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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random process and random function

Are random/stochastic process and random function both mapping-valued random variables, as mentioned in random elements? If yes, how is the $\sigma$-algebra defined on the set of sample paths for a ...
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If $P$ a probability of a sentence to be true, then $\{P(\phi | T_i)\}_{i \in \mathbb{N}}$ is a martingale over constructed theories $T_i$

I am reading Section 2.1 of Definability of Truth in Probabilistic Logic. For a language $L$, fix a probability distribution $P:L \to [0,1]$. Enumerate sentences $\phi_1, \phi_2, ...$ of a language ...
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20 views

Martingales proof

Can someone help me show that if $X_1,X_2,\ldots$ are i.i.d with mean $\mu$, then $Y_n=\sum_{i=1}^n (X_i-\mu)$ is a martingale? In general how do i verify if a sequence is a martingale other than just ...
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19 views

Steps: How to derive Probability density function for geometric functions

I am not from Mathematics background and hence lack awareness of many basic knowledge. So, please pardon if this sounds too trivial. I would like to know the steps with which I can obtain the pdf of ...
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21 views

How to develop a probability distribution/density function of an issue?

Assume that a health insurance company has $1000$ customers. It is estimated that the probability of a customer making a claim is $p = 0.2$ per year, independently of previous claims and other ...
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Reference requests for an opt-cited result in Jennrich (1969)

Lemma 2 on page 637 of Jennrich (1967) states that: Let $Q$ be a real-valued function on $\Theta\times Y$ where $\Theta$ is a compact subset of a Euclidean space and $Y$ is a measurable space. ...
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14 views

Kaplan Meier Derivation

Can someone please help me follow this proof of KP: http://data.princeton.edu/pop509/NonParametricSurvival.pdf My problems are the assumptions at the beginning: If a subject is censored at t its ...
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18 views

Mean time for the renewal process

The system is as below. Energy keeps coming at a node with a constant rate ρ. Node has files of size exponential(λ) to be transmitted (with fixed rate of transmission $r$, time for transmission would ...
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22 views

Independence of Bernoulli r.v. and product

Let $X_1,X_2$ be independent random variables each assuming only the values $+1$ and $-1$ with probability $1/2$. Are $X_1,X_2,X_1X_2$ pairwise independent ? Are $X_1,X_2,X_1X_2$ an ...
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11 views

An example shows the difference between inference in Bayesian network and Junction Tree

Why inference in Junction tree is more efficient? There are directed graph BN and the corresponded undirected graph transformed by Junction tree algorithm. The literature describes that inference in ...
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15 views

What is the limiting distribution of this Markov Chain?

Take a Markov Chain with state space $\left\{ 0, 1, \dots, 20 \right\}$. Then we have the rule that given $X_n$: Compute $Z = X_n + 1$ or $Z = X_n - 1$ with probability $\frac{1}{2}$ each (if the ...
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34 views

Convergence of $|f_n(x+n)-f(x+n)| \le \frac{\log\log \left(1+\frac{x+n}{\gamma}\right)}{\log(1+n)}$ on $x \in [0,1]$

I have the following relationship between $f_n$ and $f$ on $x \in [0,1]$ \begin{align*} |f_n(x+n)-f(x+n)| \le \frac{\log\log \left(1+\frac{x+n}{\gamma}\right)}{\log(1+n)} \end{align*} for all $x \in ...
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8 views

Understanding “Latent Variables”

I'm having troubles understanding the method of calculating/estimating a latent variable. I know that a latent variable is something unobserverd and therefore unknown that is ought to explain ...
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22 views

If the difference of two i.i.d. random variables is normal, must the variables themselves be normal?

I previously asked a similar question about the sum of two i.i.d. random variables, thinking the two cases to be equivalent. But I can't see how to apply the proof of that case to this one. It is ...
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17 views

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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12 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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10 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 ...
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20 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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36 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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7 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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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 ...
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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 ...
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22 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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15 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, ...
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28 views

What is the Vapnik-Chervonenkis dimension of sigmoidal functions?

Consider the following class of functions: $F=\{f_w:R^d \rightarrow [a,b], f_w(x)=\sigma(w^Tx), \forall x\in R^d\}$, where $\sigma(\cdot)$ is a sigmoidal function (e.g. tanh, or sigmoid so it has ...
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18 views

Measurability and random variables

Let $(\Omega, \mathcal{B})$ be a measurable space and $X$ a r.v. taking values in $\mathbb{R}$. Let $\sigma(X)$ be the sigma-field generated by $X$ and $\mathcal{B}( \mathbb{R})$ the Borel ...
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23 views

increasing functions are borelian functions

Let $X : \Omega \rightarrow \mathbb{R}$ a discrete random variable and $f : [0, \infty) \rightarrow (0, \infty)$ an increasing function. Show that $f$ is a borelian function. It suffices to show that ...
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15 views

Statistical probability of a team scoring in soccer using historical data

New to the forum, and I am not that clued up on Statistical probability (I have just bought "Understanding and calculating the odds:...") so thought the questions is best asked here: Given the ...
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44 views

Poisson distribution and idd random variables - Proof of an equality

I want to solve the following task The first part was very easy for me. But I dont know how to solve the second one. I guess I even understood it completely. I am thankful for any kind of help!
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36 views

Indepenent variables and functions

Random variables $x_1, x_2,...,x_n$ are independent. Then, how to prove whether these functions $$y_1=f_1(x) \\ y_2=f_2(x) \\ ... \\ y_n=f_n(x)$$ are independent or not . where, $x=(x_1,...x_n)$ ...
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14 views

Principle of deferred decisions: How to model this?

It is quite easy to see that if we have to pick a random string $x \in\{0,1\}^n$, that we can pick each symbol $x_i\in \{0,1\}$ uniformly at random. Moreover, if we have to pick a random subset of a ...
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18 views

A joint density with positive dependence

Suppose $f:[0,1]^2\to R_+$ is a joint density function for X and Y satisfying (i) $f(x,y)=f(y,x)$ and (ii) $X,Y$ are positive correlated, i.e. $f$ is a copula or log-supermodular Denote ...
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27 views

How does one explicitly compute the rate function for a given probability distribution?

For a given probability distribution $\mu$, the associated rate function is $I_\mu(x) = sup \left \lbrace x \lambda - ln \left( \int e^{\lambda t} \mu (dx) \right) \right \rbrace$, and if there ...
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18 views

The probability of a function whose components obey exponential distribution

Thanks for your attention, here are the details: $${x_i} \sim \exp \left( {{\lambda _i}} \right) , i = 1,2, \cdots ,6 ,a > 0$$ $$\Pr \left\{ {1 + \frac{{a\frac{{{x_1}}}{{{x_3}}} \cdot ...
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32 views

Counterexample in Probability Theory: non trivial property satisfied by indicators but not for all measurable functions.

When one has done or studied a few proofs in Probability Theory of results of the type 'The property P is satisfied by all measurable functions realizes that usually it is enough to prove that all ...
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26 views

To obtain the probability density function of a complicated function of six independent random varibles

Here is the super complicated function: $$Z = \frac{{\left( {CE + DF + 1} \right)\left( {CA + DB + 1} \right)}}{{ABCD + \left( {CA + DB + 1} \right)}}$$ where $A,B,C,D,E,F$ are independent random ...
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15 views

Example of caculation the cummulative probability

I have a question about probability that I am confusing. I have k bit symbols. Now, I want to calculate the successful decoding probability for k bits. It can defined by this equation ...
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21 views

Same limit for a different convergence

How can you show that if a sequence $(x_n)$ converges in $L^2$ to $X$, and converges almost surely to $Y$, then $X$ and $Y$ are almost surely the same. Can you give me a hint?
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16 views

Show that if mean of $X_i = m,$ Then $P(\frac{X_1 + \ldots + X_n}{n} < m/2) \to 1$ as $n$ is large.

So mean of $X_i = m, Var(X_i) = \sigma^2.$ I tried to use Weak Law of large numbers but can't seem to finish it. I want to show that $P(\frac{X_1 + \ldots + X_n}{n} - m/2 > 0) \to 0$, which is an ...
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27 views

Determining the joint density of W and Z when W = X/(1-X) and Z = Y/(1-Y)

The exercise relates to joint density $$f(x,y)= Cx^{\alpha-1}y^{\beta-1}(1-x-y)^{\gamma -1}$$ for $x > 0, y > 0, x + y \leq 1$ and $$C = ...
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Show that $X_n$ is likely $0$ for large $n$ if $P(X_n = e^n) = 1/n, P(X_n = 0) = 1 - 1/n.$

As the title states, we have a sequence of random variables $X_1, X_2, \ldots$ such that $$P(X_n = e^n) = 1/n, P(X_n = 0) = 1 - 1/n.$$ Show that as $n$ gets very large then $P(X_n = 0) \to 1.$ So ...
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21 views

M/M/1 queue, probability of time spending in queue..

Let $W$ be the time $nth$ customer spends in the queue when $n$ go to $\infty$. How do we write down the formula for $P[W \le t | N = k]$ ? where $N$ is the number of customer in the system.
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31 views

Probability of going from one character to another in k steps?

You are given a starting lowercase alphabet (b/w 'a' and 'z' inclusive) and a target lowercase alphabet. You are also given a 26*26 matrix denoting the probability of going from one character to ...
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21 views

What is the expectation of the number of collisions when inserting elements into a hash table?

If the table is of size $N$, and the hash function is $f(x) = x\mod N$, and linear probing is used to solve collisions, what is the expectation of the number of total collisions when inserting $k$ ...
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31 views

Doob's decomposition Thm_ Got stuck applying induction in proving $Z_{n+1}$ is $F_n$ measurable?

Already know $Z_0=0$, and $$Z_{n+1}=E(X_{n+1}|F_n)-X_n+Z_n$$ $X_n$ is $F_n$ measurable, $F_n$ is a filtration. How to prove $Z_{n+1}$ is $F_n$ measurable? I tried to prove by induction. Since $Z_1$ ...
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12 views

Simplifying $\sum_i[z_i'(\delta-\hat{\delta})]^2x_ix_i'$ to apply a law of large numbers

I'm in the context of linear regressions. Let $n$ be the sample size and for $i=1,\ldots,n$, let $$ \underbrace{x_i}_{K\times 1},\quad \underbrace{z_i}_{L\times 1}\quad(n>K\geq L) $$ be column ...
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30 views

A Counter Example of Doeblin Condition

The question is to prove that the following Markov process doesn't satisfy the Doeblin Condtion. Let $X=\{\ldots,-n,\ldots,-1,0,1,\ldots,n,\ldots \} $, The Markov Transition Matrix $P$ is defined as ...
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21 views

Sum converging a.s. (cont'd)

Let $X_k$ be independent random variables with zero expectation s.t. $\sum_{k=1}^nX_k\rightarrow_{a.s.} X$ and $X$ has a nondegenerate distribution. Is there a way to estimate $E[(\sum_{k=1}^\infty ...
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5 views

Expected number of distinct nodes visited in a directed bipartite graph

Let $G = (V,E)$ be a directed bipartite graph with $V = \{I \cup O\}$ where $\left\vert{I}\right\vert = n$ and $\left\vert{O}\right\vert = m$. All the edges start from a vertex in $I$ and end on a ...
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17 views

Is there a unique tilted measure with specified marginals?

Suppose $\mathcal{A},\mathcal{B}$ are finite sets and $\mu_{A,B}(a,b)$ is a probability measure on the product set $\mathcal{A}\times \mathcal{B}$ so that $\mu_{A,B}(a,b)>0$ for each $a\in ...