Tagged Questions

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
12 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 ...
-3
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0answers
28 views

what is the PDF of this random variable? [on hold]

please i want to find the PDF of this random variable :according to the center limit theorem i have random variable as expressed below which is summation of exponential of many random phases and i ...
0
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0answers
14 views

The distribution of a function of uniform and a complex Gaussian random variable

Hope this question is clear and straight to the point, if not than I can edit it accordingly. Given the following independent random variables with distribution $$X_i\sim \mathcal{CN}(0,1) $$ ...
0
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0answers
7 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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2answers
26 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 $g$ ...
1
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1answer
20 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 ...
2
votes
1answer
20 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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0answers
14 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 ...
0
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1answer
7 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 ...
2
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0answers
14 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) ...
2
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1answer
20 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 ...
2
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0answers
31 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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0answers
16 views

conditional expectation convergence in L1 [on hold]

Let $\mathcal{F}_n \uparrow \mathcal{F}_{\infty}$ and $Y_n \rightarrow Y$ in $L^1$. I'm stuck on how to show that $E(Y_n | \mathcal{F}_n ) \rightarrow E(Y | \mathcal{F}_{\infty})$ in $L^1$?
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0answers
16 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 | ...
3
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0answers
37 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 ...
3
votes
1answer
39 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 ...
0
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0answers
9 views

conditional dependence and sum of random variables

I know that $Y \perp\!\!\!\perp (X,Z)|A \Rightarrow Y \perp\!\!\!\perp X|A$, but is the following true? 1) $Y \perp\!\!\!\perp (X+Z)|A \Rightarrow Y \perp\!\!\!\perp X|A$ I feel like the addition ...
2
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0answers
27 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 distribtution as $aX$ for some real $a$, what are the possible characteristic functions of $X$. Let $\varphi_X(t)$ be ...
0
votes
1answer
23 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 ...
1
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0answers
28 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 ...
0
votes
1answer
13 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 ...
0
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0answers
14 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 ...
1
vote
1answer
32 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 ...
6
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0answers
38 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 $(\frac{X_i}{n^{\alpha}})_{i=1}^n$? More ...
1
vote
1answer
19 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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0answers
9 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 ...
0
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0answers
33 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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0answers
6 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 ...
1
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1answer
27 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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0answers
12 views

Finding the infinitesimal generator of a M/M/2 queue [on hold]

I have a M/M/2 queue with a total population of 5. The arrival times are independent exponential random variables with mean of $\lambda$ and the service times have a mean of $\mu$. The initial number ...
4
votes
1answer
42 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 ...
5
votes
1answer
43 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 ...
0
votes
1answer
28 views

How to generalize the solution of the problem? [on hold]

We have 8 boxes. Each box holds a maximum of 4 balls. We randomly place 15 balls into the boxes. What is the probability that the balls are distributed in 6 boxes?
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0answers
29 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 ...
0
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0answers
20 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 ...
0
votes
1answer
14 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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votes
0answers
12 views

Calculate the conditional expectation of an exponential r.v. [duplicate]

The question is like this : given that $Y$ is an exponential random variable with $E(Y)=1$, calculate $E(Y|min(Y,t))$. Honestly I have no idea. Can anyone help? Many thanks.
0
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1answer
38 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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0answers
13 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 ...
1
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0answers
22 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 ...
0
votes
2answers
25 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) = ...
0
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1answer
14 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 ...
2
votes
1answer
32 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 $\delta>0$ ...
0
votes
2answers
10 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 ...
1
vote
1answer
27 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 ...
0
votes
1answer
20 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?
0
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1answer
27 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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0answers
20 views

Acceptance-Rejection Method [on hold]

Consider the PDF of a random variable $X$ defined as follows: $$ f(x) = \begin{cases} x(1-x/2) & \text{ if $0 \leq x<1$} \\ 0 & \text{ otherwise} \\ \end{cases} $$ Using ...
0
votes
0answers
21 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 ...
0
votes
2answers
23 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 ...