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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1answer
16 views

Calculating difference between two probability distributions.

What is a good measure of the difference between two probability distributions other than Kullback–Leibler divergence?
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5 views

L1 error for scale/translation classes

This is an example given in the article about testability (Devroye and Lugosi 1990). First will introduce my problem, given that we have a density class $\mathcal{F}$ that consists of density ...
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0answers
14 views

Properties of the essential supremum

In the article about testability of densities (Devroye and Lugosi 1990). They use some properties of the essential supremum I cannot find. First we have to assume that we have a class of density ...
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2answers
20 views

Limit value of a product martingale

This question came from a problem i was solving for self-study. I'll state the problem first: Let $Y_n \sim \mathcal N(0,\sigma^2)$ be independent normally distributed variables, $X_n = ...
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0answers
23 views

Uniform distribution on convex hull

Let $X=\{ x_1,\dots,x_n \} \subset \mathbb{R}^m$. Let $H(X)$ be the convex hull of $X$. Assume that $X$ is a convexly independent set, i.e. none of the $x_i$ are a convex combination of the others. ...
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2answers
54 views

At a party $n$ people toss their hats into a pile in a closet.$\dots$ [duplicate]

Question: At a party $n$ people toss their hats into a pile in a closet. The hats are mixed up, and each person selects one at random. What is the expected number of people who select their own hats? ...
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0answers
20 views

Weak convergence with respect to the uniform topology on cadlag functions

Suppose I have a random sequence $X_n$ of cadlag functions on $[0,1]$ that converge weakly to $X$. In general this is meant with respect to the Skorkhod metric but suppose here I have that $X$ is ...
3
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1answer
50 views

Example of non continuous random variable with continuous CDF

Can someone provide an example of $X$ being a non-continuous random variable with continuous cumulative distribution function? For instance: $X$ is discrete if it takes (at most) a countable number ...
2
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1answer
23 views

Local convexity of the topology of weak convergence of probability measures

Let $X$ be a Polish or standard Borel space, and $\mathcal P(X)$ be the space of all Borel probability measures on $X$ endowed with the topology of weak convergence. I am thinking of using ...
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2answers
10 views

Find the probability of each outcome when a biased die is rolled, if rolling a 2 or 4 is three times as likely as rolling each of the other$\dots$

Question:Find the probability of each outcome when a biased die is rolled, if rolling a $2$ or $4$ is three times as likely as rolling each of the other four numbers on the die and it is equally ...
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0answers
38 views

What did I do wrong when using Jacobian transformation

A device containing two key components fails when, and only when, both components fail. The lifetimes, $T_1$ and $T_2$, of these components are independent with common density function $f (t) = ...
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1answer
35 views

Question about a change of variable used to compute $E(X)$ from the CDF of $X$

I was studying a proof where the author shows that if the range of x is $\mathbb R_+$ and $F$ is the cumulative distribution function then: $$E[X] = \int_{0}^\infty (1-F(x))dx $$ However on one ...
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23 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 ...
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0answers
18 views

Applying Ito lemma to multi dimensional semimartingales

Let $X_t=(X^1,\dots,X^d$) be a d-dimensional semimartingale. Then the formula for Ito's lemma I have found in several places, including wikipedia, is: $f(X_T)-f(X_0)=\sum_{i=1}^d\int_0^T ...
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2answers
31 views

probability of a flipped coin

A fair coin is flipped three times. Let $A$ be the event that a head occurs in the first flip and $B$ be the event that exactly one head occurs. a) Find $p(A/B)$ b) Are $A$ and $B$ independent? ...
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2answers
30 views

If pages in a book have an iid Poisson number of errors, in 10 pages what is the probability that exactly 3 pages have exactly 1 error?

Suppose the number of spelling error on any given page in particular book can be modeled by a Poisson distribution with $\lambda=2$, and assume that the number of errors on different pages is ...
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0answers
39 views

Choosing random marbles until one is divisible by $X$ [on hold]

A box contains twelve marbles on which they are numbered by $1,2,3,...,12$. Now let $X$ represent the number of marbles you must choose with replacement until you obtain one with a number that is ...
1
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1answer
31 views

Proving and visualizing $\mathbf 1_{(x,x+a]}(y) = \mathbf 1_{[y-a,y)}(x)$

Here is a trick from one of the proofs in probability: $$\iint \mathbf 1_{(x,x+a]}(y) \ \lambda(dx) \ \mathbb P(dy) = \iint \mathbf 1_{[y-a,y)}(x) \ \lambda(dx) \ \mathbb P(dy)$$ for $a>0$. So ...
2
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1answer
64 views

Usual convex combination and the one with measure

Let $X$ be a Borel measurable subset of $\Bbb R^n$ and let $\nu$ be a probability measure on $X$. Can we always find an integer $m$, points $x_1,\dots,x_m\in X$ and coefficients $a_1,\dots,a_m \geq ...
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1answer
31 views

Reverse Fatou's lemma on probability space

Let $(\Omega, \mathcal{F},\mathbb{P})$ be probability space and $E_{n \in \mathbb{N}}$ be $\mathcal{F}$-measurable sets. Show example that reverse Fatou's Lemma, $\mathbb{P}(\limsup_n E_n) \geq ...
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1answer
35 views
+100

Extensions of universal measures

Let $(\Omega,\mathcal F)$ be a measurable space, and let $\mathcal P$ be the set of all probability measures no this space. Let $\mathcal F^p$ denote a completion of $\mathcal F$ w.r.t. $p\in P$ and ...
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0answers
25 views

What is the optimal prize for a prize ticket in a raffle [on hold]

What, if any is the optimal price for a prize ticket given the value of a prize? For example if you were to raffle a TV and wanted to cover the cost of the prize? Let say the people were aware of how ...
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0answers
24 views

probability theory books

since I'm going to deepen my knowledge of math stats I was wondering what book I should start from...I need one that covers in the very fine details the following topics trasformations among ...
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12 views

To show that if T is a Weibull then aT also have a Weibull distribution [on hold]

(a)Show that if $T$ is a Weibull then $aT$ also have a Weibull distribution (b)$S(aT)$ is also Weibull Survival function
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1answer
36 views

Weak convergence of distribution family

I know the convergence in distribution and the weak convergence. but I have two questions: First one: does weak convergence implies pointwise convergence or it is the same? And second one: I have ...
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0answers
31 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
29 views

Maps that preserve Brownian motion law

I am looking for a list of maps that take Brownian motion to Brownian motion: Here are some: Any rigid transformation ...
3
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1answer
64 views

Sum of normally distributed independent random variables, where one has a different (exponential) unit

$$X \sim \mathcal{N}(\mu_X,\,\sigma_X^2)$$ $$Y \sim \mathcal{N}(\mu_Y,\,\sigma_Y^2)$$ $\mu_X$ and $\sigma_X$ have unit decibel watt ($\text{dBW}$); $\mu_Y$ and $\sigma_Y$ have unit watt ($\text{W}$). ...
2
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1answer
37 views

A question about Malliavin calculus

An application of Malliavin calculus is to calculate the sensitivity of financial Greeks. However, as in the theory of Malliavin calculus, to take the derivative of a random variable, we need to ...
0
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1answer
22 views

Polynomial joint pdf $f(x,y)$ such that of $f(x) \neq f(y)$

How can I build a polynomial joint pdf $f(x,y)$ for $x \in [x_1, x_2]$ and $y \in [y_1, y_2]$ such that of $f(x) \neq f(y)$ or equivalently, $x$ and $y$ are depended on each other?
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2answers
31 views

An $L^1$ convergence problem

Is the following true? If $X_n$ converges almost surely to a non-negative random variable $X$ having finite expectation, and if $E(X_n)$ converges to $E(X)$, then $E|X_n - X|$ converges to $0$? ...
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4answers
125 views

What is the difference between $E[X\mid Y]$ vs $E[X\mid Y=y]$ and some of the properties of $E[X \mid Y]$?

I was trying to understand both intuitively and rigorously what the difference between $E[X\mid Y]$ vs $E[X\mid Y=y]$. Let me tell you first the things that do make sense to me. $E[X\mid Y=y]$ makes ...
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0answers
18 views

How to estimate the covariance matrix if the unnormalized pdf is known but integral is intractable? [duplicate]

Assume a $d$-dimensional random vector $x$, whose unnormalized pdf is known as the product of N multivariate t-distribution: $$Pr(x)\propto\prod_{i=1}^nt_{\nu_i,\mu_i,\Sigma_i}(x)$$ Is there any ...
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2answers
32 views

Resolvent operators and inverses proof

I am trying to prove for myself that $A(R_{\alpha}g)=\alpha R_{\alpha}g-g$ which is proving problematic. The definition of $A$, the generator, is $\displaystyle Af(x)= \lim_{t \rightarrow 0} ...
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0answers
25 views

Question on generators in the proof of Kolmogorov's Backward Equation

Here is a part of the proof of the Kolmogorov's Backward Equation. I cannot see why $Y_t$ has been picked as it has. In particular, I cannot see why you would want to subtract t in the first bit of ...
3
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2answers
33 views

Always null recurrence at the boundary between positive recurrence and transience?

I have the following theorem: Let $\rho$ be the traffic intensity. a) If $\rho<1$, then $X$ is positive recurrent. b) If $\rho>1$, then $X$ is transient. c) If $\rho=1$, ...
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0answers
21 views

Convergence of Beta Distribution to Bernoulli Distribution [on hold]

How will I show that the $$\beta\left(\frac{a}{n} , \frac{b}{n} \right)$$ distribution converges to the $$\operatorname{Bernoulli}\left( \frac{a}{a+b} \right)$$ distribution?
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1answer
38 views

Space of Gaussian Functions is Closed in $L^2$

Let $\Omega, \mu$ be a probability space. A measurable function $f: \Omega \rightarrow \mathbb{R}$ is called Gaussian if $$\mu (f^{-1}(A))=\frac{1}{\sigma \sqrt{2\pi }}\int_Ae^{-x^2/2\sigma ^2} dx$$ ...
2
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1answer
27 views

Finding the probability of ever visiting a transient state for a zero-seeking device for a Markov Chain?

A zero-seeking device operates as follows: if it is in state $j$ at time $n$, then at time $n+1$, its position is $0$ with probability $\frac{1}{j}$ or $k$ with probability $\frac{2k}{j^2}$, where $k$ ...
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1answer
46 views

Questions about expectation of stochastic integrals

I am considering the following SDEs: $$dX_1=-\theta(X_1-a_1)dt+\sqrt{X_1}(1-X_1)dW_1-X_1\sqrt{X_2}dW_2$$ $$dX_2=-\theta(X_2-a_2)dt-X_2\sqrt{X_1}dW_1+\sqrt{X_2}(1-X_2)dW_2$$ Here $W_1$ and $W_2$are ...
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1answer
36 views

Show that $Pr[X \gg Y]\approx 1$

Can one show (and how) that $$Pr[X \gg Y]\approx 1$$ for $$X:=\sum_{i=1}^k Bin\left(n\left(\frac{1}{2}\right)^i,i\right)$$ and $$Y:=\sum_{i=k+1}^{\infty} ...
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0answers
61 views

Prove $Pr[X + Y \geq x] \sim Pr[X \geq x]$

We have two independent random variables $X_n$ and $Y_n$, where $$X_n=\sum_{i=0}^n x_i$$ and $$Y_n=\sum_{j=0}^n y_j,$$ where $x_i$,$y_j$ are (non-identically) Bernoulli distributed and independent. ...
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0answers
16 views

Sequence of independent RVs [duplicate]

Let $(X_n)$ be a sequence of independent RVs that converges in probability to some RV $X$. Show that $X$ is constant almost everywhere.
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2answers
90 views

What is the probability of going bankrupt in roulette?

Imagine that the bank has the money $M_1$ and the player has the money $M_2$. The rules are the following: You win with a chance of $\frac{17}{36}$ and lose with $\frac{19}{36}$ each round. Now you ...
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0answers
29 views

What is meant here with $\omega _k \in A _j=A_{n_k } $ [closed]

I'm trying to understand what is meant by the following from Borovkovs probability theory. "Denote by $n_k $ the number of events $A_j $ such that $\omega _k \in A _j=A_{n_k } $, $n_k =0 $ if $\omega ...
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56 views

Random Variables Proof Help [closed]

I have two random variables $X$ and $Y$. Each of them having the conditions of Markov´s Inequality. So $X$ and $Y$ take on only nonnegative values, then for any value $a > 0$ $$P\{X \ge a\} \le ...
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30 views

On an existence of a real-valued measurable function on a non-atomic probability measure space [duplicate]

Suppose that $(X,E,μ)$ is a non-atomic probability measure space. Let $\xi :X \to \mathbb{R}$ be a random variable. A Borel measure $\mu_{\xi}$ in $\mathbb{R}$ defined by ...
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0answers
38 views

Joint probability of 3 random variables when their pairwise difference is given

Consider 3 discrete random variables $X_1,X_2,X_3$ defined over $\{0..T\}$, which are identically and uniformly distributed.They are correlated in the sense that their pairwise difference has a unique ...
2
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2answers
41 views

Probability that exactly 2 of 3 objects are in 1 of 3 baskets with sizes 5, 8, 2

I want to calculate the probability that some mutation occurs on a certain DNA section by a given number mutations. I rephrased it into this problem: Three (identical) persons enter a train (section ...
1
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2answers
73 views

Partial sum of binomial

I 'm trying to figure out a closed form solution for the following summation: $\sum_{j=0}^{\omega} j{n \choose j}p^{j}(1-p)^{n-j}$ where $\omega < n$ Is there any closed form solution?