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 [tag:probability-...

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2
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1answer
43 views

$Z_n \stackrel{a.s.}{\to} 0$ and $E(|Z_n|) \to 0$

Let $Z_n$ be a sequence of random variables with finite expectation. Is the following statement true? i) $Z_n \stackrel{a.s.}{\to} 0$ implies $E(|Z_n|) \to 0$ ii) $E(|Z_n|) \to 0$ implies $...
2
votes
1answer
95 views

Prove that $X_n = e^{(\theta S_n-n \ln \phi(\theta))}$ is martingale.

Let $\xi_1 \neq const.$ be a random variable with moment-generating function $\phi(\theta) = Ee^{(\theta \xi_1)}$. Let $S_n = \xi_1 + \dots \xi_n$. Prove that $X_n = e^{(\theta S_n-n \ln \phi(\theta))}...
2
votes
1answer
39 views

Conditional expectations and random vectors.

Let $(\Omega, \sigma,P)$ a probability space and $Y$ a random variable on it, with $E|Y|<\infty$. Let $X_1,X_2$ random vectors with $\sigma(Y,X_1)$ independent of $\sigma(X_2)$. The problem is to ...
1
vote
1answer
41 views

Recurrent Set and i.i.d. random sequence

Consider an i.i.d. discrete random sequence $\{X_i\}$, suppose $EX_1 \neq 0$ and define $R:=\{x: \text{ $x$ is recurrent value for $S_n$}\}$. I was trying to show the set $R = \emptyset$ where $S_n := ...
3
votes
0answers
80 views

Let $X_n \sim U[0,1]$. Let $A_n$ count the number of local maxima of the sequence unto $n$. Prove a suitable central limit theorem for $A_n$.

Let $X_n $ be uniformly distributed on $[0,1]$. We say $X_k$ is a local maximum if $X_k> X_{k\pm 1}$. Let $A_n$ count the number of local maxima of the sequence unto and including $n$. Find $a_n, ...
1
vote
1answer
49 views

The definition of terms in Doob's decomposition theorem for submartingales

The definition of $Z_n$ in the Doob's Decomposition Theorem, I think it is a predictable submartingale starting at $0$. Is that right? Thanks for your help!:)
0
votes
1answer
30 views

Question about filtration of sigma algebra?

If $F_n$ is a filtration, which means $F_n \subset F_{n+1}$. Then if $Z_1$ is $F_0$ measurable, then is it true that it is $F_m$($m\ge 2$) measurable? Thanks so much! Your help will be appreciated!
1
vote
1answer
77 views

Why does $\sigma (X_t) \subset \sigma (X)$ hold?

We have a random process $X=\{X_t\;,t\in T\}$, where $X_t:(\Omega,\mathcal{A})\to(S_t,\mathcal{S}_t)$ are random variables. I am confused as to why does $$\sigma (X_t) \subset \sigma (X)$$ hold $\...
1
vote
1answer
20 views

Conditional expectation if the events of the sigma field have probability 0 or 1.

Let $(\Omega, \sigma, P)$ a probability space, Y a $\sigma|\beta$ measurable function with $E|Y|<\infty$ and $\sigma'$ any $\sigma$-field, $\sigma' \subset \sigma$. If we also have that $$P(A) \in \...
1
vote
1answer
57 views

Survival Probabilities

Consider an article whose lifetime $X$ takes integral values $0,1,2,...$ and define $p_j=P(X=j)$. If we define also $$ b_j=P(X=j+1|X>j)=\frac{p_{j+1}}{\sum\limits_{k=j+1}^{\infty} p_{k}} $$ ...
1
vote
1answer
132 views

continuous probability: average power of a Gaussian random variable $\mathcal{N}(0, \sigma^2)$

My book states that $E[X^2]$ is the average power. It then says for $\mathcal{N}(0, 1)$, the average power is $\frac{1}{2}$ and for $\mathcal{N}(0, \sigma^2)$ is $\frac{\sigma^2}{2}$. How can this be?...
4
votes
2answers
410 views

Weak Law of Large Numbers for a non-iid, non-ergodic sequence

I have a somewhat open-ended question. Let's say I have a sequence of random variables $(X_n: n \geq 1)$ which are neither independent, ergodic, nor identically distributed. Normally I would say that ...
0
votes
1answer
29 views

Easy Question about Computing the probability of properties of random subsets

I want to solve the following exercise: Suppose that two sets $X$ and $Y$ are chosen independently and uniformly at random from all the $2^n$ subsets of $\{1, \dotsc, n\}$. Determine $P[X \subseteq ...
3
votes
0answers
216 views

the continuity theorem with respect to Laplace transform

Let $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of probability measures on $\mathbb{N}$, such that the Laplace transform $\phi_n(\lambda)=\int e^{-\lambda x}\mu_n(dx)$ converges pointwise to a limit $\...
1
vote
0answers
32 views

Weak convergence of probability measure on $\mathscr{C[0,\infty]}$

What is the reason for considering the spaces of probability measures on the space of all continuous functions and then considering weak convergence there ? Is it that we can then use Skorohod's ...
3
votes
1answer
132 views

How to calculate the characteristic function of compound Poisson random variable?

Let $\phi_X(t)$ be the characteristic function of $X$. Let $N$ be a Poisson random varivale with mean $1$ and $(X_i)_{\in\mathbb{N}}$ be i.i.d. copies of $X$. Then how to derive the charactersitics ...
0
votes
1answer
55 views

A basic question on weak convergence of measures

Why do we need separability of the space to talk about weak convergence of measures ?
2
votes
1answer
134 views

probability: continuous uniform distribution mean by symmetry

I am trying to show that the mean of the uniform continuous distribution is $(b+a)/2$ by symmetry. The direct method is fairly simple but, for some reason, I cant get this one. \begin{align} E[X] &...
1
vote
1answer
35 views

Conditional Coin Probability:Will The Decision Change

A decision making problem will be resolved by tossing $2n + 1$ coins. If Head comes in majority one option will be taken, for majority of tails it’ll be the other one. Initially all the coins were ...
2
votes
2answers
198 views

An example for conditional expectation

A factory has produced n robots, each of which is faulty with probability $\phi$. To each robot a test is applied which detects the faulty (if present) with probability $\delta$. Let X be the number ...
1
vote
0answers
150 views

Rotational invariance and distributions

Let $k\leqslant n$ denote two positive integers, $A$ an $n \times k$ matrix with $A'A = I_k$, and $X$ and $Y$ two independent random variables on $\mathbb R^n$, each rotationally invariant (that is, ...
5
votes
0answers
131 views

How to model this easy problem as sum of indicator random variables in order to apply Chernoff bound

Do you have an idea how I could model the following process somehow as a sum of independent indicator random variables? I have given a grid of size $n \times n$ for $n \rightarrow \infty$. Now I ...
2
votes
1answer
107 views

Martingale Transforms and quadratic variation

Let $M$ be a martingale with $\mathbb{E}M_{n}^2<\infty$ for all $n$. Let $C$ be a bounded predictable process and set $X=M\cdot C$. Show that $\mathbb{E}X_{n}^2<\infty$ for all $n$ and that $\...
2
votes
1answer
275 views

What does “taking expectation w.r.t some random variable” mean in this probability calculation?

I am trying to calculate the following probability $$\mathbb{P} \big(\sum_{i=1}^{m} (A_i + S_i) \le L < \sum_{i=1}^{m+1} (A_i + S_i) \big)$$ where, $$A_i \sim \exp(\lambda), \quad S_i \sim \exp(...
2
votes
1answer
67 views

Using Levy-Khintchine representation theorem to prove the following theorem

Let $(X_{n,i})_{1\le i\le n}$ be a triangular array of independent random variables, satisfying the uniform infinitesimality condition $$\lim_{n\rightarrow\infty}\max_{1\le i\le n}P(|X_{n,i}>\delta|...
0
votes
2answers
73 views

Simple formula on $X_n^{(k)} = \sum_{1 \le i_1 < … < i_k \le n} Y_{i_1} \cdot \dots \cdot Y_{i_k}$ (to show $X_n^{(k)}$ is martingale)

Let $$X_n^{(k)} = \sum_{1 \le i_1 < ... < i_k \le n} Y_{i_1} \cdot \dots \cdot Y_{i_k}$$ If I take $k=2$ and $S_n = Y_1 + \dots + Y_n$ I have of course: $$X_n^{(2)} = \frac{1}{2} (S_n^2 - \...
1
vote
1answer
59 views

Sum converging a.s.

Let $X_k$ be independent random variable s.t. $\sum_{k=1}^nX_k\rightarrow_{a.s.} X$. So, $$X=\sum_{k=1}^\infty (X_k^+ -X_k^-)$$ Is it true that $X=\sum_{k=1}^\infty (X_k^+)-\sum_{k=1}^\infty (X_k^-)...
0
votes
2answers
218 views

Pick two points on the perimeter of a circle of radius 1. Find the expected value of the length of the shortest arc.

Pick two points uniformly randomly on the perimeter of a circle of radius 1. This divides the circle into two pieces. Find the expected value of the length of the shortest piece. I have no idea to ...
3
votes
1answer
59 views

Bounding $P(X \le \tau)$

I am trying to upper bounding $P(X \le \tau)$ where $X$ is non-negative r.v. and where $\tau \le 1$. I have become aware of the Reverse Markov inequality that says that, if $P(|X|\le a)=1$ then for $...
1
vote
0answers
41 views

Moments of $|ax-by|$

Suppose that $X$ and $Y$ are independentr.v. uniform on $[0,1]$. What is the $E[|aX-bY|^p]$ for some constants $a,b,p>0$? What I did. \begin{align*} E[|aX-bY|^p]=\int_0^1\int_0^1|ax-by|^p dx dy ...
2
votes
3answers
55 views

Simple question about random walk with stopping time

I was reading a book and stuck with one line as follows: $$\sum_{m=1}^\infty E[X_m] P(T \ge m) = EX_1 ET$$ where $\{X_m\}$ is i.i.d. with $EX_m < \infty$ and $T$ be discrete stopping time ...
1
vote
2answers
77 views

Soft Question: Are sigma fields, fields?

I'm sorry if this is a foolish question but: Is a $\sigma$-field (of sets) a field (in the sense of algebra) if we only consider finite intersections and finite unions?
5
votes
1answer
91 views

SLLN with dependent Bernoullis implies convergence of sum with conditional means?

Suppose I have $n$ bernoulli (values zero or one), possibly dependent and nonidentically distributed, random variables (like the generalized binomial model), where a law of large numbers holds. Let $...
1
vote
0answers
45 views

Does the system of equations always have a nontrivial solution?

$f:[0,1]^2\to R_+$ is a continuous conditional density function. For $g,h\in C$ on $\{(x,y)\in [0,1]^2|x\geq y\}$, the system of equations is given by$$ \frac{\partial g}{\partial x}\leq 0,\...
0
votes
2answers
43 views

Large deviation in relation with Wishard matrix

I try to prove the following fact. Let $A$ be an $m\times n$ matrix with iid standard normal random variable. Then $B:=A^{t}A$ is a $Wishart$ matrix with $m$ degrees of freedom and covariane $Id_{m}$...
3
votes
0answers
71 views

“Uniform” Convergence in Distribution (bounded Lipschitz metric)

I have been thinking about the following problem. Let me know if the notation below makes sense. Let $\mathcal{P}$ denote the set of Borel probability measures on a metric space $(\mathbb{R}^{k}, ||\...
4
votes
1answer
338 views

Derivative of conditional expectation

Let $\left( {{X_t}:t \in \left[ 0 \right.\left. {, + \infty } \right\rangle } \right)$ be a continuous time Markov chain on a probability space $\left( {\Omega ,\mathcal{F},\mathbb{P}} \right)$ with a ...
0
votes
0answers
31 views

Expected number of distinct nodes visited in a directed bipartite graph

Let $G = (V,E)$ be a directed bipartite graph with $V = \{I \cup O\}$ where $\left\vert{I}\right\vert = n$ and $\left\vert{O}\right\vert = m$. All the edges start from a vertex in $I$ and end on a ...
0
votes
1answer
195 views

How to derive this pdf?

I understand how to find the pdf for the sum of $N$ exponentially distributed random variables, but how do I find the pdf when $N$ is also an independent random variable. Here is the problem: Let $N,...
0
votes
1answer
56 views

Probability that some that m points are probable given the probability of subsets.

Working in a problem in analysis I came across this combinatorial/discrete probability question. I would appreciate if someone knows how to approach this problem or knows if this problem is related ...
2
votes
1answer
317 views

Probability Density Function of Scaled Gamma Random Variable

Assume we have a Gamma Random Variable $X$ with the following pdf $$ \frac{m^mx^{m-1}}{\Gamma(m)}\text{exp}(-mx)$$ If I am asked to find the distribution of the following $$Y= aX$$ where a is non-...
1
vote
1answer
58 views

“With high probability” statement from CLT

Suppose $X_1,X_2,..,X_n$ i.i.d. with mean $\mu$ and variance $\sigma^2$, so that \begin{equation} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\longrightarrow N(0,1)~~~~~~~~~~~~...
0
votes
1answer
46 views

Calculation of PDF of derived multivariate random variables?

Let we have $X$, $N$ dimensional vector of independent random variables. If we multiply this vector by some matrix $V$ with size $r\times N$, with property $V*V'=I$, where $I$ is identity matrix, and '...
1
vote
0answers
71 views

Are there probability density functions with the following properties?

Given two distinct and continuous probability density functions on real numbers, $f_0$ and $f_1$ consider the following set of density functions: $$\mathcal{G}_0=\left\{g_0:\int_{\mathbb{R}}g_0(y)\...
0
votes
1answer
88 views

Double-formula for the expectation

Let $(\Omega,\mathcal{F},P)$ a probability space and $X$ a continuous random variable with density $f(x)$. In probability theory, how to prove that \begin{gather*} \int_{\Omega} X(\omega)dP(\omega) = \...
1
vote
1answer
31 views

Independence between random variables and set

Let $(\Omega,\mathcal{F},\mathbb P)$ be a probability space, and $X$ is a random variable in this space. For a set $A\in \mathcal F$, can we conclude that "$X$ is independent with $A$" $<=>$"$...
2
votes
1answer
150 views

Why are these variables not conditionally dependent given 'active triplets' and the 'explaining away' effect?

I'm following the Udacity Intro to AI course. This quiz gives the following Bayes network and asks whether different variables are conditionally independent or not. (The explanation of the nodes, ...
3
votes
1answer
92 views

Prove that $S_n^2-s_n^2$ is martingale

Let $(X_i)$ be iid such that $EX_i = 0$ and $\operatorname{Var}X_i = \sigma_i^2$. Let $s_n^2 = \sum_{i=1}^n \sigma_i^2$ and $S_n = \sum_{i=1}^n X_i$. Prove that $S_n^2 - s_n^2 $ is martingale. My ...
0
votes
4answers
397 views

mutually exclusive event vs independent event

Can you illustrate with examples, what is "mutual exclusive event" and what is "independent event". Without math equations, please elaborate it.. Thanks in advance
0
votes
1answer
60 views

The independence between stochastic integral and sigma-algebra

Let $(\Omega, \mathcal{F}, \mathbb{P} )$ be the probability space, and {$W_t,0\leq t\leq T$} is a Brownian motion and $\mathcal{F_t}^W$ is the canonical filtration. For the $f(t)\in L^2([0, T])$(a ...