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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12
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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 ...
5
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4answers
2k views

Question on the 'Hat check' problem

The famous 'Hat Check Problem' goes like this, 'n' men enter the restaurant and put their hats at the reception. Each man gets a random hat back when going back after having dinner. The goal is to ...
8
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1answer
2k 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 ...
6
votes
1answer
544 views

Limit of sums of iid random variables which are not square-integrable

The Central Limit Theorem tells us that for an iid sequence of random variables $(X_n)_{n\geq 0}$ of finite variance $\sigma^2$ and zero mean $$\lim_{n\to\infty}\frac{S_n}{\sqrt{n}}=^d ...
4
votes
2answers
1k views

Conditional expectation for a sum of iid random variables: $E(\xi|\xi+\eta)=E(\eta|\xi+\eta)=\frac{\xi+\eta}{2}$

I don't really know how to start proving this question. Let $\xi$ and $\eta$ be independent, identically distributed random variables with $E(|\xi|)$ finite. Show that ...
3
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2answers
120 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
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
270 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
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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 % ...
2
votes
1answer
176 views

Probability of asymmetric random walk returning to the origin

Consider the random walk $S_n$ given by $ S_{n+1} = \left\{ \begin{array}{lr} S_n+2 & with & probability & p\\ S_n - 1 & with & probability & 1-p \end{array} ...
2
votes
1answer
530 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 ...
1
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0answers
89 views

optimization problem gaussian maximizes entropy

Let $X_1, X_2, Z_1$ be random variables and define $$Y=aX_1+bX_2+Z_1$$ I have the following optimization problem of difference of entropies, $$f=\max_{p(x_1x_2)} h(Y) - h(Y|X_2)= \max_{p(x_1,x_2)} ...
1
vote
1answer
247 views

When random walk is upper unbounded

Consider a random walk $S_n = a_1+\dots+a_n$ where $a_n$ are iid random variables with $Ea_1 = a$ and $E|a_1|<\infty$. I am interested in the case when $\sup\limits_n S_n>M$ for all $M$ a.s. ...
0
votes
1answer
83 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 ...
0
votes
2answers
332 views

Convergence in probability and almost surely

Let $X_n$ be a sequence of independent random variable which converges in probability to $X$. Prove $X$ is a constant. Can someone give me a hint how I should go about proving this? I tried proving ...
41
votes
8answers
56k 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.
33
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4answers
1k views

Rain droplets falling on a table

Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...
51
votes
7answers
8k views

Chance of meeting in a bar

Two people have to spend exactly 15 consecutive minutes in a bar on a given day, between 12:00 and 13:00. Assuming uniform arrival times, what is the probability they will meet? I am mainly ...
14
votes
1answer
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 ...
10
votes
2answers
2k 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 ...
9
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?
22
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1answer
931 views

How far can probability intransitivity be stretched?

Once upon a time I read about nontransitive dice - sets of dice where "is more likely to roll a higher number than" is not a transitive relation. After the surprise wore off, I wondered - just how far ...
13
votes
3answers
392 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 ...
11
votes
4answers
846 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? ...
7
votes
1answer
595 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, ...
7
votes
2answers
3k 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 ...
10
votes
3answers
689 views

Finitely Additive not Countably Additive on $\Bbb N$

Does there exist a function defined on the power set of the natural numbers to the interval from $0$ to $1$, $p:2^{\Bbb N}\rightarrow [0,1]$, such that $p$ is finitely additive, i.e. ...
5
votes
1answer
2k views

Another question on almost sure and convergence in probability

Convergence in probability implies convergence on a subsequence almost surely. But this means we fix a subsequence, such that $X_{n_k}$ converges for almost every $\omega$, right? The subsequence we ...
8
votes
2answers
1k views

Prokhorov metric vs. total variation norm

Let $(S,d)$ be a metric space and let $\mathcal P(S)$ denote the space of Borel probability measures on $S$ endowed with the Prokhorov metric $\pi:\mathcal P(S)\times \mathcal P(S)\to \mathbb R_+$ ...
2
votes
1answer
216 views

Problems on expected value

I'm self studying probability theory and I'm stuck in the following problems 1) Prove the following for a random variable $X$ with cdf $F$ $$E(x)=\int_0^\infty (1-F(x)) dx - \int_\infty^0 F(x) dx$$ ...
1
vote
1answer
388 views

Borel-Cantelli lemma problem

Practice problem for exam: Let ${A_n}$ satisfy $\sum_{n=1}^\infty P(A_n \cap A^c_{n+1}) < \infty$ and $\lim_{n\to \infty} P(A_n) = 0$. Show that $P(\lim \sup A_n) = 0$. I can see that it is ...
6
votes
1answer
918 views

Convergence in law and uniformly integrability

I'm looking for an elementary way of showing the following. If $(X_n)$ and $X$ are random variables such that $X_n \to X$ in distribution and such that $\{X_n\mid n\geq 1\}$ are uniformly integrable, ...
5
votes
2answers
990 views

Combinations of characteristic functions: $\alpha\phi_1+(1-\alpha)\phi_2$

Suppose we are given two characteristic functions: $\phi_1,\phi_2$ and I want to take a weighted average of them as below: $\alpha\phi_1+(1-\alpha)\phi_2$ for any $\alpha\in [0,1]$ Can it be proven ...
4
votes
1answer
877 views

Weak convergences of measurable functions and of measures

My question is "how weak convergences of measurable functions is defined?" There seems to be two different definitions which are both based on weak convergence of measures generated by the measurable ...
4
votes
1answer
1k views

Understanding the relationship of the $L^1$ norm to the total variation distance of probability measures, and the variance bound on it

I am trying to find a bound for variance of an arbitrary distribution $f_Y$ given a bound of a Kullback-Leiber divergence from a zero-mean Gaussian to $f_Y$, as I've explained in this related ...
4
votes
2answers
2k views

application of strong vs weak law of large numbers

By definition, the weak law states that for a specified large $n$, the average is likely to be near $\mu$. Thus, it leaves open the possibility that $|\bar{X_n}-\mu| \gt \eta$ happens an infinite ...
3
votes
1answer
74 views

For a distribution function $F(x)$ and constant $a$, integral of $F(x + a) - F(x)$ is $a$.

For any distribution function and any $a \geq 0$, $\int_{-\infty}^{\infty} (F(x+a)-F(x))dx = a$. In this case, "distribution function" means a right continuous function F with $F(-\infty) = 0$, ...
3
votes
2answers
219 views

Let $X_1$ and $X_2$ are independent $N(0, \sigma^2)$ random variables. What is the distribution of $X_1^2 + X_2^2$?

Let $X_1$ and $X_2$ are independent $N(0, \sigma^2)$ which means (mean = 0, variance = $\sigma^2$) random variables. What is the distribution of $X_1^2 + X_2^2$? My approach is that $X_1\sim N(0, ...
3
votes
1answer
217 views

How expected value is related to density function?

Let $X$ be a random variable on $(\Omega, \Sigma, P)$. The expected value of $X$ is defined as $$EX = \int X \,dP.$$ But when we calculate $EX$, we often use $$ EX = \int_{-\infty}^\infty xf(x) dx ...
2
votes
3answers
140 views

How to express $E[\max(x,y)]$ as an integral?

In Hull (2008, p. 307), the following equation is found (Eq. 13A.2): $$E[\max(V-K,0)]=\int_{K}^{\infty} (V-K)g(V)\:dV$$ Where $g(V)$ is the PDF of $V$ and $V,K>0$. I'd like to extrapolate from ...
2
votes
1answer
133 views

Probability that a sequence of random variables converges to 0 or 1

Fix $\alpha , \beta > 0$ with $\alpha + \beta = 1$ and consider the sequence of random variables defined by $X_0 = \theta \in (0,1)$ and $$P(X_n = \alpha + \beta X_{n-1} \mid X_{n-1} , ... , X_0) ...
2
votes
1answer
1k views

Rule with independent random variables and conditional expectations

I want to use a rule for conditional expectation I found in (German) wikipedia, not in my script/textbook of probability theory, I guess it should be simple and follow more or less straight from the ...
1
vote
1answer
136 views

What kind of f(n)'s make the limsup statement is true? What kind don't?

What kind of $f(n): \mathbb{N} \to \mathbb{N}$'s make the ff statement true? What kind don't? $\limsup A_{f(n)} \subseteq \limsup A_n$ where $n \in \mathbb{N}$ (*) Well obviously the answers to ...
1
vote
1answer
128 views

(Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$

The purpose of this question is to complete my personal exposition on the rigorous proof of Ito's lemma. I have consulted more than half a dozen mathematical finance texts and not a single one, for ...
1
vote
1answer
740 views

Likelihood Function for the Uniform Density.

Let the random variable $X$ have a uniform density given by $$ f(x;\theta)=I_{[\theta-\frac{1}{2},\theta+\frac{1}{2}]} $$ where $-\infty\leq\theta\leq\infty $ the likelihood function for a sample of ...
7
votes
4answers
356 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.
6
votes
1answer
350 views

Can you make money on coin tosses when the odds is against you?

The strategy Given an initial investment $n$ dollars and a "bet buffer" $b$. Calculate the bet size $x=\lfloor\frac{n}{2^b-1}\rfloor$ dollars. Wager $x$ dollars on random variable $C$ that $C=1$ ...
6
votes
1answer
342 views

Feller continuity of the stochastic kernel

Given a metric space $X$ with a Borel sigma-algebra, the stochastic kernel $K(x,B)$ is such that $x\mapsto K(x,B)$ is a measurable function and a $B\mapsto K(x,B)$ is a probability measure on $X$ for ...
5
votes
2answers
14k 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
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1answer
222 views

Prove that the maximum of $n$ independent standard normal random variables, is assyptotically equivalent to $\sqrt(2\log n)$ almost surely.

Lets $(X_n)_{n\in\mathbb{N}}$ be an iid sequence of standard normal random variables. Define $$M_n=\max_{1\leq i\leq n} X_i.$$ Prove that $$\lim_{n\rightarrow\infty} \frac{M_n}{\sqrt{2\log ...