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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3
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1answer
148 views

CLT for independent, but non-identically distributed exponential variables

This problem is practice for my qualifying exam and comes from Resnick, chapter 9. Could anyone comment on my solution(s)? Problem Suppose ${e_n, n\ge 1}$ are independent exponentially distributed ...
3
votes
3answers
71 views

Complete convergence not happening but convergence in probability occurs

So today I created a counterexample to "Convergence in Probability implies Almost Sure Convergence". I considered a sequence $\{X_n\}$ of independent random variables defined by: ...
8
votes
1answer
87 views

On the central limit theorem

The Central Limit Theorem states for a sequence of i.i.d. random variables $\{X_i\}$, $$\frac{\overline{X} - \mu}{\sigma/\sqrt{n}} \to N(0,1)$$ in distribution as $n \to \infty$. I saw in some ...
1
vote
1answer
37 views

Is every continuous CDF the limiting distribution of some sequence of discrete CDFs?

Note: I know that for various measure-theoretic reasons (that I don't fully understand) this does NOT apply to the underlying probability density. I'll accept as answers either a proof, paper to a ...
1
vote
2answers
41 views

Do two almost surely equal random variables necessarily have the same probability?

Let $\Omega$ be a probability space with $\sigma$-algebra $\mathcal{A}$, and let $\mathcal{B}$ be the Borel $\sigma$-algebra. Let $X:(\Omega,\mathcal{A}) \to (\mathbf{R},\mathcal{B})$ and ...
-1
votes
1answer
29 views

Mean Squared Displacement of Biased Random Walk [closed]

If $x_t=x_{t-1}+\mathcal{N}(\mu,\sigma)$ and $x_0=0$ what's the value of $\langle x_t^2\rangle$?
1
vote
1answer
42 views

Is conditional entropy ever taken to be a random variable?

In probability theory, the conditional expectation $E(X|Y)$ and variance $V(X|Y)$ er usually taken to be random variables, st. the value of $E(X|Y)$ depends on what value $Y$ ends up taking. I've ...
1
vote
0answers
52 views

Example tripped Kolmogorov and Wiener

Assuming the hint is true, I attempt to prove the latter prop: Assume on the contrary that $\mathscr{L} = \mathscr{R}$. If $\sigma(Y_0) \subseteq \mathscr{L}$, then $\sigma(Y_0) \subseteq ...
1
vote
1answer
43 views

Name when two functions are equal under integration (expectation)?

What is it called when $E[X] = E[Y]$? That is, $$\int x f(x)\,dx = \int y g(y)\,dy.$$ What I want to say is not that the expectation of $X$ is equal to that of $Y$ but rather (the equivalent ...
0
votes
1answer
42 views

Definition of independence of infinite random variables

When random variables $Y_1, Y_2, ... Y_n$ are independent, we say that $$P\left(\bigcap_{i=1}^{n} (Y_i \in B_i)\right) = \prod_{i=1}^{n} P(Y_i \in B_i)\tag{F1}$$ or for any distinct indices $i_1, ...
1
vote
0answers
29 views

Where to find “non-standard” characteristic functions?

Well, the title says it all. I need the characteristic function of the (generalized) arcsine distribution. I desperately searched the internet for it but haven't found anything. Is there some standard ...
0
votes
1answer
29 views

What is the distribution of the sum of several normally distributed random variables?

Let's say we have n normally distributed random variables all with the same median and variance. Do we have a possibility to estimate the distribution law of the sum of those variables? I assume ...
1
vote
0answers
32 views

Infinitely Many Bernoulli Sums

Suppose $Y = X_1 + X_2 ...$ where $X_n$ is Bernoulli with $p_n$ and that $\sum_{i=1}^{\infty}p_n < \infty$. What is the expectation of $Y$? Since the expectation of $X_n = p_n$ and the ...
0
votes
1answer
39 views

Does aperiodicity needed for ergodic theorem to hold true?

Given a irreducible and positive recurrent countable state Markov Chain with unique stationary distribution $\pi$, is the following true $$\frac{1}{n}\sum_{i=1}^{n}X_i \to \sum j \pi_j$$ Or, to ...
4
votes
2answers
43 views

Urn with increasing number of distinct balls

Suppose we have an urn that initially has only one labelled ball inside. At each time step, we flip a biased coin with probability $p$ $(\in(0,1))$ of landing on heads and probability $1-p$ of landing ...
1
vote
1answer
38 views

Brownian Motion is almost surely unbounded, and a proof for the discrete Random Walk

If $B_t$ is a Brownian Motion, why we have $$ P(\liminf_{t\to\infty} B_t = -\infty) = 1 $$ and $P(\limsup_{t\to\infty} B_t = +\infty) = 1$? I guess for the following "discrete version" of ...
0
votes
2answers
28 views

Continuous marginal distributions do not imply continuous joint distribution

I already proved the other implication. I need to find an explicit example that shows that if there is some random vector $(X,Y)$ and $X$ and $Y$ have both continuous marginal distributions, then ...
3
votes
1answer
47 views

Expected number of return for a random walk on a graph

Let $G$ be a simple, connected undirected graph of order $n$ and vertex set $\{v_1,\ldots,v_n\}$ and let $P = (p_{i,j})$ be a $n \times n$ matrix where $$p_{i,j} = \left\{ \begin{array}{ll} ...
1
vote
1answer
22 views

Simple Set Operation with Random Variable

Consider $X(\omega) \ge -1$ be a discrete random variable and define an event $$ \{\omega: 1+a X(\omega) \le \varepsilon\} $$ where $a \in [0,1]$ and $\varepsilon \in [0,1]$. I was wondering ...
1
vote
1answer
23 views

Convergence in probability of a composite function.

Question: Let $f$ be a continuous function on $\mathbb{R}.$ If $X_n \to X$ in probability, then $f(X_n) \to f(X)$ in probability. The result is false if $f$ is merely Borel measurable. [Hint: ...
1
vote
1answer
42 views

prior probability vs a priori probability

What is the difference between "Prior probability" and "a priori probability" Wikipedia have two distinct pages for them. As of my inference i thought "Prior" and "a priori" are same, i.e., P(y) in ...
2
votes
0answers
50 views

equivalent form of almost sure convergence

Consider random variables $X_1, X_2, \dots$ and $X$ on $(\Omega, \mathcal F, \mathbb P)$. We say that $X_n$ converges to $X$ almost surely if $$\mathbb P\left(\lim_{n \to \infty} X_n =X\right)=1.$$ It ...
0
votes
1answer
50 views

On proving that a infinite intersection of truth sets is empty and on the usefulness of almost surely.

I am trying to solve the exercise at the end of this page, the framework is that of measure theory where we are tossing a coin infinitely often so we are working with a probability triple $( \Omega, ...
4
votes
0answers
87 views

Conditional probability as a primitive concept

Most of the popular axiomatizations/theories of probability define conditional probability as a ratio involving unconditional probability. Therefore, conditional probability is second class to ...
4
votes
1answer
54 views

Uniform continuous distribution for cycles.

Let there be $n$ people standing in a circle and holding hands with probability $p$. What is the expectation value $E(X)$ for the number of 'chains' when $p=.5$? For what $p$ is $E(X)$ largest? ...
1
vote
0answers
23 views

Empirical Quantilfunction as Integral Bound

This is my first post, so please be nice ;) I'll try to outline my problem correctly and whilst keep it as short as possible! I have to deal (for my master thesis) with the integral ...
1
vote
2answers
27 views

Variance and Covariance of Linear Transformations

Suppose $X$ and $Y$ are random variables with $E(X)=2, E(Y)=3 Var(X)= 4, Var(Y)=10$ and $Cov(X,Y)=-5$ Find $Var (5X+2Y)$ From my book I know $$Var(5X+2Y)= Var(5X)+Var(2Y)+2Cov(5X,2Y)$$ but after ...
1
vote
1answer
33 views

Introduction to Lebesgue Integration for Statistical Use

I am studying statistics at the graduate level and have a moderate background in real analysis however I unfortunately have no experience with Lebesgue integration. Does anyone have some recommended ...
0
votes
1answer
27 views

Build a Markov process from a transition semigroup

Suppose that we have a Markov process $X$ with time-homogeneus transition probability function given by $p(t,x,A)$, with $t>0, x\in E, A \in \mathcal E$, where $(E,\mathcal E)$ is the state space. ...
0
votes
1answer
36 views

Kullback - Leibler Divergence and the Triangle Inequality

The KL Divergence, or relative entropy for two probability distributions $p,q$ on $\Omega$ is defined as: $$ H(p|q) = \int_{\Omega} p(\omega) \log \frac{p(\omega)}{q(\omega)} d\omega $$ This is a ...
1
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0answers
28 views

Characteristic function of $a/\sqrt(X+b)$ given characteristic function of $X$

Given that one knows the characteristic function of a rv $X$ how can we write the characteristic function of a function of $X$ when that function is $$\frac{1}{\sqrt{X+b}},$$ for $b$ constant? So, if ...
0
votes
0answers
21 views

Finding the distribution of a dependent function given the mean and coefficient of variation (COV) of the independent parameters.

It is given that $A$ is log-normally distributed with a mean of 0.19 and coefficient of variation of 0.10, and $B$ is log-normally distributed with a mean 0.040 and COV equal to 0.10. Assume that $A$ ...
1
vote
1answer
47 views

Existence of a global maximum of a function defined with the moment-generating function

Can someone give me an idea how to prove the following exercise? Let $Z$ be a real-valued random variable whose moment-generating function $m_Z$, with $m_Z(\gamma)= E\left[ \exp(\gamma Z) \right]$, ...
2
votes
0answers
58 views

If $x_n\in S$ and $x_n\to x$ then $x\in S$

Suppose $S$ is the support of a univariate cdf $F$. If $x_n\in S$ is a sequence of reals such that $x_n\to x$ then show that $x\in S$. I believe the question is actually very very simple, in the ...
2
votes
1answer
29 views

limit of gaussian process

If I have a sequence of gaussian random process $X_{t}^{n}$ which converge in $L^2$ norm to a process $X_t$ for every $t$. can I say that $X_t$ is also gaussian process? thank you
3
votes
1answer
68 views

When do we say independence is the probability of intersection being equal to the product of probabilities? [duplicate]

This is something I never really got in either Elementary Probability Theory or Advanced Probability Theory because my professors mainly discussed independence between 2 objects. Please tell me if my ...
7
votes
0answers
80 views

A matrix with a dense submatrix - application of Chernoff’s Inequality

I am trying to solve an exercise from this book, which I will post here for convenience. I have a bit of a problem understanding how the hint of using Chernoff's bound implies the claim. Specifically ...
-1
votes
1answer
74 views

Proof of Itō's lemma for the Brownian motion

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $B=(B_t)_{t\ge 0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ $\mathcal P$ be a sequence of countable subsets ...
2
votes
0answers
33 views

Integration bounds with $P(XY < z)$

In the following problem, why are the bounds of integration what they are? I think $x$ should be integrated between $0$ and $\infty$. If $X$ is greater than $z$ it's OK as long as $y < \frac z x$. ...
2
votes
0answers
24 views

Stochastic order of the minimum of a stopped sequence of random times

Let $S_k$ and $S'_k$ be two sequences of independent random times such that for each $k \in \mathbb{N}$ it holds that $S'_k \leq_{st} S_k$, that is $S'_k$ is stochastically smaller than $S_k$ which ...
1
vote
1answer
41 views

$(X_n)_{n\in\mathbb{N}}$ independent Cauchy-distributed random variables. Convergence of $n^{-\gamma}(X_1+\cdots+X_n)$

I want to solve the following exercise but i am unsure if my ideas are correct or not. Let $(X_n)_{n\in\mathbb{N}}$ be i.i.d. random variables with probability density $$ ...
4
votes
1answer
48 views

Arc Length of largest arc when $n$ points are chosen at random on the circumference of the unit circle.

I am currently doing a little self-study of A Probability Path by Sidney Resnick, and am having trouble with the following problem: Points are chosen at random on the circumference of the unit ...
0
votes
2answers
36 views

Given probabilities of a horse beating each other horse. What is the probability that the horse will finish in a particular position? [duplicate]

I have computed probabilities that horse A beats horse B, horse A beats horse C and so on. I want to find out the probability that the horse will finish in a particular position, say 2nd. Edit: In a ...
1
vote
1answer
45 views

Probability density function of a ratio with summation?

Let $X_i$ be a random variable that is exponentially distributed with parameter $\lambda$ for all $i\in\{1,\dotsc, n\}$ for some $n$. Let $Y_i$ for all $i$ be the random variable defined by: ...
2
votes
0answers
42 views

Unique stationary distribution (or measure?) of a Markov Chain

Let $(X_n)_{n \geq 0}$ be a irreducible, positive recurrent Markov chain. We have a theorem that states that the unique stationary distribution is then given by $$\pi(x)= \frac{1}{E_x[H_x]},$$ where ...
0
votes
1answer
15 views

MCMC( Markov Chain Monte-Carlo Simulation )

Suppose $Q=[q(i,j)]$ is a transitional probability matrix for an irreducible Markov chain. Suppose also $\lbrace X_{n}, n \geq 0 \rbrace$ is Markov Chain such that if $ X_{n}=i$, generate $Y=j$ such ...
0
votes
0answers
29 views

What is the proof of $\mathbb{E}\Phi (X) = \Phi\left(\mu\sqrt{\frac{1}{1+\sigma^2}}\right)$, where $X \sim \mathcal{N}(\mu,\sigma^2)$?

Let $X \sim \mathcal{N}(\mu,\sigma^2)$. I think it's true that $$\mathbb E \Phi(X) = \Phi \left(\mu\sqrt{\frac{1}{1+\sigma^2}}\right)$$ where $\Phi$ is the cdf of standard normal. This holds up ...
1
vote
1answer
34 views

$E[|X(t)|]\leq K\implies E[|X(\tau)|]\leq K $?

Let $X(t)$ be a stochastic process. Assume that, for every $t\leq M$, it holds $$E[|X(t)|]\leq K, $$ for some constant $K$. Let now $\tau\in[0,M]$ be random (stopping time). Is it true that also ...
2
votes
3answers
49 views

The limit of the integral when the set is decreasing in probability to zero.

This is an exercise problem(#2 in section 3.2) from 'A course in probability theory'. If $E(\vert X \vert ) < \infty$ and $\lim_{n \to \infty} P(A_n) = 0,$ then $\lim_{n \to \infty} \int_{A_n} X \ ...
0
votes
1answer
24 views

How does the probability change when changing the decission?

Let's say you play roulette and you set 2 coins of 2 of the 3 columns or dozens. Then your chance is 2/3rd, so about 66% to win one coin. Let's assume that you loose because the other dozen or column ...