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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26 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} | ...
2
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1answer
27 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 ...
1
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1answer
26 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
15 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
24 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*} ...
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0answers
8 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
23 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
25 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 $\boldsymbol\Sigma\otimes \mathbf{I}$. ...
0
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0answers
32 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
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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
21 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
78 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
44 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 ...
3
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0answers
56 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
16 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 ...
0
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0answers
27 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) = ...
0
votes
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
14 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
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1answer
22 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
41 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 ...
1
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1answer
16 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
57 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 ...
0
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0answers
25 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 ...
0
votes
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
45 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 ...
-1
votes
1answer
63 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
vote
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
40 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
61 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
36 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
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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 ...
0
votes
0answers
29 views

Calculating the Shannon information of drawing equal no. of cards

One card is drawn each from a $k$ deck of 52 cards where $k$ is a multiple of $52$. I need to prove that information of an outcome where each card appears the same number of times tends to ...
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0answers
14 views

exponential martingale [on hold]

If any one can help, I would really appreciate that. If I have random walk Sn constructed from summation of iid random variables Xi such that each Xi equals (Ai-Bi), where Ai is exponentially ...
0
votes
1answer
15 views

Bounding probability based on binary values

I've been reading this paper on probabilistic logic: http://ai.stanford.edu/~nilsson/OnlinePubs-Nils/PublishedPapers/problogic.pdf On page 76 theres a 3d diagram and Nilsson mentions the bounds on ...
1
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1answer
28 views

Calculating conditional probability of discrete uniform r.v.

X is a discrete uniform random variable on $\{a, a+1, a+2, ... , b\}$ with mean 7 and variance 4. Find $Pr[X \leq 6| X > 4]$ I'm not familiar with the discrete uniform distribution. I was ...
2
votes
1answer
15 views

Why $\Bbb{E}f$ is $\cal F_0$ measurable if $f$ is independent of $\cal F_0$?

In my professor's lecture note there is a remark saying that "$\Bbb{E}[f]$ is $\cal F_0$ measurable if $f$ is independent of $\cal F_0$". I think this should be easy, but I just don't see why. Can ...
0
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0answers
25 views

Can we simplify the conditional covariance $\mathbb{V}[(X\:Y\:Z)|X+Y+Z=1]$?

Given random variables $X,Y,Z$ on a probability space, can we write the conditional covariance matrix $$\mathbb{V}\left[ \left(\begin{array}{c}X\\Y\\Z\end{array}\right) \Bigg|X+Y+Z=1\right]$$ as a ...
2
votes
2answers
49 views

What does the conditional expectation look like when the $\sigma$-algebra is infinite

Given a probability space $(\Omega,\cal F,\Bbb P)$, when $\sigma$-algebra $\cal F_0$$\subseteq \cal F$ is finite (which is generated by a finite partition $\Gamma \subseteq \cal F_0$), the conditional ...
0
votes
1answer
33 views

Markov processes: Hitting times for a point form an i.i.d. sequence

Say that I have a recurrent time-homogenous diffusion process X (continuous strong markov process) and two points $x,y$. If $X_t$ goes from $y$, to $x$ and then back to $y$ again we denote it as a ...
1
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3answers
47 views

Random variable with 2 distribution functions

Just a question here, Given a random variable $X$ defined in a probability space, is it possible to have more than one distribution function $F$ ?
2
votes
1answer
28 views

Law of a random variable (characterization)

If $X$ is a real random variable defined on $(\Omega,\mathcal{F},\mathbf{P})$ then there exist several characterizations of the law of $X$ being $\mu$ : $X \sim \mu$ if and only if for every ...
0
votes
1answer
21 views

Terminology - “Sample space” vs “sample set”?

Given that a "sample space" is defined as the set of possible outcomes of a given random experiment, is there a fundamental reason to use the term "sample space" instead of "sample set" in probability ...
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0answers
28 views

Is continuous random variable always an “onto function” [on hold]

Is continuous random variable always an "onto function"? If yes, why?
1
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1answer
30 views

Determining bounds for change sum of continuous r.v.'s

I'm trying to understand how to determine the bounds when computing the sum of continuous random variables. Here is a sample question: X and Y have the following joint pdf: $f_{X,Y}(x,y) = 4xy, 0 ...
0
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1answer
25 views

Is probability mass function (PMF) the “law of X”?

Are they two the same? If not, what's the differences between these two? In continuous case, is PMF also equal to the integration of probability density function?
2
votes
2answers
58 views

Is every $\sigma$-algebra generated by a partition?

I know that every finite $\sigma$-algebra is generated by a finite partition, but is every infinite $\sigma$-algebra also generated by "kind of" partition? Can anyone help provide a explanation or ...