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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10
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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?
11
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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 ...
22
votes
1answer
973 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
2answers
851 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 ...
8
votes
3answers
306 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, ...
8
votes
3answers
1k views

What is the relation between weak convergence of measures and weak convergence from functional analysis

To keep things simple, we assume $X$ to be a polish space (think of $X$ as $\mathbb{R}^n$ for example). Let's denote with $P(X)$ the space of all Borel probability measure on $X$. We say ...
7
votes
1answer
688 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, ...
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? ...
5
votes
3answers
5k views

1D random walk-probability to go back to origin

Suppose There is a random walk starting in origin while probability to move right is 1/3 and probability to move left 2/3.What is the probability to return to the origin. Thank you
10
votes
3answers
845 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. ...
6
votes
2answers
135 views

Approximation of conditional expectation

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space. Let $\mathcal{A}$ be a complete sub-$\sigma$-algebra of $\mathcal{F}$. For the moment assume that $X$ is a random variable with ...
6
votes
1answer
437 views

Right continuous version of a martingale

This is an exercise in chapter 2 of the book "Continuous Martingales and Brownian Motion" by Revuz and Yor: Consider the probability space $([0,1], \mathcal{B}([0,1]), dx)$, where $dx$ denotes ...
2
votes
1answer
562 views

Borel-Cantelli lemma problem [duplicate]

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 ...
11
votes
2answers
505 views

Why do we want probabilities to be *countably* additive?

In probability theory, it is (as far as I am aware) universal to equate "probability" with a probabilistic measure in the sense of measure theory (possibly a particularly well behaved measure, but ...
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 $\pi:\mathcal P(S)\times \mathcal P(S)\to \mathbb R_+$ ...
6
votes
1answer
1k 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
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
4k 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 ...
4
votes
2answers
1k views

A criterion for independence based on Characteristic function

Let $X$ and $Y$ be real-valued random variables defined on the same space. Let's use $\phi_X$ to denote the characteristic function of $X$. If $\phi_{X+Y}=\phi_X\phi_Y$ then must $X$ and $Y$ be ...
4
votes
1answer
1k 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 ...
3
votes
1answer
147 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) ...
3
votes
1answer
237 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
1answer
138 views

“Fair” game in Williams

In David Williams' Probability with Martingales, $\exists$ this exercise. What's fair about a fair game? Let $X_i$, i = 1, 2, ... be indp RVs s.t. $X_i = i^2 - 1$ with prob $1/i^2$ and $-1$ with ...
2
votes
1answer
221 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
145 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 ...
0
votes
1answer
157 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 ...
6
votes
1answer
347 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
3answers
1k views

A Good Book for Mathematical Probability Theory [duplicate]

I am from mathematical background, and I always hated the way they teach elementary probability theory in schools without giving any clue about measure theory. I want a theoretical book in ...
5
votes
2answers
1k 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
2answers
101 views

How can I show that the “binary digit maps” $b_i : [0,1) \to \{0,1\}$ are i.i.d. Bernoulli random variables?

In this post What is the Lebesgue measure of the set of numbers in $[0,1]$ that has two thirds of ones in their infinite base-2 expansion? we needed the fact that if we let $b_i (x) \in \{0,1\}$ for ...
4
votes
1answer
440 views

If X and Y are equal almost surely, then they have the same distribution, but the reverse direction is not correct

Show that if two random variables X and Y are equal almost surely, then they have the same distribution. Show that the reverse direction is not correct. If $2$ r.v are equal a.s. can we write ...
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
2answers
218 views

Is it correct to say that ($\color{red}{(} \limsup |W_k|/k\color{red}{)} \le 1) \supseteq \limsup \color{red}{(}|W_k|/k \le 1\color{red}{)}$?

Let $W_0, W_1, W_2, \dots$ be random variables on a probability space $(\Omega, \mathscr{F}, \mathbb{P})$ where $$\sum_{k=0}^{\infty}P(|W_k|>k) <\infty$$ Prove that $$\limsup ...
3
votes
1answer
96 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
255 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, ...
2
votes
3answers
317 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
3answers
1k views

Conditional mean on uncorrelated stochastic variable

I know that $E[X|Y]=E[X]$ if $X$ is independent of $Y$. I recently was made aware that it is true if only $\text{Cov}(X,Y)=0$. Would someone kindly either give a hint if it's easy, show me a reference ...
2
votes
1answer
635 views

Sum of two independent normal distributed random variables

If $X_i$, $i =1,2$ are independent and have normal distribution with mean $0$ and variance $\sigma_i ^2$. Show that $X_1 + X_2$ has a normal distribution with mean $0$ and variance $\sigma_1^2 + ...
2
votes
1answer
1k views

Is continuous L2 bounded local martingale a true martingale?

I can prove it briefly, but I found a "counter" example. (There must be a mistake in the following words...) I can prove: X is a continuous local martingale, with $X_0=0$ a.s, then X is $L_2$ bounded ...
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
144 views

Is it correct to say that $P(A) = 1 \to P(A|B) = 1$? $P(A) < 1 \to P(A|B) = P(A)$?

Is it correct to say that $P(A) = 1 \to P(A|B) = 1$? Precisely, consider a probability space $(\Omega, \mathscr{F}, \mathbb{P})$. Let $A \in \mathscr{F}$ and let $B_1, B_2, ... \in \mathscr{F}$ ...
1
vote
1answer
161 views

polynomial approximation on compacts

Let's say $f:\mathbb{R}^d\rightarrow \mathbb{R}$ is of class $C^k$ with $k \geq 0$. How do I know that I can find a sequence of polynomials such that all its derivatives up to order $k$ converge ...
1
vote
1answer
918 views

What is the distribution of $Y = e^X$ when $X$ is normal?

What is the distribution of $Y = e^X$ when $X$ is normally distributed? Am I supposed to use characteristics function of normal random variable ?
1
vote
1answer
925 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 ...
0
votes
1answer
68 views

On “for all” in if and only if statements in probability theory and stochastic calculus

1 In my friend's Probability Theory long test there was this question: Let $(\Omega, \mathfrak{F}, P)$ be a probability space on which is defined all sub-$\sigma$-algebras, events and random ...
-1
votes
1answer
49 views

Prove a thm on stopped processes given fundamental principle 'you can't beat the system'?

How does the principle below imply the thm below? From Williams' Probability w/ Martingales: Principle: Thm: What I tried: $$E[X_{T \wedge n} - X_0 | \mathscr{F_m}] =/ \le X_{T \wedge ...
-2
votes
2answers
491 views

Is first order moving average a Markov process?

Given first order moving average $$ x(n) = e(n) + ce(n-1) $$ where $e(n)$ is a sequence of Gaussian random variables with zero mean and unit variance which are independent of each other, and $c$ is ...
9
votes
1answer
411 views

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

The strategy Given an initial investment $n$ dollars and a "bet buffer" $b$. Calculate the bet size $x=\left\lfloor\frac{n}{2^b-1}\right\rfloor$ dollars. Wager $x$ dollars on random variable $C$ ...
5
votes
1answer
554 views

When the sum of independent Markov chains is a Markov chain?

I try to find as much as possible cases, when the chain $Z(t) = |X_1(t)-X_2(t)|$ is Markov, where $X_1(t)$ and $X_2(t)$ are independent, discrete-time and space, preferably non-homogeneous Markov ...
4
votes
1answer
407 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 ...