and $Y$ be independent, real random variables. Show that $X + Y \overset{\mathcal{D}}{=} X$ implies that $\mathbf{P}[Y = 0] =1$. Note: $U \overset{\mathcal{D}}{=} V$ means that the ...
3
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1answer
69 views

Random walk of geometric random variables

I was wondering if there's a more advanced theory that can fit with the following context: Let $\tau_{n} = \sum_{i=1}^{n}{T_{i}}$ be a sum of iid geometric random variables with parameter $p$ and for ...
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1answer
45 views

$X\in L^1$, then $\int_{|X|>n}XdP\to 0$ and $P(A_n)\to 0 \Rightarrow \int_{A_n}XdP\to 0$

I'm trying to prove the following: 1. Suppose $X\in L^1$, then $\int_{|X|>n}XdP\to 0$ Attempt: $$\int_{\Omega}|X|dP = \int_{|X|≤n}|X|dP+ \int_{|X|>n}|X|dP = M<\infty \space \forall n$$ ...
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1answer
18 views

Joint Density of Z

Find the density of $Z$ given that $Z=Y/X$ for $f(xy)=8xy$. Let $x \in [0,1]$ and $y$ is between 0 and $x$. I think this is $$\int^1_0\int^{x/4}_0f(x,y)dydx$$ Correct? I do not think it is because I ...
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1answer
27 views

Distribution of sine composed with a random variable

Could you tell me if my calculations are correct? We are given a random variable with the following discrete distribution $$P(X=n) = \frac{2^n}{3^{n+1}}, \ \ n \in \mathbb{N}.$$ Find the ...
2
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1answer
265 views

If I flip a coin $n$ times, what is the expected maximum number of heads or tails in a row?

Question: If I flip a coin $n$ times, what is the maximum number of heads or tails in a row that I should expect?
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0answers
46 views

Prove the Fourier Inversion Formula for a Multivariate Distribution

Question: Prove the Fourier Inversion Formula for the specific function $\phi_{\Sigma, \mu}(x)$: $$\phi_{\Sigma, \mu}(x) = (2\pi)^{-k} \int_{R^k}\hat{\phi}_{\Sigma,\mu}(\xi)e^{-i\xi\cdot x}d\xi$$ ...
1
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1answer
52 views

Convergence in Distribution and Exponential Function

There's a well known fact that if a sequence of real numbers, $\{x_{n}\}$ converges to $x$, then: \begin{equation*} \lim\limits_{n\rightarrow\infty}\left(1+\dfrac{x_{n}}{n}\right)^{n} ...
2
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1answer
144 views

Proving Events belong to a tail sigma field

I'm really confused about tail sigma fields and how to prove that a set is or is not a tail event (belongs to the tail sigma field). I was wondering if anyone has seen examples of proving that a set ...
2
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1answer
186 views

Prove this liminf is a tail events

Let $A_{k}$,$k\geq1$ be [0,$\infty$)-valued random variables on a common probability space. I want to prove the following events are in/not in tail $\sigma$-field T($A_{k}$:$k\geq1$). First, event ...
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0answers
27 views

Does Convergence in Total variation plus existence of moments imply convergence of moments?

If $\{P_n\}$ is sequence of probabilities distributions converging to $\mathcal N(0,1)$ in Total variation norm. Suppose variances of the distribution $P_n$ exist for all $n$.. Can we say then ...
3
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1answer
75 views

Stochastic continuity

Let $(X_t)_{t \in \mathbb{R}}$ be a square-integrable real-valued process with a continuous mean value function $\mu:\mathbb{R}\rightarrow\mathbb{R}$ and a continuous covariance function ...
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0answers
17 views

Exchangeable filtration $\mathcal{E}_n$ is the same as $\sigma(S_n,X_{n+1},\dots)$?

It should be a really intuitive conclusion but I kind of missing in the detail treatment. Suppose we have $\mathbb{R}^{\mathbb{N}}$ and equipped it with the sigma-algebra $\mathcal{B}_{\infty}$ ...
2
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0answers
42 views

Complex moment problem

Let $X$ be a complex-valued random variable. For simplicity, let us assume that $X$ is bounded. If I know all of the $\mathbb{E}[X^k]$ then do I know the distribution of $X$? I know it suffices to ...
0
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1answer
46 views

Maximum likelihood estimator when log likelihood equals constant

I am trying to find the MLE for a parameter $\theta = \alpha$ when $\alpha = 0$ (I am doing hypothesis testing, and $H_0 : \alpha = 0$, $H_1 : \alpha \not = 0$). Under the null hypothesis, the log ...
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2answers
132 views

Difference between conditional expectation and conditional probabilty

These are known definitions: We have a probability space $(\Omega, A, P)$ Conditional probability is defined through $P(A|B) = \frac{P(A \cap B)}{P(B)}, P(B) > 0$. This is a real nunmber. Then ...
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1answer
89 views

Exponential Probability Question

A college buys 70% of dorm light bulbs from Company A with an exponential lifetime $f_A(x)~ exp(\lambda = 2)$. The other 30% come from company B have lifetime $f_B(x) ~exp(\lambda = 4)$. At the start ...
0
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1answer
62 views

What kind of distribution is this? Use Moment Generating functions

Let X Pois($\phi$) and Y Pois($\tau$) be independent poisson random variables. a) Use moment generating functions to show that Z = X + Y Pois($\phi +\tau$ ) b) Find the conditional distribution of X ...