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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2
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1answer
130 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
293 views

Convergence of characteristic functions to $1$ on a neighborhood of $0$ and weak convergence

Prove the following statement: $ X_n \Rightarrow 0 $ (convergence in distribution) if and only if $ (\exists\; \epsilon>0: |t|<\epsilon) \;\; \phi_n(t) \rightarrow 1 $, where $\phi_n(t)$ is ...
2
votes
1answer
501 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
vote
0answers
81 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
3answers
295 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 ...
1
vote
1answer
231 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
2answers
278 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 ...
35
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8answers
48k 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 ...
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
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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?
22
votes
1answer
900 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 ...
12
votes
3answers
385 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
768 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
552 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, ...
10
votes
3answers
609 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. ...
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 ...
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
215 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 $F$ $$E(x)=\int_0^\infty (1-F(x)) dx - \int_\infty^0 F(x) dx$$ ...
6
votes
1answer
859 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
931 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
999 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 ...
3
votes
2answers
107 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. ...
3
votes
2answers
209 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, ...
1
vote
1answer
684 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
331 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
304 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 $C=1$ ...
4
votes
2answers
13k views

Finding probability P(X<Y)

How can I find this probability $P(X<Y)$ ? knowing that X and Y are independent random variables.
4
votes
1answer
813 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
325 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 ...
3
votes
1answer
195 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 ...
3
votes
1answer
210 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 ...
3
votes
5answers
208 views

Probability distribution for the perimeter and area of triangle with fixed circumscribed radius

Given a circle with radius R = 1, I'm trying to find either the probability distribution function or the density function for the space of triangle, which is randomly selected on this circle. The same ...
2
votes
2answers
267 views

conditional expectation of brownian motion

Let $(B_t)_{t\geq 0}$ be a standard Brownian motion in $\mathbb{R}^d$. It is intuitive that, for fixed $s<t<u$ $$\mathbb{E}[B_t\mid \sigma(B_s,B_u)]=B_s+\frac{t-s}{u-s}(B_u-B_s).$$ However, I ...
2
votes
1answer
511 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
2answers
1k views

Expected value of the minimum (discrete case)

Maybe related to this question In the comments of this question they say that it gets easier if the variables are identically and independently distributed. But i don't see how because in my case the ...
2
votes
1answer
929 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
4answers
2k views

Why isn't there a uniform probability distribution over the positive real numbers?

Apparently, the solution to the Card Doubling Paradox is that a uniform probability distribution over the positive real numbers doesn't exist. Can anyone explain why this is the case and what ...
1
vote
1answer
125 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 ...
0
votes
1answer
92 views

Prove $\mathbb{P} \left( \sup_{t \geq 0} M_t > x \mid \mathcal{F}_0 \right) = 1 \wedge \frac{M_0}{x}$ for a martingale $(M_t)_{t \geq 0}$

Let $M$ be a positive, continuous martingale that converges a.s. to zero as $t$ tends to infinity. I now want to prove that for every $x>0$ $$ P\left( \sup_{t \geq 0 } M_t > x \mid \mathcal{F}_0 ...
-2
votes
2answers
374 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 ...
26
votes
3answers
28k views

Probability density function vs. probability mass function

I've an confession to make. I've been using pdf's and pmf's without actually knowing what they are. The idea that I've been having so long is that density = area under the curve but if I look at it ...
48
votes
6answers
6k 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 ...
9
votes
2answers
165 views

Probability of getting A to K on single scan of shuffled deck

Let us say we have a regular 52-card well-shuffled deck. We scan through the deck (first to last) till we find an Ace. Then we continue (from that Ace) till we find a 2. Then we scan (from the 2) ...
4
votes
1answer
617 views

Martingale and bounded stopping time

A theorem of submartingale and bounded stopping time says: Theorem 5.4.1. If $X_n$ is a submartingale and $N$ is a stopping time with $\mathbb P (N \le k) = 1$ then $\mathbb EX_0 ≤ \mathbb EX_N ≤ ...
8
votes
2answers
220 views

Weak topologies and weak convergence - Looking for feedbacks

I am currently trying to get exactly what the weak and the weak* topologies are, in particular in connection to the concept of weak convergence in measure, however I am not completely sure on what I ...
8
votes
8answers
3k views

Applications of Probability Theory in pure mathematics

My (maybe wrong) impression is that while probability is widely used in science (for example, in statistical mechanics), it is rarely seen in pure mathematics. Which leads me to the question - Are ...
5
votes
3answers
1k views

Is expectation Riemann-/Lebesgue–Stieltjes integral?

In probability theory, when having $ E(f(X))=\int_{-\infty}^\infty f(x)\, dg(x) $, an expectation of a measurable function $f$ of a random variable $X$ with respect to its cumulative distribution ...