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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2
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1answer
59 views

Proving that three events are mutually independent

Suppose that: the events $A$ and $B\cap C$ are independent. the events $B$ and $A\cap C$ are independent. the events $C$ and $A\cap B$ are independent. the events $A$ and $B\cup C$ ...
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0answers
58 views

Banach Fixed Point Theorem. Measurable version.

The Banach fixed point theorem has the following statement THEOREM ( Banach contraction principle). Let $(Y,d)$ be a complete metric space and $F:Y\to Y$ be contractive . Then $F$ has a uniqe ...
14
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1answer
235 views

The problem of the most visited point.

Represents the set $R_{n\times n}=\{1,2,\ldots, n\}\times\{1,2,\ldots, n\} $ as a rectangle of $n$ by $n$ as points in the figures below for exemple. How to calculate the number of circuits that visit ...
0
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1answer
34 views

Can the transition probabilities of an inhomogeneous Markov chain be written as an exponential?

If $Z_t$ is a homogeneous continuous-time Markov chain with finite state space $E=\{1,\ldots,p\}$, transition matrices $(P(t))$ and intensity matrix $Q$, it holds that $$ P(t) = \exp(tQ), $$ see for ...
4
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2answers
110 views

A counter example of Brownian Motion

Here is an example in my textbook to illustrate why we need the continuous sample path in the definition of Brownian motion. Let $(B_t)$ be a Brownian motion and $U$ be a uniform random variable on ...
0
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1answer
19 views

Probability densities and Absolute continuity

I've not deep knowledge in measure theory/real analysis but just few concepts given me during this second year probability course. I'm trying by myself to understand more, but I don't want to dive in ...
0
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1answer
25 views

Asymptotic Equivalence implies same asymptotic distribution?

A book I'm reading stated that if we have nonnegative random variables, and if $X_n\to X > 0$ in distribution and $\frac{Y_n}{X_n} \to 1$ in probability then $Y_n \to X$ in distribution. However, ...
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0answers
34 views

Chernoff Binomial Bound

I am reading a paper and the following Chernoff-type bound is presented: For X~Bin(n,p) and a>0, the following bounds for lower and upper tail, respectively, hold: $$\Pr[X\le np-a]\le ...
0
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1answer
30 views

Simple conditional probability inequality

I'm reading on some branching process theory in Harris' Theory of Branching Processes and encountered an inequality which looks simple but is eluding me. The full version is a bit complicated to ...
3
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1answer
40 views

Uniform sampling with replacement item frequency

Suppose we are sampling from $N$ distinct items uniformly with replacement $M$ times. What can be said about the distribution of frequencies of items drawn? For example, if I sort all the frequencies ...
1
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1answer
24 views

Probability Game Question

I am new to probability. I am trying to solve the following problem. In a game, probability of winning the game is $w$ & losing the game is $l$ & probability of game continuing is $(1 - w - ...
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0answers
34 views

Finding an example where the sum of iid rv's is infinite almost surely

$Y_1, Y_2, \dots$ are iid non-negative random variables. Let $S = \sum_{n=1}^\infty \alpha^n Y_n$, where $\alpha \in (0,1)$. Now, $EY < \infty$ implies that $S < \infty$ almost surely. Can ...
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1answer
18 views

Filtration from a Brownian Motion

The textbook I am reading defines the filtration induced from a Brownian Motion as follows. Let $\{B(t): t \geq 0\}$ be a Brownian Motion defined on some probability space, then we can define a ...
0
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0answers
23 views

Normal approximation with dependent variables

I have a sequence of $N$ dependent random variables $$y_i = \frac{x_i}{||\vec x||_2} \quad \mathrm{for} \quad \vec x \sim \mathcal N(0,\mathbb{1}_N),$$ where the $x_i$ are the iid elements of $\vec ...
2
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3answers
40 views

Finding the expected value of a function of random variables

I'm having troubles with finding marginal density functions and expected values in my probability theory class. I was hoping someone would be able to walk me through the solution to this question (I ...
-6
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0answers
40 views

NUMBER OF ATOMS IN A SIGMA-ALGEBRA [on hold]

I have been trying to solve the followIng question. DESCRIBE THE SMALLEST SIGMA ALGEBRA CONTAINING 'n' ARBITRARY SUBSETS OF THE SAMPLE SPACE.GIVE AN UPPER BOUND FOR THE NUMBER OF SETS IN THIS SIGMA ...
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0answers
29 views

Geometric series: Convergence under which conditions?

For which functions $p(n)$ does $$\sum_{i=0}^{n} i (p(n))^i \rightarrow \infty$$ but $$ \frac{1}{n} \sum_{i=0}^{n} i (p(n))^i \rightarrow 0$$ Or stated differently: I want $$ 1 \ll \sum_{i=0}^{n} i ...
0
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2answers
33 views

Finding independence of two random variables

We're learning about independent random variables in the context of multivariate probability distributions and I just need some help with this one question. If $f(y_1, y_2)=6y_1^2y_2$ when $0\leq y_1 ...
1
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1answer
40 views

Poisson, Gamma distribution example.

Can someone explain me answer for these questions? Suppose customers arrive at a store as a Poisson process with λ = 10 customers per hour. The Poisson process of X ∼ Poisson(λ) the time until k ...
0
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0answers
23 views

Testing for the independence of random variables

In probability theory, $X$ and $Y$ are independent if: $f_{X|Y}(x|y)=f_X(x)f_Y(y)$ If I have sample $Y_1,...,Y_n$ and I would like to test if $Y_i$ is independent from the rest of the sample, I ...
2
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1answer
33 views

Using Taylors to show convergence in probability

I'd like to show that \begin{equation} \sqrt{n} \left( (1-\frac{1}{n})^{n\bar{X}} - e^{-\bar{X}} \right) \to 0 \end{equation} in probability for a random variable with mean $\mu$ and finite variance ...
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0answers
41 views

When can one represent the conditional expectation $E[X|Y]$ as $g(Y)$ with continuous $g$?

Given two random variables $X$ and $Y$ we know that $E[X|Y] = g(Y)$ where $g$ is a Borel function. Is it a good question to ask under which condition there exists a function $g$ which will be ...
0
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1answer
43 views

solving a simple inverse problem related to elliptic pde

Suppose that I have the elliptic PDE $\nabla(\nabla A(x)\cdot U(x)) = 0$ where $x \in [0,l_1]\times [0,l_2]$ with boundary conditions $U(0,x_2) = 0, U(l_1,x_2)=1$ and $U_{x_1}(x_1,0)=0, ...
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0answers
44 views

Is my interpretation of Bayesian probability and inference correct?

I have the following interpretation of the Bayesian probability and inference (without referring to Measure Theory, I am still at the very beginning of learning it): Let's say we have five random ...
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1answer
31 views

Derive/ prove: p(a,b|c) = p(a|b,c).p(b|c)

How can this expression be derived? p(a,b|c) = p(a|b,c).p(b|c) where a,b,c are random variables. UPDATE: from the following ...
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0answers
16 views

Category theoretic view of coupling measures/RVs

Here is a general definition of the word "coupling" that covers every use I've seen of it. (And this generality is necessary because sometimes one does not define a coupling on an exact product space, ...
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1answer
38 views

Proof of “continuity from above” and “continuity from below” from the axioms of probability

One of the consequences of the axioms of probability ($\sigma$ field and probability axiom) is the "infinite subset" and "infinite union" property, I can't figure out how it follows from them. if ...
1
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1answer
523 views

Correlation between Beta distributions

I have a Computer Science background and not very knowledgeable in Probability and Statistics. So excuse me if my question,notation, or language is flawed. Anyways, the problems is that we have two ...
0
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0answers
34 views

Uniform intergrability of the maximum of a series of iid random variables [on hold]

Suppose $X_n$ are independent and identically distributed $L_1$ random variables. Let $T_n = \max_{1\leq j\leq n} X_n$. Is $\frac{T_n}{n}$ uniformly integrable?
2
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2answers
65 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 ...
0
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0answers
20 views

Jacobian for a matrix transformation: Example of Cholesky decomposition

I would like to generally understand how the Jacobian of a matrix transformation can be computed. As a concrete example, consider the Transformation from a (correlation) matrix to its Cholesky ...
3
votes
1answer
106 views

What's $\mathbb E[\mathbb E[Y|X]|Y]$? Is it well-defined?

Consider two real random variables $X,Y$ and the conditional expectation $\mathbb E[Y|X]$, also a random variable. What is the conditional expectation $\mathbb E[\mathbb E[Y|X]|Y]$? Is it $=Y$? Is it ...
2
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1answer
46 views

$\mathbb E[\mathbb E(X|Y, Z)|Y]$ or $\mathbb E\{\mathbb E[(X|Y)|Z]\}$?

To begin with, the standard iterated law of probability is as follows. $$ \mathbb E X = \mathbb E [\mathbb E(X|Y)]. (1) $$ I am perfectly happy with $(1)$ and there is also some quite good ...
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0answers
50 views

Writing probability as log

I have a question regarding the log probability and I am confused on this. The question is: $$\hat P^{(t)}(x)=\sum_{i=1}^N v_i^{(t)}P_i^{(t)}(x)$$ which is some function of size $N$. The question ...
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2answers
141 views

The distribution of barycentric coordinates

Let $\mathcal{X} = \{x_1,\dots,x_n\} \subset \mathbb{R}^d$ and let $Z$ be a random variable uniformly distributed over convex hull of $\mathcal{X}$, denoted as $\text{conv}(\mathcal{X})$. Assuming ...
1
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2answers
59 views

Proof of, and requirements for, the reverse of Jensen's Inequality for concave functions

As I understand it, Jensen's Inequality states $$\int_{U}f_{V}\left(h(u)g(u)\right)du\geq f_{V}\left(\int_{U}h(u)g(u)du\right)$$ For a convex function $f_{V}$, a probability distribution $g(u)$ on ...
0
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1answer
819 views

Finding joint cdf and pdf of independent random variables

Let $X$ and $Y$ be independent random variables. Each has an exponential distribution with parameter $\lambda$. Define two new random variables by $W = \min({X,Y}) $ $Z = \max({X,Y})$ Find the ...
0
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2answers
25 views

Show that $A \cup B$ and $C$ are independent as well

Show that if 3 events ($A$, $B$ and $C$) are independent, that the events $A \cup B$ and $C$ are independent as well. It seems pretty logically straightforward but how do you show this statistically. ...
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0answers
37 views

Characteristic functions

Here $E(Y)$ means the expected value of $Y$. 1) Could any one explain for me how to get from (2.7) to (2.8) ? 2) Why does the author know to define $\phi_1(u)$ and $\phi_2(u)$ in such a way? ...
1
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1answer
48 views

Kelly criterion for 3 outcomes

I have been exploring the Kelly criterion for optimizing the bet size for a two outcome bet situation. I'm having trouble applying this to a three outcome bet. I may refer to this excellent thread: ...
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0answers
83 views
+50

Unbiased asymptotic variance

Problem: Let $X_1,...,X_n$ be indep. r.v.'s that satisfy, for $i = 1,...,n$, $E(X_i) = \mu_i(\theta)$ & $\mathrm{Var}(X_i)= \sigma_i^2(\theta)$. $\theta$ is the parameter of interest and the ...
0
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2answers
42 views

Please just tell me if my working is correct.

It is known that $P(X = n) = p_n = \frac{2}{3^n}$ $X$ is the number of attempts needed to win the lottery. The question: Find $P(X>5)$ My take: $P(X>5) = 1 - P(X\leq 5)$ => is this correct? ...
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1answer
31 views

Show that $k = 2$ (Use the fact that $\sum_{n=1}^{\infty}p_n = 1$)

The set of numbers $p_1, p_2, p_3, ..., p_\infty$ such that $P(X = n) = p_n = \frac{k}{3^n}$ define an infinite probability space associated with the number of attempts, X needed to win the lottery. ...
0
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1answer
23 views

Total law of probability in continuous space

I am finding little difficulty in the following definition of total probability specified in a NLP related paper. Say $q^i$ is a partition of my continuous sample space. The authors have defined the ...
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13 views

Stopped supremum of the Brownian local time still $L^p$ bounded in space?

Let $B_t$ be a standard Brownian motion and $L_t^x$ its local time in $x$ at time $t$. For fixed $t$ and $p>1$, it holds that $$ \sup_{x \in \mathbb{R}} \operatorname{E} [ (L_t^x)^p ] < ...
2
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1answer
41 views

Relation between uniformly distributed random variable and i.i.d Bernoulli sequence (Cantor space)

(Uniform RV <==> i.i.d Bernoulli sequence) (1) Let $(X_n)_n$ be a sequence of i.i.d. Bernoulli random variables($P(X_n=0)=P(X_n=1)=\frac 12$) on a probability space. Then show that $\xi:= \sum_n ...
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0answers
28 views

Probability Distributions and Random Discrete Variables

How do you read this? For (a) do we let $X= 1/6, 1/2, 1/5$ and $2/15$ and sub into the equation, $$ Y=X^2-2X. $$ How do we go about solving this?
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22 views

Intuition about generating functions

I am trying to gain some intuition about moment generating functions. In particular, for a random variable $X$, we have $$ \newcommand{\E}[1]{\mathbf{E}\!\left[#1\right]} M_X(t) = \E{e^{Xt}} = ...
1
vote
1answer
39 views

Why is this statement false?

If $P(A^C) = \alpha$ and $P(B^C)=\beta$ then $P(A \cup B) < 1 - \alpha - \beta$ It is false. I know that and I can visualize it but how can I show it statistically?
0
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1answer
38 views

Suitable martingales and optional stopping theorem

Starting at value 0, the fortune of an investor increases per week by 200 with probability 3/8, remains constant with probability 3/8 and decreases by 200 with probability 2/8. The weekly increments ...