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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-2
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1answer
107 views

Given $B \cup A = B$ and probability and set theory axioms, prove $\mathbb{P}(A) \leq \mathbb{P}(B)$.

I need to prove that $\mathbb{P}(A)$ is less than or equal to $\mathbb{P}(B)$ using only this three things: $B \cup A = B$ The three axioms of probability: a) $\mathbb{P}(A)$ is greater or equal to ...
1
vote
1answer
35 views

Conditional Distributions

Choose a random integer $X$ from the interval $[0, 4]$. Then choose a random integer $Y$ from the interval $[0, x]$, where $x$ is the observed value of $X$. Make assumptions about the marginal pmf ...
5
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1answer
203 views

Set of all probability measures with finite support

Let $X$ be an uncountable set endowed with the discrete topology. Let $\mathcal{P}(X)$ be the set of all Borel probability measures on $X$, and consider the subset $A$ of $\mathcal{P}(X)$ consisting ...
0
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1answer
96 views

Meaning of $\mathcal A_{\tau}$ for stopping time $\tau$.

Let $(X_n)$ be a stochastic process, adapted to a filtration $\mathcal A_n$, and let $\tau$ be a stopping time, then $$ \mathcal A_{\tau} := \left\{ A \in \sigma\left(\bigcup_n A_n\right) : A \cap \{ ...
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2answers
46 views

An $m$ sided dice is rolled $n$ times what is the chance of getting an average of $\frac{m+1}{2}$?

If I roll a six sided dice twice there is a $1$ in $6$ chance that the results will sum up to $7$ (giving an average of $3.5$ per dice). And if I only roll it once it is not possible to get a $3.5$. ...
1
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1answer
97 views

Optional Stopping Theorem for Stochastic Processes with Constant Mean (and not a Martingale)

Let $(X_n)_{n\geq 1}$ be a martingale with respect to $(Y_n)_{n\geq 1}$, i.e., the martingale condition $$ \mathbb{E}[X_n|Y_1, \ldots, Y_{n-1}] = X_{n-1} $$ holds. From this condition, it follows ...
0
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1answer
436 views

Expected value of the product of i.i.d random variables

Assume we have random variables $$X_i \,\,\,\ \text{ i.i.d } \,\,\ i\in[1:n]$$ with expected value $$\mathbb{E}[X_i] = \frac{1}{2}$$ Now let us compute the following expected value of the product of ...
0
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2answers
123 views

Expected delay problem on expectation based on uniform distribution

At a traffic junction, the cycle of traffic light is 2 minutes of green and 3 minutes of red. What is the expected delay in the journey, if one arrives at the junction at a random time uniformly ...
0
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1answer
226 views

Is $W^3(t)$ a martingale if $W(t)$ is a Brownian motion

Is $W^3(t)$ a martingale if $W(t)$ is a Brownian motion? The answer seems like no to me. Using Ito's lemma I can write $$W^3(t)=\frac{3}{2}W^2(t)+\int_0^t3W(u)dW(u)$$ The second piece on the LHS is an ...
2
votes
1answer
104 views

Example: convergence in distributions

Give an example $X _n \rightarrow X$ in distribution, $Y _n \rightarrow Y$ in distribution, but $X_n + Y_n$ does not converge to $X+Y$ in distribution. I got a trivial one. $X_n$ is $\mathcal ...
3
votes
1answer
39 views

Does wikipedia state the definition of probability correctly?

In the wikipedia article on probability http://en.wikipedia.org/wiki/Probability it says: To qualify as a probability, the assignment of values must satisfy the requirement that if you look at a ...
7
votes
0answers
374 views

The Expectation of a function of independent random variables

Assume we have for some index $i>n$ ($n \in \mathbb{N} $) the following ${\it Independent \ Random \ Variables}$ $$h_i \sim \text {i.i.d }\ \ \mathcal{CN}(0,1) \ \ \text{ Complex Gaussian}$$ ...
0
votes
1answer
30 views

if a sequence of random variables $X_n \to Y$ in distribution and $X_n \to Z$ in distribution $\Rightarrow$ $F_Y=F_Z$?

In handouts provided by a professor I read: if a sequence of random variables $X_n \to Y$ in distribution and $X_n \to Z$ in distribution $\Rightarrow$ $F_Y=F_Z$. It does not feel right to me. $X ...
0
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2answers
50 views

What is the meaning of the below probability equation?

Can someone explain the intuitive idea behind this probability equation (especially the part where the limit of epsilon downarrow zero notation).
2
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1answer
82 views

Showing convergence by using Levy's continuity theorem

Let $X_n \sim \mathrm{Po} (n)$. I want to show that $Y_n = \displaystyle \frac{X_n -n}{ \sqrt{n}}$ for $n \rightarrow \infty$ converges in distribution to a random variable that is standard normal ...
1
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0answers
53 views

Using Stirling's formula to uniformly bound Bernoulli success probabilities (part 2)

In this paper, the authors say that for any $\gamma \in [1/2,1)$, there is a positive constant $A=A(\gamma)$ such that for any $n$, $$ \sum_{n\gamma\leq k \leq n} \binom{n}{k} \geq A n^{-1/2}2^{n ...
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3answers
36 views

Is this a conditional probability or not?

Suppose that the telephone calls during one minute time follow a Poisson distribution with mean=4. If people can handle at most 6 calls per minute, what is the probability that the people will receive ...
1
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2answers
46 views

infinite sum of normal r.v. is still a normal r.v. when given $\sum \limits_{i=1}^\infty a_i^{2}$ is finite

If $X_1, X_2, ...$ are i.i.d.standard normal random variables and for real constants $a_1, a_2, ...$, given $\sum \limits_{i=1}^\infty a_i^{2} $ is finite, then $Y_n =\sum\limits_{i=1}^n a_iX_i$ ...
4
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1answer
3k views

Proof of the Box-Muller method

This is Exercise 2.2.2 from Achim Klenke: »Probability Theory — A Comprehensive Course«. Exercise (Box–Muller method): Let $U$ and $V$ be independent random variables that are uniformly ...
1
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1answer
48 views

Convergence of probability measures problem

Let $\mu_n$ be sequence of probability measures on a polish space $S$ such that for any bounded and continuous $f:S \to \Bbb R$ we have $$\int fd\mu_n \to \int fd\mu$$ Then I have seen in some place ...
0
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1answer
45 views

What are the possible values of Z=X+Y

If I have two independent probability mass function, where $P_{x}(0)=\frac{1}{2}$ , $P_{x}(2)=\frac{1}{2}$ and $P_{y}(1)=\frac{1}{6}$ , $P_{y}(2)=\frac{1}{3}$ , $P_{y}(3)=\frac{1}{2}$ I am asked ...
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0answers
38 views

“Expected value” of Thirteen card game !! [duplicate]

Thirteen cards numbered $1$ to $13$ are shuffled and dealt one at a time. "Match" occurs on deal $k$ if $k$th card revealed is card number $k$ Let N be the total number of matches that occur in the ...
2
votes
1answer
53 views

Convergence in distribution conditions

I'm reading the weak convergence section (Ch.$18$) in "Probability Essentials" by Protter and Jacod. Theorem $18.7$ states that "$Xn \overset{D}{\to} X \iff \underset{n\to\infty}{\lim} E[g(X_n)] = ...
4
votes
0answers
100 views

Almost surely either $X_n=0$ for some $n$ or $\lim_{n\to\infty}X_n=\infty$

How I came to this: Let $\{X_n\}_{n\in\mathbb{N}}$ be a sequence of non negative random variables, $\mathcal{F}_n=\sigma(X_l,l\leq n)$ the sequence of corresponding sigma-algebras and define ...
2
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0answers
99 views

Calculate probability of nodes in a graph

I have the following graph: A and F post the same joke on Facebook. Now there is a probability of 0.6 that node b will post the joke too. and so on... So the weights on the edges say how likely it ...
9
votes
2answers
491 views

Weak topologies and weak convergence - Looking for feedbacks

I am currently trying to get exactly what the weak and the weak* topologies are, in particular in connection to the concept of weak convergence in measure, however I am not completely sure on what I ...
2
votes
1answer
336 views

Uniformly integrable martingales and stopping time

I want to prove the statement below: Theorem: Let $(Y_n,\mathfrak{F})$ be a uniformly integrable martingale. Show that $(Y_{T\wedge n},\mathfrak{F})$ is a uniformly integrable martingale for any ...
4
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1answer
186 views

Techniques for proving asymptotic normality by Taylor expansion?

Suppose I have a sequence of densities $$ f_{X_n}(x) = \exp[\ell_n(x)], \qquad (x \in A). $$ My goal is to prove a statement like $\sqrt n (X_n - \mu) \to N(0, \sigma^2)$ in distribution, for an ...
3
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0answers
44 views

Calculating $\mathbf{P}[X < Y]$ for $X, Y$ exponentially distributed?

This is exercise 2.2.1 from Achim Klenke: »Probability Theory — A Comprehensive Course« Let $X$ and $Y$ be independent random variables with $X \sim \exp_\theta$ and $Y \sim \exp_\rho$ for certain ...
0
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1answer
29 views

An equality from the well-known analysis of variance formula

Suppose that we have a parametric model $(\mathcal{Y},\{P_\theta:\theta\in\Theta\})$ dominated by some measure $\mu$. That is, each $\theta$ is associated with a density $l(y;\theta)$. Let $S(Y)$ ...
2
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0answers
78 views

Deriving mean and variance of a function of Gaussian process

Suppose $\mathbb{G}$ is a tight zero mean Gaussian process and $F$ is an absolutely continuous CDF $$Y=\int_a^b\frac{d\mathbb{G}}{1-F}+\int_a^b\frac{\mathbb{G} \, dF}{(1-F)^2}$$ I know that $Y$ is a ...
3
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0answers
127 views

How to prove product of measurable set is measurable using dynkin's $\pi-\lambda$ theorem?

My question: If $E_1$ and $E_2$ are measurable subsets of $R^1$, I want to show that $E_1 \times E_2$ is a measurable subset of $R^2$ and $|E_1 \times E_2|=|E_1||E_2|$. My attempt: First I tried to ...
1
vote
1answer
93 views

Unbounded stopping time

Suppose we have a sequence of i.i.d. random variables $(X_n)_{n \in \mathbb{N}}$ with $\mathbf{P}(X_n = -1) = \frac{1}{2}, \mathbf{P}(X_n = 0) = \frac{1}{3}, \mathbf{P}(X_n = 1) = \frac{1}{6}$. Denote ...
1
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2answers
107 views

Proving completeness of the average of a random normal sample

Suppose that $n$ is a fixed positive integer and $\theta$ is a parameter belonging to $\Theta=\mathbb{R}$. Suppose that we are given that $Y_1,\ldots,Y_n$ are i.i.d. $N(\theta,1)$. I'm trying to show ...
4
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1answer
183 views

Exercise in Probability/Measure Theory

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Let also $A_{n,j}\in\mathcal{F},n\in\mathbb{N}_0,j\in\{1,2,3,...,2^n\}$, be such that for all ...
0
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0answers
56 views

Create a Martingale out of a Markov Chain.

Consider a homogeneous finite state Markov chain $\{X_n\}$ with transition matrix $P$ and state set $S$ consisting of real numbers. How to choose the elements of $S$ so that $\{X_n\}$ be a ...
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0answers
35 views

About moments in a quantile processes

Let $q_{n}(t)$ be the $nth$ quantile processes ($t\in (0,1)$) based on the distribution F: $$q_{n}(t) = \{\sqrt{n}[F^{-1}_{n}(t)-F^{-1}(t)]\}.$$ In this case, $F^{-1}$ is the (generalized) inverse of ...
1
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0answers
34 views

Convergence in probability of the Fisher information

Given a family $\{\mathbb{P}_\theta\}_{\theta\in\Theta}$ on $\mathcal{B}(\mathbb{R})$, where $\Theta\subset\mathbb{R}$ and each member of this family is absolutely continuous w.r.t. $\lambda^1$, and ...
0
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1answer
28 views

Does law of total probability apply here?

Let $X$ and $Y$ by positive independent random variables. Let $f(x,y)=\frac{ax}{y^2+ay}-\frac{ab}{y}$ where $a>0$ and $b>0$ are constants. I am wondering if the following is true: ...
2
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1answer
27 views

Condition in a theorem in Probability theory.

I passed by a simple theorem in Probability theory , yet it really bugs me that I think that 1 condition in the hypothesis is not necessary. After checking the proof for many times, I still can't ...
0
votes
1answer
278 views

Expected value of a complicated function of more than one random variable.

Assume we have random variables with Probability Density Functions (pdf) as follows $$\omega_i \sim f_{1},\,\,\,\,\ i \in[1:n]$$ $$ \gamma= \{\gamma_1,\cdots,\gamma_n\} \sim f_2: \text{joint pdf of ...
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0answers
39 views

Inverse Bessel Process

Is there any reference on this process? For example, analytical derivations for the hitting times, density, etc? Im studying local martingales and am interested in the density of hitting times for ...
0
votes
1answer
149 views

PDF of distance from the center of a random point in the unit disk

I found in a certain website (also in an IEEE paper) that the probability function for the distance mentioned in the title is given by the following: $P(d)=2d$, but no one is giving the way to derive ...
3
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1answer
123 views

Exercise on Martingales

I have been struggling with the following exercise and I was wondering whether my solution is correct or not. I am pretty sure about the second part of the question (the martingale part) but not so ...
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0answers
37 views

Question about Lebesgue Dominated Convergence Theorem involving a Markov Time / Stopping Time

I am trying to understand the proof of the following lemma: Let $W$ be an arbitrary random variable satisfying $\mathbb{E}[|W|] < \infty$, and let $T$ be a Markov time (or stopping time) for which ...
0
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1answer
45 views

Dependence of RVs exponentially distributed

Let $X$ and $Y$ be random variables that are both exponentially distributed with different parameters. I dont get the idea how can they be dependent then... and we can only manipulate one parameter in ...
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1answer
29 views

Does $P(X\leq a) = P(X^2\leq a^2)$ if $X$ is a positive random variable and $a>0$?

The answer looks positive to me, since $$P(\omega:X(\omega) \leq a) = P(\omega:X(\omega)^2\leq a^2)$$ Am I right?
1
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1answer
46 views

Characteristic function, how to integrate?

Find the characteristic function $\phi_X(t)$ of an absolute continious r.v. $X$ with density: $$f_X(x) = \frac{a}{2}e^{-a|x|} \qquad (a>0; x\in \mathbb{R})$$ Notation I have some ...
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0answers
43 views

given distribution function find the density

I am confused about one trivial thing. We have independent variables $X,Y,\beta$ where $X$ has a distribution function $F(x)$, $Y$ has $G(x)$ and $\beta$ is $Bin(1,p)$ distributed. Also $X,Y$ have ...
2
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0answers
89 views

Is a limit of a sequence of distribution functions is necessarily a distribution function

I got a question: "is a limit of a sequence of distribution functions is necessarily a distribution function". My answer is NO. I have a counterexample: $F_n(x)=0$ if $x<n$ and $1$ otherwise. It ...