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 [tag:probability-...

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7
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4answers
514 views

How variance is defined?

The variance of a random variable $X$ is defined as $E[(x-\mu )^2]$. Why can't it be defined as $E[|x-\mu |]$. i.e., What is the basic idea behind this definition. Thank you.
5
votes
1answer
610 views

Generalized Second Borel-Cantelli lemma

A generalized version of the second Borel-Cantelli lemma says Theorem 5.3.2. Second Borel-Cantelli lemma, II. Let $\mathcal F_n, n \ge 0$ be a filtration with $F_0 = \{\emptyset, \Omega\}$ and $...
2
votes
1answer
397 views

Questions on Kolmogorov Zero-One Law Proof in Williams

Here is the proof of the Kolmogorov Zero-One Law and the lemmas used to prove it in Williams' Probability book: Here are my questions: Why exactly are $\mathfrak{K}_{\infty}$ and $\mathfrak{T}$...
20
votes
2answers
3k views

Beta function derivation

How do I derive the Beta function using the definition of the beta function as the normalizing constant of the Beta distribution and only common sense random experiments? I'm pretty sure this is ...
19
votes
1answer
4k views

Interpretation of sigma algebra

My question is how to interpret sigma algebra, especially in the context of probability theory (stochastic processes included). I would like to know if there is some clear and general way to interpret ...
21
votes
2answers
575 views

A simple way to obtain $\prod_{p\in\mathbb{P}}\frac{1}{1-p^{-s}}=\sum_{n=1}^{\infty}\frac{1}{n^s}$

Let $ p_1 <p_2 <\cdots <p_k <\cdots $ the increasing list in set $\mathbb{P}$ of all prime numbers . By sum of infinite geometric series $\sum_{k=0}^\infty r^k = \frac{1}{1-r}$ for $0<...
36
votes
3answers
4k views

Why is this coin-flipping probability problem unsolved?

You play a game flipping a fair coin. You may stop after any trial, at which point you are paid in dollars the percentage of heads flipped. So if on the first trial you flip a head, you should stop ...
20
votes
2answers
2k views

What is meant by a continuous-time white noise process?

What is meant by a continuous-time white noise process? In a discussion following a question a few months ago, I stated that as an engineer, I am used to thinking of a continuous-time wide-sense-...
12
votes
4answers
7k views

Intuitive explanation of the tower property of conditional expectation

I understand how to define conditional expectation and how to prove that it exists. Further, I think I understand what conditional expectation means intuitively. I can also prove the tower property, ...
14
votes
3answers
434 views

Limit of a Wiener integral

How to show that $$ \lim _{\alpha \rightarrow \infty } \sup_{t \in \left [0,T \right]} \left | e^{-\alpha t} \int _ 0 ^t e^{\alpha s} ~ dB_s \right | =0, \ \ \text{a.e.} $$ where $\left (B_s \right)...
13
votes
2answers
2k views

Algebra of Random Variables?

I've been looking online (and in teaching journals) for a good introduction to Algebras of Random Variables (on an undergraduate level) and their usage, and have come up short. I know I can find the ...
9
votes
2answers
4k views

Meaning of non-existence of expectation?

When reading another post, I was wondering about the definition of existence of expectation of a random variable. From Kai Lai Chung, We say a random variable $X$ has a finite or infinite ...
5
votes
3answers
6k views

Showing that Y has a uniform distribution if Y=F(X) where F is the cdf of X

Let X be a random variable with a continuous and strictly increasing c.d.f. function F (so that the quantile function F^−1 is well-defined). Define a new random variable Y by Y = F(X). Show that Y has a ...
5
votes
1answer
1k views

$n$ balls are thrown randomly into $k$ bins - how many are empty?

A large number of variants of this question were already asked here, including these - one, two, which are close, but none seem to answer my question. Assume that $n$ balls are thrown randomly and ...
6
votes
3answers
3k views

Showing that ${\rm E}[X]=\sum_{k=0}^\infty P(X>k)$ for a discrete random variable

Let $X$ be a discrete random variable whose range is $0,1,2,3,\ldots$. Prove that $$ {\rm E}[X]=\sum_{k=0}^\infty P(X>k). $$ How to prove this? I tried a bit but unable to post due to formatting ...
9
votes
1answer
3k views

Asymptotics of binomial coefficients and the entropy function

I found a question while I was trying to practice Combinatorics and Probabilistic methods.I tried to solve it with no success.. this is the question: Use the Stirling approximation of the ...
5
votes
2answers
2k views

Tightness condition in the case of normally distributed random variables

Let $(X_n)_{n\geq 1}$ be a sequence of random variables such that $X_n\sim N(\mu_n,\sigma_n)$ for all $n\geq 1$. Then i'm trying to deduce that if $(X_n)_{n\geq 1}$ is tight in the sense that $$ \...
5
votes
2answers
21k views

Finding probability P(X<Y)

How can I find this probability $P(X<Y)$ ? knowing that X and Y are independent random variables.
3
votes
1answer
177 views

Asymptotics of $\max\limits_{1\leqslant k\leqslant n}X_k/n$

I found an assertion in this paper at the beginning of page 6, but i can't see how to justify it: Let $X_n \geq 0$ i.i.d. with finite expectation then: $$ \frac1n\max\limits_{k \leq n}X_k \to 0 \quad\...
0
votes
2answers
681 views

If X,Y and Z are independent, are X and YZ independent?

If yes: I know that f(X) and g(Y) are independent if X and Y are independent and f and g are "measurable".* If that is to be used, is g(Y) = YZ measurable? If not, how else to approach this? If ...
3
votes
2answers
207 views

Conditional Expectation of Functions of Random Variables satisfying certain Properties

Suppose that we have a probability space $(\Omega, \mathcal{F}, P)$. Let $X,Y$ be real-valued random variables defined on this space, and let $\mathcal{H} \subset \mathcal{F}$ be a sub-sigma-algebra. ...
2
votes
2answers
484 views

Distribution of sums

I'm really having a hard time with this topic in probability theory and I was wondering if someone has any tricks, tips or anything useful to help me understand it. In my notes I am told that $X\sim$...
2
votes
1answer
754 views

Prove that $f(X)$ and $g(Y)$ are independent if $X$ and $Y$ are independent [duplicate]

Let $X$ and $Y$ be independent random variables. Prove that $f(X)$ and $g(Y)$ are independent for any choice of measurable functions $f$ and $g$. This sounds very obvious, but I have no idea how to ...
2
votes
3answers
556 views

Understanding the Gamma function? [duplicate]

I'm working my way through a probability textbook, and i have encountered the Gamma function through the Gamma distribution. I understand that the Gamma function is an interpolating function that ...
2
votes
1answer
602 views

Alternative Expected Value Proof

I am currently tasked with proving an alternative definition of the expected value function. Considering X to be a random variable that takes all positive integers, I have to prove that $E[X]=\sum\...
2
votes
1answer
4k views

Tower property of conditional expectation

I'm trying to prove the "tower property" of conditional expectations, $$ E[V\mid W] = E[\ E[V\mid U,W]\ \mid W\ ], $$ where $U$, $V$ and $W$ are any random variables. $E[X \mid Y]$ is itself a ...
1
vote
2answers
331 views

Find the pdf of $\prod_{i=1}^n X_i$, where $X_is$ are independent uniform [0,1] random variables.

How do I find the pdf of $\prod_{i=1}^n X_i$, where $X_is$ are independent uniform [0,1] random variables. I know X~U[0,1], -ln(x) is exponential(1). I also know the sum of two or more independent ...
0
votes
3answers
2k views

{Thinking}: Why equivalent percentage increase of A and decrease of B is not the same end result?

original post the examples here are, the most important word -- fundamentally -- the same. example1: the most abstract way to present this example. Why equivalent % increase of A in event1 and % ...
5
votes
2answers
193 views

Sequence satisfies weak law of large numbers but doesn't satisfy strong law of large numbers

Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of independent random variables such that $$P(X_n=n+1)=P(X_n=-(n+1))=\frac{1}{2(n+1)\log(n+1)}$$ $$P(X_n=0)=1-\frac{1}{(n+1)\log(n+1)}$$ Prove that $X_n$ ...
3
votes
2answers
1k views

How can I show that the conditional expectation $E(X|X)=X$?

I tried to show that $E(X|X=x)=x$, which would lead me to get $E(X|X)=X$ but I am having trouble doing so. I know that the definition of conditional expectation (continuous case) is: $$E(X|Y=y)=\int_{-...
1
vote
1answer
250 views

Expected Value for number of draws

There are $n$ types of balls in an urn; $a$ balls of type $1$ , $b$ balls of type $2$ , $c$ balls of type $3$ and so on. Now balls are drawn until a ball of type $1$ is obtained with condition that if ...
0
votes
1answer
69 views

Asymmetric Random Walk / Prove that $T:= \inf\{n: X_n = b\}$ is a $\{\mathscr F_n\}_{n \in \mathbb N}$-stopping time

Given random variables $Y_1, Y_2, ... \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \...
0
votes
1answer
131 views

Mutual Independence Definition Clarification

Let $Y_1, Y_2, ..., Y_n$ be iid random variables and $B_1, B_2, ..., B_n$ be Borel sets. It follows that $P(\bigcap_{i=1}^{n} (Y_i \in B_i)) = \Pi_{i=1}^{n} P(Y_i \in B_i)$...I think? If so, does ...
-1
votes
2answers
155 views

Do Kolmogorov's axioms really need only disjointness rather than pairwise disjointness?

According to 1 2, the third Kolmogorov axiom is for disjoint sets $(A_n)_{n \in \mathbb{N}}$ $P(\cup_n A_n) = \sum_n P(A_n)$ Is that really disjoint rather than pairwise disjoint? If we ...
52
votes
8answers
81k views

What is the best book to learn probability?

Question is quite straight... I'm not very good in this subject but need to understand at a good level.
22
votes
8answers
7k views

Good books on “advanced” probabilities

what are some good books on probabilities and measure theory? I already know basic probabalities, but I'm interested in sigma-algrebas, filtrations, stopping times etc, with possibly examples of "...
10
votes
7answers
5k views

Best measure theoretic probability theory book?

I'm looking for a clear way to learn measure theoretic probability theory. Any suggestions?
21
votes
3answers
2k views

What is the importance of the infinitesimal generator of Brownian motion?

I have read that the infinitesimal generator of Brownian motion is $\frac{1}{2}\small\triangle$. Unfortunately, I have no background in semigroup theory, and the expositions of semigroup theory I have ...
12
votes
4answers
3k views

Intuitive explanation of variance and moment in Probability

While I understand the intuition behind expectation, I don't really understand the meaning of variance and moment. What is a good way to think of those two terms?
12
votes
2answers
3k views

How should I understand the $\sigma$-algebra in Kolmogorov's zero-one law?

I'm learning Kolmogorov's zero-one law in probability theory: Let $(Ω,{\mathcal F},P)$ be a probability space and let $F_n$ be a sequence of mutually independent $\sigma$-algebras contained in $\...
14
votes
2answers
936 views

Existence of independent and identically distributed random variables.

I often see the sentence "let $X_1, X_2, \ldots$ be a sequence of i.i.d. random variables with a certain distribution". But given a random variable $X$ on a probability space $\Omega$, how do I know ...
13
votes
3answers
617 views

Are polynomials dense in Gaussian Sobolev space?

Let $\mu$ be standard Gaussian measure on $\mathbb{R}^n$, i.e. $d\mu = (2\pi)^{-n/2} e^{-|x|^2/2} dx$, and define the Gaussian Sobolev space $H^1(\mu)$ to be the completion of $C_c^\infty(\mathbb{R}^n)...
11
votes
5answers
1k views

Probability of having zero determinant

Given a matrix $A_{n \times n}$, which has elements $a_{i,j} \sim \mathrm{unif} \left[a,b\right]$, what is the probablity of $\det(A)$ being zero? What if $a_{i,j}$ have any other distribution? ...
8
votes
3answers
379 views

Maximum of a sum of random variables

Let $X_1, \dots, X_n$ be independent and identically distributed random variables with $E(X_i) = 0$ and $$S_k = \sum_{i \leq k} X_i$$ What is the probability distribution of $M_2 = \max \{ X_1, ...
7
votes
2answers
3k views

Prove the time inversion formula is brownian motion

Let $B=(B_t)_{t\geq 0}$ be a brownian motion. Show the time inversion formula $\hat{B}=(B_t)_t\geq0$ is a brownian motion, where for $t \geq 0$ we set $\hat{B}=0$ for $t=0$ and $\hat{B}=tB_{1/t}$ for $...
7
votes
2answers
19k views

how to derive the mean and variance of a Gaussian Random variable?

How do we go about deriving the values of mean and variance of a Gaussian Ransom Variable $X$ given its probability density function ?
7
votes
1answer
762 views

For symmetric stable distributions, why is $\alpha \le 2$?

I'm preparing a lecture on stable distributions, and I'm trying to find a simple explanation of the following fact. Suppose we are trying to come up with stable distributions. From the definition, ...
5
votes
2answers
4k views

Why does the median minimize $E(|X-c|)$?

Suppose $X$ is a real-valued random variable. Let $P$ be the probability measure of $X$. Then $$ E(|X-c|) = \int_\mathbb{R} |x-c| dP(x). $$ Its median is defined as a number $m \in \mathbb{R}$ ...
3
votes
0answers
126 views

Compound Distribution — Log Normal Distribution with Log Normally Distributed Mean

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Log Normal Distribution whose mean (actually, the mean ...
12
votes
3answers
1k views

Is there a possibility to choose fairly from three items when every choice can only have 2 options

Me and my wife are often not knowing which DVD to watch. If we have two options we have a simple solution, I put one DVD in one hand behind my back and the other DVD in the other hand. She will ...