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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199 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
36 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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17 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
18 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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46 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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10 views

de Finetti's theorem in two dimensions?

We know that for an array of exchangeable Bernoulli r.v.s $X_i, i\in \mathbb{N}$, de Finetti's theorem can be rephrased to be that $$\exists f: \mathbb{R\times \mathbb{R}}\rightarrow \{0,1\}, \; ...
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34 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} $ ...
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1answer
32 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
26 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 ...
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1answer
18 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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0answers
18 views

Is it a Risk-averse utility function?

I'm a little unsure whether this utility function represents a risk-averse attitude, while it's not wholly concave: Would you define it as both risk-averse and risk-neutral as it seems to have ...
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39 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
24 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 ...
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2answers
119 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 ...
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32 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
47 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 ...
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1answer
35 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
35 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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0answers
26 views

Laplace transform for Bernoulli with exponential distribution

If we have, say a power $P_{ri}$, which is received power from node-$i$, distanced-$r$ and distributed exponentially with mean $r_i^{-\alpha}$, and the PDF of that power is equal to ...
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0answers
28 views

independence copula diagonal

I'm reading Nelsen's Instruduction to copulas, and there is (probably very simple) excersice I cannot deal with. It says that if the diagonal section of the copula equals the diagonal of independence ...
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1answer
29 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
24 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 ...
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1answer
52 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
39 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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0answers
30 views

What is the difference between weak and strong law of large numbers? [duplicate]

Why is the Weak Law and Strong Law always stated separately? From my textbook, weak law is convergence in probability and strong law is about convergence almost surely. But doesn't 'almost surely' ...
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1answer
16 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
10 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 ...
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1answer
71 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
18 views

Prove the Fourier Inversion Formula for a Multivariate Distribution

Question: Prove the Fourier Inversion Formula for the specific function $\phi_{\Sigma, \mu}(x)$: $$\phi_{\Sigma, \mu}(x) = (2\pi)^{-k} \int_{R^k}\hat{\phi}_{\Sigma,\mu}(\xi)e^{-i\xi\cdot x}d\xi$$ ...
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1answer
24 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} ...
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1answer
26 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
71 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
17 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 ...
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1answer
63 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
10 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
26 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 ...
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1answer
20 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
55 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
66 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 ...
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1answer
51 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 ...
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1answer
67 views

An application of the Central Limit Theorem

Suppose $X_i$ are independent random variables uniformly distributed on $[1,3]$. We are interested in the product $W=X_1X_2\cdots X_{10}$. Each $X_i$ is centered about $2$ so we might think $W$ should ...
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1answer
34 views

Pattern Recognizing in Integrals and Probability Distirbutions

This is a basic pattern recognizing question to begin with, but asks about probability densities. Solve the following integrals and find a pattern. I was able to solve them all and they are all ...
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1answer
38 views

Marginal Densities

I just have a few questions about joint density and marginal density questions. Q1: Joint Distribution $f_1=2x+4y$ on triangle with vertices $(0,0), (0,1),(1,0)$. Sketch the region and compute ...
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0answers
28 views

Coalescent theory - Why are coalescent times independent?

I am reading from this book and I want to make sure I understand what is going on. What I get from the book Consider a population of $N$ individuals. The population size ($N$) is constant. select ...
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0answers
17 views

Dsitribution of $|Ae^{j\phi} + W(t)|$, where $\phi \sim unif[-\pi,\pi]$

Let $Y(t) = Ae^{j\phi} + W(t)$, where $\phi \sim unif[-\pi,\pi]$ and $W \sim \mathcal{N}(0,\sigma^2)$. What is the probability distribution of $|Y(t)|$ ? If $\phi$ was deterministic, i.e. a constant ...
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1answer
26 views

Distribution of the sum of squared independent normal random variables

How do I go about finding the the pdf of the statisitc $\sum_i x_i ^2$ such that each $x_i$ is iid from a $N(\sigma , \sigma)$ distribution? I've searched, but cannot find a straightforward answer. ...
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1answer
14 views

On convergence in probability given a bound on the random variable.

I am dealing with the end of a proof: Could somebody please clarify for me the extra steps needed to show that $P(||\bar{W}_n - \mu||_{\infty} > 3\varepsilon) \rightarrow 0$ as $n \rightarrow ...
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1answer
28 views

Characteristic function of a product of two dependent random variables such that one is continuous the other is discreet

If you're given the characteristic function of a continuous random variable, say $X$, and the distribution of another discreet random variable, say $U$, which is dependent of $X$, how do you ...
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1answer
22 views

Inheritance of independence of random variables

I want to show the following statement: let $(X_n)$ and $(Y_n)$ be sequences of random variables and $X_n\perp Y_n$ for each $n$. If $X_n\to X$ and $Y_n\to Y$ in probability respectively, then $X\perp ...
1
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1answer
19 views

If a sequence of random variables all have the same mean, is the sequence tight?

Suppose $(X_n)$ are almost surely non-negative random variables all with the same finite mean $\mu$. Is this sequence necessarily tight?