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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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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13 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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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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18 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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8 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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12 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 $\frac{a_{k+1}}{a_k}$? ...
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30 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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15 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 ...
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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, ...
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1answer
24 views

How is this an application of the independence property of events?

I'm currently working my way through Klenke's book on probability theory and do not understand a step in his proof of the Borel-Cantelli lemma (Theorem 2.7): The assertion is that for an independent ...
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1answer
12 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 ...
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1answer
353 views

Characteristic function and probability density function: Fourier or Inverse Fourier?

I have come across two contradicting definitions of characteristics function (CHF). In wikipedia CHF is defined as the inverse Fourier transform (FT) of probability density function (PDF) and at some ...
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1answer
115 views

Soft question: what are some elementary motivations of using functional analysis to study probability theory?

Recently I've become curious about the links between functional analysis and probability theory. What are some simple reasons why a functional analytic approach is preferable to a measure-theoretic ...
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52 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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1answer
62 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\}] ...
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23 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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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 ...
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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) ...
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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 ...
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4answers
60 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 ...
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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 ...
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3answers
57 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 $0,1, ... ,9$. ...
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2answers
57 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 ...
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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. ...
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12 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 ...
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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 ...
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17 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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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 ...
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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 ...
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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 ...
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1answer
89 views

Monty Hall problem again (from Grimmet and Stirzaker)

Grimmet and Stirzaker Exercise 1.4.5.2 In a game show you have to choose one of three doors. One conceals a car, 2 conceal goats. You choose a door but the door is not opened immediately. Instead ...
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1answer
12 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 ...
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1answer
56 views

Origin of $\sigma$-algebra

In what paper, article or book was the notion of an $\sigma$-algebra first defined or mentioned? Or at least how far could this concept traced back?
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1answer
32 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 ...
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1answer
109 views

Interpretation of a tail event

I am currently reading about tail events wikipedia. And I was wondering: Where does the interpretation come from that events in this sigma algebra are independent from the behaviour of any finite set ...
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2answers
48 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 ...
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1answer
24 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?
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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 $F$ ?
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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 ...
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3answers
71 views

$\sum_{m=1}^{\infty }\frac{m}{2^n} \mu (E_{n,m}) \uparrow \int f d\mu$

Let $f \geq 0$ and $E_{m,n}=\{x :m/2^n \leq f(x) < (m+1)/2^n \}$ I need to show that as $n \uparrow \infty$ $\sum_{m=1}^{\infty }\frac{m}{2^n} \mu (E_{n,m}) \uparrow \int f d\mu$ My attempt: I ...
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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 ...
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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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27 views

Interpretation of the negative binomial and geometric distributions

I am having trouble putting together the way these distributions work. It doesn't matter whether we speak of the support space in terms of number of trials or failures. Specifically what variable is ...
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2answers
34 views

How to prove convergence in $L^p$ imply convergence in $L^r$ when $p>r$?

$X_n$ converges to $X$ in $p$th mean. Show that $X_n$ also converges to $X$ in $r$th mean when $p\ge r$. I have tried conditioning on $|X_n-X|\ge1$ and $|X_n-X|<1$ but no luck.
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3answers
73 views

How do mathematician make sense of “outcome” and “events” in probability?

One of the biggest challenge for me to understand probability is to make sense of this concept of outcomes and events. To put it plainly, it just doesn't feel like mathematics anymore when we talk ...
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39 views

How $\sigma$-algebra determines random variable?

In my probability textbook there is a statement saying that Knowing the $\sigma$-algebra $\sigma(X)$ generated by a random variable $X$ is equivalent to knowing $X$ itself. We equate $\sigma(X)$ ...
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1answer
357 views

Distribution of sum of multiplication of i.i.d. exponential random variables.

I have two questions: A) Suppose that we have $Z=c\Sigma (X_i-a)(Y_i-b) $ where $X_i$s and $Y_i $s are independent exponential random variables with means equal to $\mu_{X}$ and $\mu_{Y}$ (for ...
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1answer
192 views

KL divergence of multinomial distribution

Consider $q(x)$ be a Multinomial distribution over $\{1, \ldots, k\}$ with parameters $\{\theta_1,\ldots, \theta_k\}$. And p(x) over $\{1,\ldots, k\}$ with distribution $p(x)=\frac{1}{k}$. Then what ...
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3answers
75 views

Can someone give me real world example of uniform distribution [0,1] of a continuous random variable.

Can someone give me real world example of uniform distribution [0,1] of a continuous random variable, because I could not make out one.