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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20 views

Number of rewards before death

I have a question regarding Poisson events with death. Assume time is continuous $t\in[0,\infty)$. A person may die with intensity $\delta$. Conditional on being alive, he may achieve a reward with ...
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1answer
10 views

Can a finite set support a $\sigma$-algebra

Assume the set $\Omega$ is finite (finite number of elements). $A$ is a collection of subsets of $\Omega$. It is clear that $A$ can be an algebra, but is $A$ then also automatically a ...
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1answer
25 views

If $w$ is a discrete random variable then is $P(w|x)$ a density or mass function? [on hold]

$w$ is a discrete random variable. $x$ is a continuous random variable. Then should I denote $P(w|x)$ as a probability density function or probability mass function, and why ?
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2answers
28 views

Borel $\sigma$-algebra, definition

This following statement is from a book: Let $C$ denote all open intervals. Since every open set in $\mathbb{R}$ is the countable union of open intervals, we have $\sigma(C)=$ the Borel ...
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1answer
20 views

Monotone likelihood property and first order stochastic dominance

I have a question regarding first order stochastic dominance. Give two pdf $f(x)$, $g(x)$, $x\in[x_0,x_1]$. For all $x$ on the support, I have $$ g(x) = f(x)\cdot H(x) $$ where $H(x)$ is continuous, ...
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1answer
26 views

Example of uncountable union of sets that is in $\sigma$-algebra but does not satisfy additivity property of probability

Let $(\Omega, \mathcal{F}, \mathrm{P})$ be a probability space. I know that $\mathrm{P}$ should satisfy: $$ \mathrm{P}(\cup_{n \geq 1}A_n) = \sum_{n \geq 1} \mathrm{P}(A_n) $$ for disjoint family of ...
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0answers
32 views

The Gambler's Ruin without using random walks

This is more of a doubt. I understand that this problem can be described with Markov chains and the recursion solved without much trouble. However I've seen that some people casually say that $$ ...
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1answer
36 views

What is a nice, clean proof to show that a fair coin toss satisfies axioms of probability?

If we assume H=Heads T=Tails and we're dealing with a fair coin what is a good way we can show that Kolmogorov Axiom has been satisfied?
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0answers
23 views

What can be said about $E(X|N)$ when $N \subset \sigma(X) $?

When $\sigma(X) \subset N $, $E(X|N) = X$ a.e.. When $N \subset \sigma(X) $, is there some result about $E(X|N)$? Or is it no more particular than an arbitrary $N$? Thanks.
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1answer
24 views

Determine measurability of E(X|N) or even $\sigma(E(X|N))$?

Suppose $(\Omega, F, P)$ is a sample space, $X$ a random variable, and $N$ a sub sigma algebra of $F$. How can we determine $\sigma(E(X|N))$? How is $\sigma(E(X|N))$ related to $\sigma(X)$ and ...
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2answers
43 views

Independence between conditional expectations

Suppose $(\Omega, F, P)$ is a sample space, $X$ and $Y$ random variables, and $N$ and $M$ sub sigma algebras of $F$. I know that $E(X\mid N)$ and $E(X\mid\{\emptyset, \Omega\})$ are independent. ...
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0answers
13 views

Source coding with 2 distinct distributions and entropies

I'm learning about source coding, and many of the books/resources I've read give examples of the source $X^n$ being defined as a sequence of iid random variables. How about when the sequence is ...
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1answer
22 views

Are the converses of the following special cases of conditional expectation also true?

Let $X$ be a random variable, and $N$ be a sub sigma algebra of the underlyign sigma algebra of the sample space. if $X$ is in $L^1$ and measurable wrt $N$, then $E(X|N)=X$ a.e.. Is it true that ...
0
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2answers
28 views

Inequality for the expected value of the sum of Bernoulli random variables

I'm stuck with this seemingly simple inequality. Suppose that $X_1,X_2,\ldots$ are Bernoulli random variables and denote $S_n=X_1+\ldots+X_n$. Let $n_k=\inf\{n:\operatorname ES_n\ge k^2\}$ for ...
0
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3answers
32 views

What is the probability that after pulling out of a card deck 3 heart cards, that the 4th card will be also a heart?

What is the probability that after pulling out of a card deck 3 heart cards, that the 4th card will be also a heart? There are 52 cards in the deck and there is no replacement. $$P(4\text{th heart} | ...
3
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1answer
38 views

$\sum_{n=1}^{\infty}P\left( |X_n-X|>\epsilon\right)<\infty$ for each $\epsilon>0$ $\implies$ $X_n\to X$ a.s.

Problem. Let $\left( \Omega, \mathcal A, P \right)$ be a probability space and $X,X_1,X_2,...$ random variables on $\Omega$. If $\sum_{n=1}^{\infty}P\left( |X_n-X|>\epsilon\right)<\infty$ for ...
6
votes
1answer
37 views

Continuity of a stochastic process

Exercise 3.11 in Oksendal's book "SDEs: an intorduction with applications". Let $W_t$ be a stochastic process satisfying 1) ${W_t}$ is stationary; 2) $t_1\not=t_2\implies W_{t_1}$ and $W_{t_2}$ are ...
1
vote
1answer
30 views

Characteristic function of a measurable set.

Let $X=L^p[0,1]$ $(1\leq p<\infty)$ be the Lebesgue space of p-integrable real functions on $[0,1]$. Let $D\subseteq [0,1]$ be measurable subset. The characteristic functions $\chi_D$ is defined as ...
0
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0answers
18 views

Max function as bounded functions

I have an algebra of bounded functions $A$ that contains the constant functions and is closed under uniform convergence. We also have that if $f \in A$ then $|f| \in A$. I'm trying to show that if $f, ...
0
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1answer
32 views

A Characterization of the Strong Markov Property

I have a question concerning the strong Markov property: For a strong Markov process $(X_u)_{u\ge 0}$, a real time $t\in \mathbb{R}$ and an optional stopping time $T$ with $t< T$ \begin{align*} ...
1
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0answers
14 views

Infinite sets of rvs equal in distribution

Assume $\{\mathcal{L}(X_{k})\}_{k\in I}=\{\mathcal{L}(Y_{k})\}_{k\in I}$ for all finite $I\subset \mathbb{N}$ i.e. equal in all finite dimensional distributions. Then I want to show that ...
1
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2answers
29 views

Equivalence of Definitions of lim inf of Sequence of Sets

Prove : $\{w : w \in A_n \text{ for all $n$ except a finite number}\}= \bigcup_{n=1}^{\infty}\bigcap_{k=n}^{\infty}A_k$. I am trying to prove these two definitions are equivalent but I am having ...
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0answers
30 views

How to prove the following

Let $\mathbf{A}\in\mathbb{R}^{p\times n} (n\ge p)$ be a positive definite symmetric matrix having a Wishart distribution with mean $\mathbf{0}$ and covariance $\boldsymbol\Sigma\otimes \mathbf{I}$. ...
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0answers
35 views

What is the probability that from 23 people 2 people have their birthday on the same day?

What is the probability that from 23 at least people 2 people have their birthday on the same day. Assume that the year has 365 days and that all the birthday combinations have the same probability. ...
1
vote
1answer
23 views

Quick question on why two measures have equal total mass.

I am following Probability with Martingales by Williams I am having troubles with why the two measures $H \rightarrow P(I \cap H)$ and $H \rightarrow P(I)P(H)$ have the same total mass $P(I)$. Is ...
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0answers
23 views

how to determine presence of an event with a degree of confidence proportional to a set of observations and conditional probabilities

My probability theory has become a bit rusty and i can't seem to figure out how to determine the presence of a malfunction within a device given a set of observations displaying a certain phenomenon ...
11
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1answer
81 views

Distribution of $\sum\limits_{i=1}^{N}X_{i}$ conditionally on $\sum\limits_{i=1}^{N}X_{i}^{2}$ for i.i.d. standard normal $X_i$s

Assume that the random variables $X_{i}$ are i.i.d $\mathcal{N}\left(0,1\right)$, then: $$S_N=\sum_{i=1}^{N}X_{i}\sim\mathcal{N}\left(0,N\right)\qquad\qquad ...
2
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0answers
46 views

Transformation of probability density function

I'd like to compute the pdf of $w= g_1(x) = \frac{x}{1+e^{-x}}$ in dependence of the density $f_x(x)$ with domain $x>0$. As I was not able to write the inverse function of $g_1(x)$, I tried the ...
4
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0answers
71 views

Max and sum of random variables

I have a set of independent random variables $\{A_1, A_2, B_1, B_2\}$. All of them have the same distribution function $F(x)$. I want to find distribution function of a variable $C$, where $C=max(A_1 ...
0
votes
1answer
20 views

Show that f is a density and find the corresponding cdf

$f(x) = \frac{(1+\alpha x)}{2} $ for $-1 \leq x \leq 1$ and $f(x) = 0$ otherwise, where $-1\leq \alpha \leq 1$. Show that $f$ is a density and find the cdf. I am mainly having trouble with finding ...
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0answers
31 views

Example where probability theory fails without $\sigma$-algebra

I have just started reading theory of probability in a measured theory based approach and was wondering if someone could give an example where probability fails without using $\sigma$-algebra (or ...
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0answers
25 views

Markov Property Definition

Let $(X_t)$ be a stochastic process on $(\Omega, \mathcal F, \{\mathcal F_t\}, \mathbb P)$. The typical definition of the Markov property is $\mathbf{P}(X_{t+s} \le x \, |\, \mathcal F_t) = ...
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1answer
7 views

Walking through the reduction of a cumulative probability function to a polynomial

Setup Define $P(p)$ as follows: $$ P(p) = \sum_{N_1-\phi \cdot N_2 \geq \theta} {n_1 \choose N_1} {n_2 \choose N_2} p^{N_1 + N_2}q^{n_1 + n_2 - N_1 - N_2}. $$ Here, $$ q = 1 - p. $$ The sum is ...
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0answers
16 views

Conditional expectation and Joint distribution

Given $(X,Y)$ and $(X',Y')$ pairs of rvs on different probability spaces $\Omega_{1},\Omega_{2}$ respectively but with equal joint distribution show that ...
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0answers
14 views

Questions about solution to finding solution to mode of a binomial distribution

So i read over the solution presented by Andre Nicolas: finding mode in Binomial distribution But i have a few questions about the whole thing: 1) why did he set the ratio as $\frac{a_{k+1}}{a_k}$? ...
0
votes
1answer
23 views

On a proof regarding the sigma algebra generated by a single random variable.

I left (b) and (c) for the sake of the curious. What I am trying to do is Exercise (a) except that I recall that $\sigma(Y):= ( \{ w : Y(w) \in B \} : B \in \mathcal{B} )$ is the definition of ...
5
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0answers
49 views

A generalization of simple random walk

Suppose $S_n, n\geq 0$ is a martingale on $\mathbb{R}$ such that $S_0=0$ and $|S_{n+1}-S_{n}|\in [\frac{1}{2}, 1]$. Prove that there exists $c,C>0$ s.t. $$ \frac{c}{\sqrt{n}} \leq P( S_1\geq ...
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1answer
18 views

Proof check on $I (\lim \sup E_n(w)) = \lim \sup I_{E_n}(w)$, where $I$ is the indicator function.

I proceed by cases: $ \lim \sup I_{E_n}(w) := \lim_m \downarrow \{ \sup_{n>m} I_{E_n}(w) \}$ this is $0$ only if $w \not \in E_n \forall n > m$. At the right of the equality we have ...
1
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0answers
63 views

How to construct $\mathbb{R}^N$ where $N$ is a random variable?

How does one rigorously construct $\mathbb{R}^N$ where $N$ is a $\mathbb{Z}^{++}$-valued random variable on some Borel probability space $(\Omega,\mathcal{B},\mathbb{P})$? Would someone be so kind ...
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0answers
29 views

how to distribute power in time intervals

Let's suppose the following game - there are two opponents with n power digits each and time frame of k time intervals.In each time interval each of the players decides how many power digits to invest ...
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3answers
29 views

Uniform PDF for continuous variable, why does the probability values increase to 1, when its normalized?

Consider a "spinner": an object like an unmagnetized compass needle that can pivots freely around an axis, and is stable pointing in any direction. You give it a spin and see where it comes to rest, ...
2
votes
1answer
46 views

Does $X ⊥ Y \leftrightarrow X ⊥ Y | Z$ implies $(X,Y) ⊥ Z$?

Let $X, Y$ and $Z$ be random variables. Let $p_1$ be the statement that $(X,Y) ⊥ Z$ (meaning $(X,Y)$ and $Z$ are independent), $p_2$ be the statement that $X ⊥ Y$ (meaning $X$ and $Y$ are ...
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votes
1answer
64 views

How to simplify the conditional expectation $E[v_3\mid v_1 < \max\{v_2,v_3\}, v_3=\max\{v_2,v_3\}]$ [on hold]

Suppose $v_1,v_2,v_3$ are three random variables drawn independently from the same distribution $\mathrm{uniform}(0,1)$, is it correct that $$E[v_3\mid v_1 < \max\{v_2,v_3\}, v_3=\max\{v_2,v_3\}] ...
1
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1answer
32 views

apply the law of total expectation

I'm a little bit confused about applying the law of total expectation. Suppose $v_1,v_2,v_3$ are three random variables drawn independently from the same distribution $\mathrm{uniform}(0,1)$, which ...
0
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0answers
31 views

Counterexample for $(Y_1 \perp Y_2) \mid (X_1, X_2) \Rightarrow (Y_1 \perp Y_2) | X_1$?

Let $(Y_1 \perp Y_2) \mid (X_1, X_2)$ mean that random variables $Y_1$ and $Y_2$ are conditionally independent on $(X_1, X_2)$. Either is there a counterexample for $(Y_1 \perp Y_2) \mid (X_1, X_2) ...
0
votes
1answer
41 views

Quick question on an example of bad probability theory.

From the text Probability with martingales by Williams. I don't understand why $p(C_n) = 0$, is it not $$P(C_n) = \lim_{n \rightarrow \infty} \# \{ k : 1 \le k \le n ; k \in C_n \} = \lim_{n ...
2
votes
3answers
64 views

Adding two discrete distributions

I am taking a probability course and I am having trouble adding two discrete distributions. The two distributions given are: $X$ has a discrete uniform distribution on the integers $0,1, ... ,9$. ...
2
votes
2answers
37 views

Is conditional expectation E(X|N) an a.e. equivalence class wrt N or underlying sigma algebra?

Let $X$ be a random variable defined on a measure space $(\Omega, F, P)$. Let $N$ be a sub sigma algebra of $F$. Then conditional expectation $E(X|N)$ is an a.e. equivalent class. Is the a.e. ...
2
votes
0answers
14 views

Wireless networks on two sequential office floors: Random partitions of a finite interval via a point process on a line

Construct a Poisson point process of density one on a line of length $L$. Allow each point in the process to "see" part of the line to their left, and part of the line the their right (such that the ...
5
votes
4answers
62 views

Supermartingale vanishing at some stopping time

Let $\left\{X_t\right\}_{t\in[0, T]}$ be a continuous and non-negative supermartingale. We define the stopping time $$\tau_0:=\inf\{t\in[0,T]:X(t)=0\}\wedge T$$ and immediately obtain by continuity ...