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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35 views

How can we apply the Borel-Cantelli lemma here?

Let $(A_n)$ be a sequence of independent events with $\mathbb P(A_n)<1$ and $\mathbb P(\cup_{n=1}^\infty A_n)=1$. Show that $\mathbb P(\limsup A_n)=1$. It looks like the problem is practically ...
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1answer
17 views

Filtration right continuity completion

I have a question about filtration. Now fix a measurable space $(\Omega,\mathcal{M})$. Let $(\mathcal{M}_{t})_{t\in[0,\infty)}$ be a filtration on $(\Omega,\mathcal{M})$. We set \begin{eqnarray*} ...
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2answers
33 views

Probability of begin the first of two players to get H when tossing a coin

Two players, $A$ and $B$, alternately and independently flip a coin and the first player to obtain a $H$ wins. Assume player $A$ flips first. Suppose that $P(H)=p$. Show that for all $p$, ...
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2answers
37 views

A and B are independent under$\mathbb{P}$ but not under $\mathbb{Q}$

As the title, how to construct such an example that 2 events from the same measurable space ($\Omega$,$\mathscr{A}$) are independent under probability measure $\mathbb{P}$ but not independent under ...
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1answer
18 views

Is there a shorter way for me to answer this joint probability question?

Suppose a box contains 10 green, 10 red, and 10 black balls. We draw 10 balls from the box by sampling with replacement. Let X be the number of green balls, and Y be the number of black balls in the ...
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0answers
18 views

joint pdf f(x1,x2,x3), find marginal

Now, given a joint pdf $f(x_1,x_2,x_3)$, I found $$f_{x_1}=\int_{\text{support of $x_3$}}\int_{\text{support of $x_2$}} {f(x_1,x_2,x_3) dx_2 dx_3}$$ $$f_{x_2}=\int_{\text{support of ...
1
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1answer
20 views

Random variable $X^2$ determined by moments

Let $X$ be a real random variable, with standard normal distribution. Is the distribution of $X^2$ determined by its moments? In general, if $n \in \mathbb N$, is the distribution of $X^n$ ...
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0answers
20 views

optimization problem gaussian maximizes entropy

Let $X_1, X_2, Z_1$ be random variables and define $$Y=aX_1+bX_2+Z_1$$ I have the following optimization problem of difference of entropies, $$f=\max_{p(x_1x_2)} h(Y) - h(Y|X_2)= \max_{p(x_1,x_2)} ...
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0answers
14 views

Exchangeability and pairwise exchangeability

Suppose we have $\{Y_{i}\}_{i=1}^{n}$ that is exchangeable so the joint distribution of this sequence is the same as $\{Y_{\sigma(i)}\}_{i=1}^{n}$, where $\sigma:\{1,...,n\}\rightarrow\{1,...,n\}.$ ...
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0answers
23 views

Continuity preserved unter expectation? Dominated convergence?

Let $0<Z<\infty, 0<\mathbb{E}[Z]<\infty$ and $Z$ be atom-less, i.e. $\mathbb{P}(Z=z)=0$. Further, let $g:\mathbb{R}^+\to\mathbb{R}^+$ be continuous and strictly decreasing and $w$ be ...
1
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1answer
13 views

Proof of convergence in distribution

I want to prove that a sequence of random variables $X_n$ converges in distribution to $N(0,1)$, if we have the following condition for an arbitrary $\epsilon>0$: $$(1-\epsilon)Y_n \le X_n \le ...
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0answers
16 views

Simulating beta random variables

Let's say I estimated the parameters of Beta distribution with a single-period dataset, and want to generate a multi-period sample. How can I do that? To be more specific, I estimate my parameters ...
0
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1answer
27 views

Prove of Stopping time

Let $(X_k)_{k\in\mathbb{N}}$ be iid random variables with $\mathbb{P}(X_1=1)=\mathbb{P}(X_1=-1)=\frac{1}{2}$. Let $Z_n=\prod_{k=1}^n(1+X_k)$, so $Z_n$ a martingale. Consider ...
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1answer
54 views

Probability Question- A Poker Game

Jesse and three of his friends are playing Poker with 32 cards, The 32 cards are of every combination of the for patterns with the numbers 1, 7-13. In this game, each player takes five cards randomly. ...
0
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1answer
22 views

Derivative of integral over part of Gaussian distribution

I am currently trying to compute the following derivative and integral: $$ P\psi_\theta = \frac{d}{d\theta}\int_{-k}^k tf_T(t)dt, $$ where $t=x-\theta$ and $X\sim N(\theta_0,\sigma^2)$. $f_T$ above ...
0
votes
1answer
29 views

The unit circle(disk), $\sigma(X)$ measurable function

If I have two measurable functions $X,Y:S \to \mathbb{R}$ (with the Lebesgue mesure) such that $X(\{x,y\})=x$ and $Y(\{x,y\})=y$ on the unit circle that is $x^2+y^2=1$. Then is $Y$ ...
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0answers
23 views

Markov property question

In every book I can find, the Markov property for ito diffusions, $E[f(X_{t+h})\mid F_s] = E^{X_t}f(X_h)$ is stated for $\textbf{bounded}$ Borel functions. However, I have the following statement ...
0
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1answer
22 views

Variance from the pdf

FOr the interpretation of the variance, it is the fluctuation of the data around the mean. So if I know that mean (say mean=0), and then there are lots of data (70%) points are greater than +/-10 away ...
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0answers
25 views

Partial sums are alternate upper and lower bounds for $\mathbb{P}(\cup A_i)$

Show that $$ \sum_{k=1}^m(-1)^{k+1} S_k \leq \mathbb{P}(\cup_{i=1}^n A_i) \leq \sum_{k=1}^{m'}(-1)^{k+1} S_k$$ where $m, m' \leq n$, $m $ is even and $m'$ is odd, and $S_k = ...
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0answers
18 views

How to prove that the distribution function Fx is left continuous if and only if the distribution law µ is non atomic [on hold]

How to prove that the distribution function $F_x$ is left continuous if and only if the distribution law $\mu$ is non atomic. Can the law $\mu$ and lebesgue measure be singular if the distribution ...
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votes
1answer
24 views

Probability and Induction help [on hold]

Let $Y=X_1+X_2+ \cdots+X_n$ where $X_1, X_2, \ldots, X_n$ are independent Bernoulli random variables, each with probability of success equal to $q$. Use induction to prove that $Y$ has a Binomial ...
1
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1answer
23 views

A linear combination of characteristic functions is a characteristic function?

Let $\phi_k(t)$ be the characteristic function of a random variable $X_k$, $k = 1,2,\dots$. Consider a set of positive real numbers $\{p_1, p_2, \dots \}$, take a function: $$\phi(t) = ...
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2answers
30 views

Let $A$ and $B$ be events, and let $I_A$ and $I_B$ be the associated indicator random variables [on hold]

Let $A$ and $B$ be events, and let $I_A$ and $I_B$ be the associated indicator random variables. Show that : $I_{A\cap B} = I_AI_B = \min(I_A, I_B)$ and $I_{A\cup B} = \max(I_A, I_B)$
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2answers
26 views

Proof Question- Need Help [on hold]

Show that if $P(A|E) \geq P(B|E)$ and $P(A|E^c) \geq P(B|E^c)$, then $P(A) \geq P(B)$. I am reviewing for test, and I came across this problem in the textbook. I need help with this question.
3
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0answers
39 views

Convergence in probability and convergence of Cesaro means

Consider the random variable $X_n$, not necessarily iid. If $X_n\rightarrow 0$ almost surely, then the Cesaro means $\frac 1n\sum_{k=1}^nX_n$ converge almost surely to 0. This cannot be weakened to ...
0
votes
1answer
15 views

Soft: Interpretation of a periodic event on circle group

Recently I've been exploring probability measures on topological groups, derived from the (essentially) unique Haar measure defined thereon. I had begun to focus on the example of the circle group ...
0
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1answer
17 views

Where do we encounter sequence of infinite events of which we would like to study probabilities?

I have come across sequence of functions and numbers in the context of approximation theory and understand that a lot of theory of functional analysis came out with the idea to approximate solutions ...
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1answer
63 views

Explanation of how probability density functions transform under the change of variable

I've just read about probability density function from this article. In that article, there is some wired concept that I can't understand, please see the section named "Dependent variables and change ...
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1answer
13 views

Related to chi-squared functions

I'm finding difficulty in finding what type of function it is in continuous distributions in probability.Mainly how can i identify whether a function is chi-squared or not?
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0answers
33 views

Distribution of balls

I want to distribute $n$ ball in $m$ cells such that each cell has the capacity $n'$( each cell could ontain at most $n'$ balls). now there are two questions: 1 - what is the probability of having ...
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2answers
34 views

Continuous and Discrete random variable distribution function

I have a very basic question in probability. It pertains to the difference between a continuous random variable distribution function and a discrete one. This question has confused me many times. ...
2
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1answer
27 views

Coin-tossing games

Suppose that you start off with $100$ dollars. You toss a coin $10$ times and guess it right $5$ times and lose $5$ times (the order of the outcomes is not known). It is known that every time you ...
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2answers
64 views

Best E-books and online-resources for Probability and its applications(especially games of chance)

I am very much interested in studying games of chance and the probabilities related to our daily life instances but I need an online resource or some e-book to study them. I am a self-learner. Can ...
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1answer
37 views

Probability: basic question and concept

I have always been struggling with the problem, in particular, I usually have great difficulty in differenting when should I multiply n! to take care of the ordering, and when should I not do so. For ...
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1answer
20 views

Comparing $X^{-1}(E(X| \mathscr{G})(A))$ and $A$

Is it true that $X^{-1}(E(X| \mathscr{G})(A)) = A$ where $A \in \mathscr{F} $ and $A\notin \mathscr{G}$ where $(\Omega, \mathscr{F}, P)$ is the probability space?
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0answers
12 views

Recovering pmf from characteristic function

I'm having some trouble trying to recover the probability mass function of a discrete random variable from its characteristic function. I have seen that some continuous cases, you can recognize that ...
0
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1answer
17 views

Formally proving $\sum_{k=1}^{\infty}P\left(-k<X\leq-k+1\right)=P\left(X\leq0\right)$?

$\sum_{k=1}^{\infty}P\left(-k<X\leq-k+1\right)=P\left(X\leq0\right)$ This fact seems pretty obvious but how would I formally prove it, is there a painless way?
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0answers
20 views

Convergence of infinite union on probability space

Let $(\Omega,F,P)$ be a probability space. Let $B_n$ be a sequence of sets in $\Omega$. I'm trying to prove that $P((\cup_{n=1}^\infty b_n)\cap(\cup_{n=1}^k b_n)^C)\rightarrow0$ as ...
0
votes
1answer
19 views

Distribution of hitting position of line by brownian motion.

What is known about the distribution of the hitting position of a line by a 2d brownian motion? I've tried to make some simulations of a 2d brownian motion where every computational step has a ...
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1answer
26 views

Probability Question - Number of boxes one should look at

Will someone help me understand how to solve the following ? Jenny has 10 boxes all containing clothes. She is looking for her white pants, but has the following problem: while searching, she can ...
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2answers
39 views

Additive but not $\sigma$-additive function

Give an example of a measure space $(\Omega, \mathit{F})$ and a function $\mu$ on $\mathit{F}$ that is additive but not $\sigma$-additive, i.e. $\mu(\cup A_i)= \sum\mu(A_i)$ for a finite collection of ...
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1answer
28 views

Probability of words

The question is as follows: A word of $6$ letters is formed from a set of $16$ different letters of English alphabet (with replacement). Find out the probability that exactly two letters are ...
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1answer
25 views

Question about the construction of lebesgue measure of $(0,1]$ (or: the Borel sigma algebra is generated by the half open intervals)

I am reading the book "Probability with Martingales" In this book, the author constructs lebesgue measure on (0,1] as follows. Let $F =$ the collection of subsets of $(0,1]$ which can be written as ...
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0answers
14 views

MAP for exponential function (Maximum a posteriori)

I am trying to find the MAP for an exponential function of the form $p(y) = \theta.e^{{-\theta}y}$ Given that $\theta$ is constant, I want to estimate maximum $y$ = $p(y).p(X=x_i|y)$ for $i = 1..n$. ...
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1answer
33 views

Does bounded in probability imply convergence in probability?

A random variable, $X_n$, is defined to be bounded in probability if there exists an $M$ and $N$ for which \begin{align*} \mathbb{P}\big(|X_n| < M\big) > 1 - \epsilon,\ \forall n > N,\ ...
0
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1answer
31 views

Show $E$ is a valid expectation operator

Hello I am working within the confines of Probability via Expectation by Whittle. In this approach the expectation operator is given certain axioms. Thus I must answer the question from this ...
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1answer
37 views

If $\mathbb P_{Y|X\in A}(B):=\mathbb P(Y\in B \ | \ X \in A)$ can we explicitly define the r.v. $(Y|X\in A)$?

When introducing conditional expectation, one can define $\mathbb P_{Y|X\in A}(B):=\mathbb P(Y\in B \ | \ X \in A)$ which is itself a law. I was wondering if there is a way to define a random variable ...
2
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1answer
62 views

determine distribution by finding the moment generating function

Mathematical Statistics and Data Analysis, Rice, Chapter 4, Problem 92 $\theta$ is Gamma($\lambda,\alpha$) distributed, $X|\theta$ follows a Poisson($\theta$) distribution. Wanted: the unconditional ...
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16 views

Dolphin in a pool - using Kolmogorov's forward equations

Problem A dolphin swims between 3 different pools, A B and C. The time is spend in each pool, before going to the next one, is Exp(1/2). The possible ways for it to travel is A to B. B to C. C ...
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2answers
55 views

Proving formally $\lim_{x \to -\infty}\mathrm{Pr}( \left \lfloor{x}\right \rfloor \le X < x) = 0$ (Proof check)

we have $$\lim_{x \to -\infty}\mathrm{Pr}( \left \lfloor{x}\right \rfloor \le X < x) $$ where X is a real random variable, and $x \in R$. My idea of a proof would be by contradiction: Assume ...