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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6
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1answer
2k views

Jensen's Inequality (with probability one)

In the following theorem, I have a problem about the second part. That is showing if $f$ is strictly convex then $X=EX$ with probability $1$. While I can see this must be true, I don't know how to ...
5
votes
2answers
175 views

Example: satisfying $E(X_{n+1}\mid X_n)=X_n$ but not a martingale

I am wondering if there is such a sequence of random variables $(X_n)_{n=0}^\infty$ such that $\mathbb{E}(X_{n+1}\mid X_n)=X_n$ for all $n\geq0$ but which is not a martingale with respect to the ...
5
votes
1answer
745 views

Conditional expectation $E[X\mid\max(X,Y)]$ for $X$ and $Y$ independent and normal

I am trying to obtain the conditional expectation $$E[X\mid Z]$$ where $Z= \max(X,Y)$ and $X,Y$ are independent Gaussian random variables.
4
votes
4answers
507 views

Expected number of frog jumps

There is frog jumping forward on a line. Each jump distance is random with a known cumulative distribution function $F$. What is the expected number of jumps to reach (or go beyond) distance $d$ from ...
3
votes
1answer
2k views

Calculating stationary distribution of markov chain

I am asked to compute the stationary distribution of the markov chain with state space $E=\{0\dots,n\}$ and transition matrix below: \begin{bmatrix} 0 & 1 \\ \frac{1}{n} ...
2
votes
2answers
842 views

First hitting time for a brownian motion with a exponential boundary

Let $B_t$ be the standard Brownian Motion. Is the distribution/density of the first hitting time of $B_t$ for an exponential decaying boundary known? Trying to be more formal, if ...
15
votes
1answer
2k views

Under what circumstance will a covariance matrix be positive semi-definite rather than positive definite?

I have a covariance matrix: $\operatorname{cov}(\mathbf{X}, \mathbf{X}) = \operatorname{E}[(\mathbf{X} - \operatorname{E}[\mathbf{X}])(\mathbf{X} - \operatorname{E}[\mathbf{X}])^T]$ According to ...
10
votes
2answers
254 views

Two-valued measure is a Dirac measure

Let $(X,\mathfrak B)$ be a measurable space such that $\{x\}\in \mathfrak B$ for all $x\in X$, and let $\mu$ be a positive measure on this space such that $$ \mu(B) \in\{0,1\} \quad\text{for all ...
8
votes
1answer
206 views

Estimates for the normal approximation of the binomial distribution

I'm interested in estimates of the normal approximation for binomial distributions, i.e. in estimates of $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - ...
8
votes
1answer
3k views

The strong and weak laws of large numbers: Why two?

The following questions are entirely based on the corresponding article from Wikipedia. The assumptions of both laws are the same, and the strong law has a more general claim than the one of the weak ...
6
votes
3answers
316 views

Do the Kolmogorov's axioms permit speaking of frequencies of occurence in any meaningful sense?

It is frequently stated (in textbooks, on Wikipedia) that the "Law of large numbers" in mathematical probability theory is a statement about relative frequencies of occurrence of an event in a finite ...
5
votes
1answer
121 views

An expectation inequality

Let $X$ and $Y$ be iid random variables, and $\mathbb E[|X|]<\infty$, prove that $$\mathbb E[|X+Y|]\geq\mathbb E[|X-Y|]$$ Let $F(x)$ denote the distribution, after calculation, I need to prove ...
5
votes
2answers
359 views

Name/significance of integral of the square of a probability density function

Background/Motivation Given a probability density function $f(x)$, the mean of the corresponding random variable is the $x$-coordinate of the centroid of the region under the graph of $f$. I ...
5
votes
2answers
314 views

Infinite Inclusion and Exclusion in Probability

Is there some way of generalizing the principle of inclusion and exclusion for infinite unions in the context of probability? In particular, I would like to say that $P(\bigcup_{n=1}^{\infty}A_n) = ...
5
votes
2answers
1k views

Showing $\cos(t^2)$ is not a Characteristic Function

Usually when we try to show a function is not a characteristic function, we would prove it is not uniformly continuous. I am wondering if there is any other way to show $\cos(t^2)$ is not a ...
4
votes
1answer
520 views

central limit theorem for a product

Given $-1\leq x_i\leq 1$ identically distributed random variables for $i=1,2,\dots n$. What is the distribution function of their product? Is there a central limit theorem for products if $n$ is ...
4
votes
1answer
673 views

IID Random Variables that are not constant can't converge almost surely

I am trying to prove the following. If $\{ X_n \}$ are iid random variables and not constant, then $R:=P\{ \omega \mid X_n(\omega)\text{ converges} \}=0$ Using independence I know that by ...
4
votes
1answer
2k views

Independent and mutually exclusive

Prove or disprove via proof that events $X$ and $Y$ can be independent and mutually exclusive if both of their probabilities are greater than $0$.
4
votes
2answers
8k views

Expected Value of a Hypergeometric Random Variable

How do you show, that the expected value of a hyper-geometric random variable X with parameters $r$,$w$, and $n$ (a box contains $r$ red balls, $w$ white balls and $n$ balls are drawn without ...
4
votes
1answer
681 views

Expectation of $TS_T$ where $T$ is the absorption time at $\{a,-a\}$ of a simple symmetric random walk $\{S_n\}$

I was trying to calculate the expectation of $T^2$ using some martingale and got that I needed the expectation of $TS_T$. Any idea?
3
votes
2answers
89 views

Hashing upper bound?

I am hashing $n^2$ objects into $n$ slots and all slots have equal probabilities of taking in the values, and I am trying to find an upper bound on the expected maximum number of objects in any slot. ...
3
votes
1answer
202 views

Is a probability of 0 or 1 given information up to time t unchanged by information thereafter?

Given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_n\}_{n \in \mathbb{N}}, \mathbb{P})$, let $A \in \mathscr{F}$. Suppose $$\exists t \in \mathbb{N} \ \text{s.t.} \ E[1_A | ...
3
votes
2answers
86 views

Convergence to $N(0,1)$ in distribution

Question is as following: $X\sim Po(\lambda)$ $$\frac{X-\lambda}{\sqrt{\lambda}} \,{\buildrel d \over \rightarrow}\, N(0,1)$$ as $\lambda \rightarrow \infty$. Obs. One is asked not ...
3
votes
2answers
171 views

Show that $\lim\limits_{n\rightarrow\infty} e^{-n}\sum\limits_{k=0}^n \frac{n^k}{k!}=\frac{1}{2}$

Show that $\displaystyle\lim_{n\rightarrow\infty} e^{-n}\sum_{k=0}^n \frac{n^k}{k!}=\frac{1}{2}$ using the fact that if $X_j$ are independent and identically distributed as Poisson(1), and ...
3
votes
0answers
227 views

Independence in infinite sequence of random variables

I was reading this page from planetmath. It states the following, Consider an infinite sequence of independent random variables $\{X_n,n\in \mathbb{N}\}$. Using the Monotone Class Theorem one ...
3
votes
2answers
211 views

The strong Markov property with an uncountable index set

The following definition of the strong Markov property, from Klenke's book, supposes an index set $I$ that is not necessarily countable. However, it is explicitly mentioned previously (following Lemma ...
2
votes
2answers
550 views

The distribution of barycentric coordinates

Let $\mathcal{X} = \{x_1,\dots,x_n\} \subset \mathbb{R}^d$ and let $Z$ be a random variable uniformly distributed over convex hull of $\mathcal{X}$, denoted as $\text{conv}(\mathcal{X})$. Assuming ...
2
votes
1answer
274 views

Conditional expectation and martingales

I have a few questions concerning martingales. Let $Y\in \mathcal{L}^1(\Omega,\mathcal{F},\mathbb{P})$ be given, and $(\mathcal{F}_n)$ a filtration, and define $X_n:=\mathbb{E}[Y|\mathcal{F}_n]$. We ...
2
votes
3answers
518 views

Uniform distribution on $\mathbb Z$ or $\mathbb R$

I was assisting once the course in Probability Theory where students learnt quite quickly that there are ways to assign the uniform distribution to any finite set - or even subsets of $\mathbb R$ of a ...
1
vote
1answer
85 views

Assume $A,B,C,D$ are pairwise independent events. Decide if $A\cap B$ and $B\cap D$ are independent

Assume A,B,C,D, are pairwise independent events. Decide if $(A \cap B)$ and $(B \cap D)$ are independent events? Then repeat this assuming the four events are mutually independent. Well, what I'm ...
1
vote
2answers
136 views

If $\{X_n\}$ converges in probability to $1$, where does $\{1/X_n\}$ converge to?

Without using the continuous mapping theorem, I want to show that, given $\{X_n\}$ is a sequence of random variables converging in probability to $1$, $\{1/X_n\}$ converges in probability to $1$. The ...
0
votes
1answer
42 views

Geometric random variables $X_1:G(p_1)$ $X_2:G(p_2)$ $X_3:G(p_3)$ are independent, prove the following :

$$P(X_1 < X_2 < X_3)= \frac{(1-p_1)(1-p_2)p_2p_3^2}{(1-p_2p_3)(1-p_1p_2p_3)}$$ To be frank I do not know where to start with this question, I would like an idea to get me going, or better yet ...
0
votes
2answers
272 views

Is a non-negative random variable with zero mean almost surely zero?

We have proven the following in class: If $X$ is a finite random variable with $X\geq 0$ then $$E(X)=0 \iff P(X=0)=1$$ (By finite I meant that the range has finitely many elements). Does it ...
9
votes
2answers
1k views

Completeness of a finite direct sum of closed subspaces of $L^2$

Let $X_1$ and $X_2$ be real-valued square-integrable random variables defined on a probability space $(\Omega, {\cal F},P)$. For $i=1,2$, set $$ A_i := \{g(X_i)\in L^2 \mid g \text{ is some Borel ...
8
votes
0answers
254 views

Uniqueness of the random variable from its distribution [closed]

Moderator's Note: This question has been put on hold due to the version over at MathOverflow having received better attention and produced an accepted answer. Interested readers are advised to ...
8
votes
8answers
6k views

Expected value is a linear operator? Under what conditions is median also a linear operator?

I have always taken for granted that expected value is a linear operator. For any random variables $X$ and $Y$: $E(aX + bY) = aE(X) + bE(Y)$. Can anyone point me to a rigorous proof of this? Also, I ...
7
votes
1answer
852 views

Hölder Continuity of Fractional Brownian Motion

I would like to prove the following theorem: Let $H\in (0,1)$. The fractional Brownian motion $B_H$ admits a version whose sample paths are $a.s.$ Hölder continuous of order strict less than $H$. ...
7
votes
1answer
711 views

stopped filtration = filtration generated by stopped process?

I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky: Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration ...
7
votes
2answers
1k views

Dual space of the space of finite measures

Since I am reading some stuff about weak convergence of probability measures, I started to wonder what is the dual space of the space consisting of all the finite (signed) measures (which is well ...
6
votes
2answers
312 views

Proof of the infinitude of primes by probabilistic methods.

I'm looking if there is proof of the infinitude of prime numbers using probabilistic method. I am motivated by the answer of my question here. The answer is based on a relationship between ...
5
votes
1answer
158 views

Uniform Integrbility

I need to show that if $|f_n|\leq g$, and $g$ is integrable, then $\{f_n\}$ is uniformly integrable, i.e., $\underset{a\rightarrow\infty}\lim\sup_n\int_{[|f_n|\geq a]} |f_n| d\mu=0$. Here is how I ...
5
votes
1answer
507 views

What is the significance of $\sigma$-fields in probability theory?

I'm an undergraduate level maths student and I've just done a number of exercises on elementary set theory and sigma fields. We've just started the courses and are discussing probability events as ...
5
votes
1answer
343 views

Surface of a sphere and cube

I have a sphere $S^2 \in \mathbb{R}^3$ with radius 1 that is painted red on the surface (90% of it), the rest is painted blue. Now I shall show for every configuration of a cube that is possible in ...
5
votes
1answer
114 views

$X+Y\in L^1$ implies $X \in L^1$ given $X$ and $Y$ are independent random variables

This problem can be found here, which is a previous prelim exam problem of UT Austin. Let $X$ and $Y$ be two independent random variables with $X+Y \in L^1$. Show that $X\in L^1$. Generally, ...
4
votes
1answer
239 views

X,Y,Z are mutually independent random variables. Is X and Y+Z independent?

X,Y,Z are mutually independent random variables. Is X and Y+Z independent? Please, give me a hint how to prove it?
4
votes
0answers
116 views

Why not defining a measure as a function on functions?

A measure $\mu$ is a function to $\left[0,\infty\right]$ on the sets belonging to a $\sigma$-algebra. Then for integrable functions $f$ the integral $\int fd\mu$ comes in, having nice properties ...
4
votes
1answer
249 views

Joint distribution by independent distributions

We have $N$ independent discrete finite random variables (RVs) $X_1,\dots,X_i,\dots,X_N$ where RV $X_i$ has $M_i$ finite number of elements. We are free to choose any distribution $f_i$ for RV $X_i$ ...
4
votes
2answers
2k views

Explanation of Lyapunov condition of CLT

I found the Lyapunov condition for applying the central limit theorem, which is useful in settings where one has to deal with non-identically distributed random variables: Lyapunov CLT. Let $s_n^2 = ...
4
votes
2answers
501 views

Let $X,Y\sim \mathcal{N}(0,1)$. Let $Z=\max(X,Y)$. Find $EZ$.

Let $X,Y$ independent random variables with $X,Y\sim \mathcal{N}(0,1)$. Let $Z=\max(X,Y)$. I already showed that $F_Z$ of $Z$ suffices $F_Z(z)=F(z)^2$. Now I need to find $EZ$. Should I start like ...
4
votes
1answer
3k views

Expectation of Stopping Time w.r.t a Brownian Motion

How do you take the expectation of a stopping time with respect to a Brownian motion? The specific question is: $$ \tau = \inf\{ t \ge 0: B(t) \in \{-a, b\}\} $$ I understand the optional stopping ...