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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4
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1answer
31 views

How to show that $(X(t)-\lambda t)^2 - \lambda t$ is a martingale, where $X(t)$ is a Poisson Process?

I am trying to show that $(X(t)-\lambda t)^2 - \lambda t$ is a martingale, where $X(t)$ is a Poisson process with rate $\lambda$. So far, what I have done is: \begin{align*} E\left((X(t)-\lambda t)^2 ...
0
votes
1answer
36 views

expected value of fisher distribution

I know that the pdf of an F-distribution is $f_{k,m}(t) = \Gamma(t)=\frac{\Gamma((k+m)/2)}{\Gamma (k/2)\Gamma(m/2)}k^{k/2}m^{m/2}t^{k/2 - 1}(m+kt)^{-(k+m)/2}$ Also, $E(F)=\int xf_{k,m}dx$. How do ...
2
votes
1answer
48 views

Does weak convergence ($X_n\Rightarrow X$) imply weak convergence of the difference to zero ($X_n-X\Rightarrow 0$)?

Let $(X_n)_{n\geq1}$ be a sequence of random variables weakly converging to $X$ ($X_n\Rightarrow X$) as $n\rightarrow \infty$. I am wondering if this implies that $X_n-X\Rightarrow 0$ as $n\rightarrow ...
2
votes
1answer
22 views
+50

Linear transform of a strictly stationary time series

First, let me clarify what I mean by a strictly stationary time series. Let $(X_t)_{t\in \mathbb{Z}}$ be a sequence of random variables on some probability space. If it holds that $$(X_t, ...
0
votes
1answer
26 views

how to calculate conditional independence

This Bayesian net (click) is given with the binary variables B, F, G and D and the following probabilities $p(B=1) = 0.9$ $p(F=1) = 0.9$ $p(G=1\mid B=1,F=1) =0.8$ $p(G=1\mid B=1,F=0) = 0.2$ ...
2
votes
3answers
27 views

Proving continuity of the following function

Let $X,Y$ be compact sets in $\mathbb{R}^n$ (with the usual topology) and let $f:X\times Y \rightarrow \mathbb{R }$ be a continues function function moreover let $P(Y)$ be the space of all ...
1
vote
0answers
19 views
+50

Convergence of moments of a sequence of random variables

I encountered this problem in my study of time series. It seemed trivial at first but I don't see the finishing move to complete the proof. The problem is as follows. Let $(X_n)_n$ be a sequence of ...
1
vote
1answer
31 views

Conditional probability involving coin flips

A coin has an unknown head probability $p$. Flip $n$ times, and observe $X=k$ heads. Assuming an uniform prior for $p$, then the posterior distribution of $p$ is $B(\alpha = k + 1, \beta = n - k + ...
1
vote
1answer
26 views

Probability proof and making sure I cannot make further simplifications to my answer

The question asks me to compute the probability (sums are ok) of the probability of having at least one of r cells empty with n>r balls thrown at the cells with equal likelihood of landing in any of ...
1
vote
2answers
18 views

Show $E(Y)-E(X) = \int_{\mathbb R} P[X<t\le Y] - P[Y< t \le X] dt$

Suppose X and Y are integrable random variables on the measure space $(\Omega,\mathcal F, P)$. Im trying to show that $E(Y)-E(X) = \int_{\mathbb R} P[X<t\le Y] - P[Y< t \le X] dt$ but I got ...
1
vote
0answers
18 views

Reshuffling the order statistic of uniform at midpoint

Let $U_1,\dots,U_n$ be i.i.d. drawn from the uniform distribution on $[0,1]$ and let $U_{(1)} \le U_{(2)} \le \dots \le U_{(n)}$ be their order statistics. Assume that $n$ is even and that we take ...
0
votes
0answers
16 views

Calculating the Distribution Function of Chord Length

The question: Choose two (different) points on the circle $S^1 \subset \mathbb{R}^2$ at random (with uniform distribution), and connect them with a straight line. Define a suitable probability space ...
2
votes
0answers
21 views

I found $E(Σ_{j=0}^{k-1}η_j-Σ_{j=0}^{k-1}E(η_j|G_j))^2=Σ_{j=0}^{k-1}(E(η_j)^2-E(E(η_j|G_j)^2)$ in a book with faulty assumptions on the objects

In Stochastic Equations in Infinite Dimensions (Second Edition) on page 109, the authors state the following: If $\eta_0,\ldots,\eta_{k-1}$ are random variables with finite second moments and ...
0
votes
0answers
28 views

Upper bound for $\frac{\|x\|_1}{\|x\|_2}$ if each entry of $x\in R^d$ is i.i.d. sampled from Gaussian distribution $N(0,1)$

In the question, $\|x\|_1=\sum_{i=1}^d|x_i|$ with $|\cdot|$ being the absolute value, and $\|x\|_2=\sqrt{\sum_{i=1}^d x_i^2}$. In general, $\frac{\|x\|_1}{\|x\|_2}\leq \sqrt{d}$ always holds for ...
1
vote
1answer
22 views

How to define $E(X)$ when X is a random variable from the sample space to an infinite-dimensional topological vector space?

Let $V$ be an infinite-dimensional vector space with a topology. For simplicity, we can assume $V$ is a Banach space. Let $(\Omega, \mathcal{F}, P)$ be a probability space. Let $X:\Omega \to V$ be a ...
0
votes
1answer
17 views

Moment generating function properties: $3φ_X (t)$ and $φ_X (t) × φ_X (6t)$

Suppose that $φ_X (t)$ is the moment generating function of some random variable $X$. Are the following functions moment generating functions of some (other) random variables? i. $3φ_X (t)$ I think ...
2
votes
1answer
16 views

Bounding expectation of a supremum process

This is exercise 3.9(c) on page 15 of Karatzas and Shreve's Brownian Motion and Stochastic Calculus. Let $N_t$ be a Poisson process with intensity $\lambda$. In particular, if $t$ is fixed, $N_t$ is ...
1
vote
1answer
37 views

two ways of counting

I'm reading Morris DeGroot's Probability and Statistics. In chapter 1.9 there's an example 1.9.3 says that suppose that 12 dice are to be rolled. We shall determine the probability $p$ that each ...
1
vote
0answers
32 views

Probability Models (Poisson Process)

There are two types of claims that are made to an insurance company. Let $\ N_{i}(t)$ denote the number of type i claims made by time t, and suppose that $\{N_{1}(t): t \ge 0\}$ and $\{N_{2}(t): t ...
1
vote
0answers
16 views

Possibility of analogy of the Bayes theorem for expectations

I recently found a scenario where I wanted to find the relation of $ E[X|Y] $ and $ E[Y|X] $ for $ X,Y $ two random variables. For probabilities and densities we have the Bayes theorem which is well ...
1
vote
1answer
13 views

A specific question in conditional expectation with mixed discrete and continuous random variables

In my probability class I have just met this seemingly difficult question: Let $ \{X_n\}_{n=1}^{\infty}, \{Z_n\}_{n=1}^{\infty} $ be two i.i.d sequences of random variables such that we know $ ...
1
vote
1answer
23 views

Proof explanation - weak law of large numbers

Let $(X_i)$ be i.i.d. random variables with mean $\mu$ and finite variance. Then $$\dfrac{X_1 + \dots + X_n}{n} \to \mu \text{ weakly }$$ I have the proof here: What I don't understand is, why ...
1
vote
0answers
25 views

Prove that a sum of random variables converges against an Itō integral

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$$^1$ be nonnegative and symmetric with finite trace $f:[0,\infty)\times H\to\mathbb R$ be ...
2
votes
0answers
24 views

Derive an Itō formula for $f(t,X_t)$ where $X_t=X_0+tY+W_tZ$ and $f:[0,\infty)\times H\to\mathbb R$ and $H$ is a Hilbert space

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$$^1$ be nonnegative and symmetric with finite trace $f:[0,\infty)\times H\to\mathbb R$ be Fréchet ...
0
votes
1answer
25 views

Convergence of random probability measures.

This question is motivated by the standard definition of convergence in probability, although applied in a different way... Suppose that I have $V_n = V(W_1,...,W_n)$ for $i=1,...,n$ where $W_i$ are ...
2
votes
1answer
42 views

Is this an adapted process?(deterministic integrator in Itô-process)

Assume you have a probability space with a filtration, $(\Omega,\mathcal{F},P,\{\mathcal{F}_t\})$. Assume that the stochastic process $X_t$ is adapted to this filtration, and is jointly measurable ...
3
votes
1answer
40 views

Example of a Continuous-Time Markov Process which does NOT have Independent Increments

1. Given a discrete-time Markov chain without independent increments, is the embedding of it into a continuous time Markov chain (i.e. via the use of exponential waiting times) an example of a ...
1
vote
1answer
29 views

probability of which alarm clock goes off first

I'm learning about continuous time markov chains, in the text I am reading, they are setting up the discussion by talking about a series of alarm clocks that are set: Suppose $T_1,...,T_n$ are ...
1
vote
2answers
30 views

Limit question on independent random variables (Exercise 4.2.4 from Grimmett and Stirzaker)

Let $\{X_r | r \geq 1\}$ be independent and identically distributed with distribution function $F$ satisfying $F(y) < 1$ for all $y$, and let $Y(y) = \min \{k | X_k > y\}$. Show that $$ ...
1
vote
1answer
17 views

Compute the distribution function of the random variable $Y:=-ln(F(X))$

I'm trying to prove this result: If $X$ is a continuous random variable with distribution function $F$, where $F$ is strictly increasing function, then find the distribution function of the random ...
0
votes
2answers
29 views

Does this property hold for these type of functions?

Assume that $X(t,\omega)$ is a stochastic process such that $\omega$. And we have that $E[\int_0^T X^2dt]<\infty$. Is then the r.v. $\int_0^TXdt$ in $L^2(P)$? This looks a lot like what happens ...
-1
votes
1answer
33 views

Probability and conditional expectation [closed]

$ X, Y, Z$ and $W $ are jointly distributed Bernoulli random variables; and each of these can assume values $0$ or $1$ only. It is known that $X = max $ {$W, Z$} and $Y = min$ ...
0
votes
0answers
17 views

Is there a theorem or law stating that the expected value of a symmetrical distribution across $f(X)$ will equal $f(\bar{x})$ iff $f'=c$ across X?

I have noticed that the expected value of a symmetrical distribution across $f(X)$ will equal $f(\bar{X})$ only if $f'$ is constant across X. For example, consider a uniform distribution X ranging ...
1
vote
1answer
13 views

A particular implication of convergence in probability

Suppose that $(X_n)_{n\in\mathbb N}$ and $X$ are random variables on a probability space $(X,\Sigma,\mathbb P)$ and $X_n\overset{\mathbb P}{\to}X$. That is, for each $\varepsilon>0$, ...
-1
votes
0answers
20 views

Why X and Y=u(X) have the same probability for the 2 sets specified below?

Let X,Y be two random variables defined on the same sample space. More specifically, Y(S) = u ◦ X(S), where u is just a simple transformation. Define the sets {y′|y′ ≤ y} and {x|u(x) ≤ y}. Why do ...
1
vote
0answers
16 views

“Local” functional central limit theorem for the empirical distribution function

Assume $(X_i)_{i=1}^{\infty}$ is a sequence of i.i.d. real-valued random variables such that $\mathbb E[X^2]<\infty$. Denote by $F_X(t) := \mathbb P(X\leq t)$ their common distribution function. ...
1
vote
0answers
33 views

Prove that $I_A$ is a random variable with $A$ being the union of events $B$ $\in$ $\mathbb{F}$ such that $0 < P(B) < 1$

Let $(\Omega, \mathbb{F}, P)$ be a probability space. Prove that $I_A$ is a random variable with $A$ being the union of events $B$ $\in$ $\mathbb{F}$ such that $0$ $<$ $P(B)$ $<$ $1$. My ...
1
vote
0answers
22 views

For Ito diffusion, what is the difference between two measures $Q^x$ and $P$?

I am confused about the difference between $Q^{s,x}$ and $P$ for the following ito diffusion: $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad t\ge s;\quad X_s=x.$$ Followings are from most books: given the ...
0
votes
0answers
11 views

Equicontinuity of stochastic approximation projection terms in Kushner and Yin

I've been learning a bit about stochastic approximation via the ODE method and have gotten stuck on a point that arises when dealing with projected/truncated SA algorithms. To be concrete let me ask ...
1
vote
0answers
19 views

Markov property of ito diffusion [duplicate]

Most books show Ito diffusions satisfy Markov property, that is, $E[f(X_{t+h})\mid F_t]=E^{X_t}[f(X_h)]$. But I was wondering whether it's true that $E[f(X_{t+h})\mid X_t]=E^{X_t}[f(X_h)]$. In this ...
0
votes
1answer
14 views

How do you compute multivariate normal distribution probabilities?

Given a vector $$ X = (X1,X2,X3)^t $$ which is multivariate normal with mean 0 and covariance matrix $$ \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & 1 \\ 0 & 1 & 3 ...
0
votes
3answers
39 views

Expectation Value of a Multiset

Imagine that I have $k$ balls randomly distributed (uniformly) among $n$ boxes. I.e., with repetition. How could I calculate the expected number of balls in a randomly chosen box?
0
votes
2answers
42 views

Show $(X_n+a)^2$ is a submartingale

Let $(X_n)$ be a martingale, and let $EX_n^2 < \infty$ - then I am told to show $E(X_n+a)^2 $ is a sub martingale. I wrote $$(X_n+a)^2 = ((X_{n-1} + a) + (X_n - X_{n-1}))^2 $$ then $$E((X_n+a)^2 | ...
0
votes
1answer
44 views

Proof of Barbier's Theorem

In the probabilistic proof of Barbier's Theorem, I'm not sure why the expected number of line crossings of a continuous curve is the limit of the expected number of line crossings of piecewise linear ...
3
votes
1answer
39 views

Borel-Cantelli exercise

I'm stucked with this exercise. Let $X_1,X_2,\ldots$ be i.i.d. random variables with $E(X_1)=0$ and $Var(X_1)=\infty$ Prove that$$P(\limsup\limits_{n\to\infty}\{|X_n|\geq \sqrt{n}\})=1$$ I need to ...
1
vote
1answer
32 views

Convergence of series of random variable without distribution

I'm trying to solve the following task and I'm struggling very much. I don't know if it is correct what I did so far. Let $(X_n,n\geq1)$ be a sequence of independent random variables such that ...
1
vote
0answers
24 views

Itō isometry in Hilbert spaces

Let $U$ and $H$ be separable Hilbert spaces $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $\mathfrak L:=\mathfrak L(U,H)$ be ...
0
votes
0answers
6 views

How can we proove that it's a Gaussian system?

$(W_1, W_2)$ are 2 independent Wiener processes and $$B_1= W_1, ~~~ B_2 = a W_1 + \sqrt{1-a^2} W_2,$$ where $a=(a(t, \omega))_t>0$ and is $(F_t=F_t^{(W_1,W_2)})$-measurable. $0<a<1$. It ...
-1
votes
0answers
36 views

Semicontinuity of stochastic kernel

Let $X$ and $Y$ be metric spaces with Borel sigma algebra and $P(B|y)$ be a stochastic kernel on $X$ given $Y$. I'm trying to proof the equivalence of the following two statements: (i) $P(\cdot,y)$ ...
0
votes
0answers
13 views

Is there such a thing as a Negative Poisson Binomial?

The Poisson Binomial distribution is a generalization of the binomial for non-equiprobable Bernoulli trials. The negative Binomial give the number of Bernoulli trials until a number of success is ...