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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fractional Brownian motion is not a semimartingale. How to apply Ergodic theorem in the proof of this theorem?

Here is the proof of the theorem. I couldn't understand how to apply Ergodic theorem in this proof. Let $X=(X_t)_{t\geq0}$ be a fractional Brownian motion with self-similar parameter $H\in(0,1)$. We ...
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1answer
72 views

How to deal with probability problems in proper class sample spaces?

Here my focus is mainly on $Ord$ and questions such as: An ordinal is chosen by random. What is the probability of the event that it is a cardinal number? A coin is tossed proper class many ...
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1answer
40 views

Martingales and variance

For a martingale $(Z_n)_{n\in \mathbb N}$ define $X_i=Z_i-Z_{i-1}$ with $Z_0=0$ Show: $$Var(Z_n)=\sum_{i=1}^nVar(X_i)$$ My attempt: We can write $Z_n=\sum_{i=1}^nX_i$, so we actually just have to ...
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what will be the PDF of magnitude and phase of this random variable?

i have a random variable as shown in the figure and i tried to find the PDF of the magnitude and phase of this random variable using central limit theory as i mentioned , i know that if we have ...
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40 views

What is the probability of this question?

On a single draw from a deck of playing cards the probability of selecting heart is 1/4 the probability of selecting a black card is 1/2. what is the probability of selecting both a heart and a black ...
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1answer
244 views

Exponential distribution: waiting time at post office

Consider a post office with two clerks. Three people, A, B and C, enter simultaneously. A and B go directly to the clerks, and C wait until either A or B leaves before he begins service. What ...
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108 views

Markov Chain: Moving on a circle

A particle moves on 12 points situated on a circle. At each step it is equally likely to move one step in the clockwise or in the counterclockwise direction. Find the mean number of steps for ...
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1answer
30 views

Notation with random variable $\overline{X}_{n^2}$ in Strong Law of Large Numbers proof.

I'm reading the proof for the strong law of large numbers. It says: Let $X_1,X_2,\ldots$ be a sequence of independent and i.i.d. random variables with finite mean $\mu$ and finite variance ...
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18 views

Posterior tail probability is absolutely continuous?

Suppose that the distribution of $X$ given $\theta$ is absolutely continuous with respect to Lebesgue measure on $\mathbb{R}$, for each value of $\theta$. Denote the conditional density with ...
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50 views

Formula the conditional probability of mables

I have a interesting question that need your help. I have two sets A and B. Set A have 10 marbles that numbered from 1 to 10. Set B have 6 marbles that numbered from 1 to 6. Randomly choose $g$ ...
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49 views

Relating a Gamma Distribution to an Exponential one?

Question related to Gamma and Exponential random variables. Suppose I have a Gamma random variables with shape and scale parameter $m$ and $\theta$ i.e $$X\sim\Gamma(m,\theta)$$ respectively. Can I ...
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1answer
35 views

On Schwarz Zippel Lemma

Theorem (Schwartz, Zippel). Let $P\in F[x_1,x_2,\ldots,x_n]$ be a non-zero polynomial of total degree $d≥0$ over a Field $F$. Let $S$ be a finite subset of $F$ and let $r_1,r_2,...,r_n$ be selected at ...
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104 views

Why do two points never 'arrive at once' in a Poisson point process

In the following, all the measure spaces are endowed with the Borel $\sigma$-algebra corresponding to their topology (we take the usual topology on $[0,\infty)$). Let $E$ be a Polish space and let ...
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1answer
20 views

Expected number of visits to a state of a Markov chain up to a certain time

Let $P=\{p_{ij}\}$ be a stochastic matrix (with rows and columns indexed by a countable set) and let $p^{(k)}_{ij}$ be the entries of $P^k$. I'm trying to prove that, if the associated Markov chain is ...
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29 views

(Billingsley, 2nd ed, 1968) D space, (12.33) inequality proof

Convergence of probability measures, Billingsley, 2nd ed, p132, Theorem 12.4 This is what I want to prove where $x \in D \equiv$ the set of cadlag functions defined on $[0,1]$ and $w_x^{''}(\delta) ...
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1answer
46 views

Help with Linear Transformation of a multivariate normal

Given X ~ $N_2$ (μ, Σ)$ Find the Distribution of $$ \begin{pmatrix} X+Y \\ X-Y \end{pmatrix} $$ Show independence if $Var(X) = Var(Y)$ Attempt: Given proper of ...
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99 views

Probability and Quantum mechanics

I don't quite understand how the probability language of sample spaces, $\sigma-$algebra, random variables, etc, fit into the quantum mechanics' formalism. To wit, we usually say that an observable ...
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49 views

martingale convergence proof

This is out of Durrett 5.5.7. Let $X_n \in [0,1]$ be adapted to $\mathcal{F_n}$. Let $\alpha,\beta > 0$ such that $\alpha + \beta = 1$. Suppose that $$ P(X_{n+1} = \alpha + \beta X_n | ...
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1answer
122 views

Find a symmetric random walk on $\mathbb{Z}$ that is transient.

I wanted to know if it is possible find a symmetric random walk on $Z$ that is not recurrent. Let $X$ have the following distribution, with a probability $1/2^{i+1}$, $X=\pm b_i$. Let ...
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1answer
51 views

Where do I go wrong?

Suppose $X,Y$ are independent Uniform$(0,1)$ random variables. Find the probability $P(Y\geq X\mid Y\geq\dfrac{1}{2})$. Please note that I know the correct answer and that I have arrived at the ...
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81 views

If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ has the distribution of $aX$, find all characteristic functions of $X$.

If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ have the same distribution as $aX$ for some real $a$, what are the possible characteristic functions of $X$? Let $\varphi_X(t)$ be ...
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1answer
29 views

Uniformly Converging Sequence but Not Normly Converging

I want to find a sequence $(f_n) \subset \mathcal L^1(\mathbb R)$ of integrable functions which uniformly converges to $0$ but with $\int f_n \nrightarrow 0$. I came up with the following example: the ...
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70 views

Show that a measure is a probability measure

I have trouble with this question: We define an arc segment $B(\theta, \eta, r, R)=\{x \in \mathbb{R}^2\vert \omega(x)\in [\theta,\eta], \Vert x \Vert_2 \in [r,R] \}$ where $0 \leq \theta \leq \eta ...
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1answer
39 views

Uniform integrability of the average of i.i.d.s

So I'm being asked to show that for an i.i.d ${X_i}$, ${n^{-1}S_n}$ is uniformly integrable provided $X_i\in L_1$ My professor keeps insisting that $S_n$ is a number and not a random variable, which ...
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27 views

Doob's decomposition of $X_n=\sum _{m=1}^n1_{B_m}$, where $B_m \in \mathcal{F}_m$.

$X_n=\sum _{m=1}^n1_{B_m}$, where $B_m \in \mathcal{F}_m$. I want to find the Doob's decomposition. I think $X_n=Y_n+Z_n$, where $Y_n$ is a martingale, $Z_n$ is a predicable process. Then ...
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1answer
62 views

Probability puzzle : Expected no. of coins in the smaller pot

There are two pots of coins having size m & n. A new coin is thrown and goes to 1st pot with probability m/(m+n) and to 2nd pot with probability n/(m+n). We start with both pots of size 1 & 1 ...
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74 views

Donsker's Theorem for triangular arrays

Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
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1answer
28 views

Rewriting Gaussian r.v. $Z$ as sum of two independent Gaussian r.v.

Suppose, $Z$ is Gaussian r.v. assume that it has mean 0 an variance 1. My question is can $Z$ be rewritten as \begin{align*} Z=\rho Z_1+(1-\rho)Z_2 \end{align*} where $Z_1$ and $Z_2$ are independent ...
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17 views

What is the Fractional Functional Central Limit theorem?

What is the statement of the "Fractional Functional Central Limit theorem (FFCLT)"? There is a Functional Central Limit Theorem, also called Donsker's theorem. Which has a Wikipedia article . I ...
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45 views

Inequality involving the inverse of a covariance matrix

Consider the following covariance function: $$ k(s_i, s_j) = e^{-|s_i - s_j|/2}. $$ Take $s_i \leq 0$ for $i = 1, \dots, n$ and construct the following matrix and vector: $$ A = (k(s_i, s_j))_{i,j ...
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17 views

Conditional Likelihood estimation

I'm reading a book called "Bayesian Reasoning and Machine Learning" and I have come across a question that gives me some problems. Unfortunately I neither have solutions, nor anyone else who I know ...
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1answer
44 views

Probability : Maximize the expected payoff

Given $2$ random variables $X, Y$ that take integer values with uniform distribution from $0$ to $100$. You play a game in which a random value of $x$ comes first & you have to decide if the ...
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1answer
96 views

What are some good references on how probability theory got mathematically rigorous?

I am working on a term paper for an analysis course and I thought it would be interesting to talk about the connection between analysis and probability theory. Honestly, it would also benefit me a lot ...
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1answer
54 views

Bounded Measurable Functions on [0,1]^2

Suppose $f(x,y),g(x,y)$ are two functions on $[0,1]^2$ that are bounded and measurable, such that: $$ \int_0^1 f(x,u)g(y,u) du \leq 1 $$ for almost every $(x,y) \in [0,1]^2$. Show that $$\int_0^1 ...
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31 views

Is there a better way to mathematically use this data than the way I am doing it?

I am trying to use math to predict NFL fantasy football scores. My current process for projecting a players score is as follows: For every team (32 teams), I list the average points it gives up to ...
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40 views

Usual augmentation filtration? (Sigma algebra generated by a descreasing family of sets)?

My aime is to understand the usual augmentation filtration. More pricesely, I want to understand the last identity in this PDF file. http://onlinelibrary.wiley.com/doi/10.1002/0470863617.app1/pdf ...
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1answer
29 views

Substitution in conditional expectation

A paper I'm reading does something like the following: Random variable $Y$ has the property that $E[e^{mY} \mid X] \leq 1$ for all $m\in\mathbb{R}$. Hence it is claimed that $E[e^{XY}] \leq 1$. How ...
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1answer
40 views

Further explanation regarding calculation of E[X^2]

I was reading over the following evaluation of $ E[X^2] $ on the following pdf: http://crab.rutgers.edu/~guyk/dmlec/lectures/lec15/l15.pdf. This part was especially confusing for me: ...
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1answer
73 views

Covariance of $Z'Vb$ given that the rows of V are i.i.d.

Suppose that we have the following entities $$ \underbrace{Z}_{n\times k},\quad\underbrace{V}_{n\times L},\quad \underbrace{b}_{L\times 1}. $$ $Z$ and $b$ are nonstochastic whereas we assume that the ...
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29 views

Applying the Law of Large Numbers recursively

If I want to apply the LLN for an estimator that uses another estimator, can I apply the LLN inside the summation and after it simplify the outer summation by using the expected value of the inner ...
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2answers
39 views

Translating expected values between two sets of related iid variables

The setting: $\mu$ is a probability measure on $\mathbb{R}$, $f: \mathbb{R} \to [0, \infty)$ so that $0 < ||f||_{L^1(\mu)} < \infty$, and $v$ is another probability measure defined by $v(A) = ...
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1answer
28 views

Calculating probabilities over longer periods of time further explanaton

I found a question posed on here regarding a 5% chance of fire per month, and how does one take that probability out to one year. I follow the original explanation (see below). My question is why do ...
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1answer
50 views

If $X \in L^{1}(\mathbb{R})$ then $\int_{B} |X| < \epsilon $ for $P(B)< \delta$

Let $(\Omega, \mathcal{B}( \mathbb{R}), P)$ be a probability space and $X \in L^{1}(\mathbb{R})$ a random variable. Show that for any fixed $\epsilon>0$, there always exists a $\delta>0$ ...
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17 views

Prove that a composition of a borel function and a random variable

Question: If $g$ is a Borel function and X is an RV, c is an atom of X. Prove that $\lbrace X=c\rbrace \subseteq \lbrace g(X)=g(c)\rbrace $ . Thoughts: This was given as a part of a proof. I don't ...
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1answer
35 views

Algebra and partions of a set

My book in mathematical finance introduces algebras and partitions of a set, in order to explain how information is modeled to the investor. But there is one thing I don't get. They say that for every ...
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1answer
33 views

How to prove Markov's inequality $P_X(X\geq t) \leq \frac{\mathbb{E}[X]}{t}$?

As the subject states, how can Markov's inequality $P_X(X\geq t) \leq \frac{\mathbb{E}[X]}{t}$ be proven? Is the proof distribution-dependent or there is a general way to prove it?
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35 views

Dealing with Conditinal Expectation

if i have E(X1a|Ft) and 1a is independent of X and Ft.However i dont know if X is independent of Ft. Can i still split the conditional expectation into E(X|Ft)E(1a)=E(X|Ft)P(A)? cheers
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29 views

Convergence in law of random variables

Let $X_1,\ldots,X_n \overset{i.i.d}{\sim} \ Uniform(0,1)$ and write $M_n = \max(X_1,\ldots,X_n)$. If we have proved that $M_n \overset{a.s.}{\rightarrow} 1$, then we know that $(M_n)$ converges in ...
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2answers
38 views

How to solve the PDF of lognormal distribution using the Normal Distribution

Let $X$ be $N (\mu,\sigma^2)$.Define the random variable $Y=e^x$ and find its probability distribution function. My solution is this, let $G(y)= P(Y\le y)=P(e^x\le y) =P(X\le ln y)$.Let ...
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1answer
24 views

Basic questions about Counting Process

I am learning Counting Process in my probability course. The following comes from Sheldon M. Ross*'s *Introduction to Probability Models: If we say that an event occurs whenever a child is born, ...