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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26 views

Probability of infected but does not show symptoms of disease?

A person moving through a tuberculosis prone zone has a $50\%$ probability of becoming infected. However, only $30\%$ of infected people develop the disease. What percentage of people moving through a ...
2
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0answers
8 views

What is the rate of convergence in the uniform local limit theorem?

Let $(X_i)_i$ be a sequence of iid random variables, s.t. for some sequences $a_n, b_n$ the normalized sum $$Z_n=\frac{X_1+\dots+X_n}{b_n}-a_n$$ converges weakly to an $\alpha$-stable distributed ...
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0answers
19 views

Is the expected value of a monotone function on a uniformly distributed random variable monotone?

Consider the following definition: A sequence of uniformly distributed random variables $(X_n)_{n \in \mathbb n}$ where $X_{n-1} \sim U[a_{n-1},b_{n-1}]$ and $X_n \sim U[a_{n},b_{n}]$ such that $a_n ...
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3answers
15 views

Basic question about the joint probability.

I have a doubt regarding the joint probability. Experiment: I have a deck of cards marked from $1$ to $4$. Two cards are drawn in sequence without replacement. Let $X$ denote the random variable ...
1
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1answer
11 views

Independence of two non-negative integer valued random variables

Let $X,Y$ be two non-negative integer valued random variables defined on a probability space $(\Omega,\cal F, \Bbb P)$. The question is, If $\Bbb P\{X=i,Y=j\}=\Bbb P\{X=i\}P\{Y=j\}$ for every ...
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0answers
10 views

Proving linear operators are Markov Generators

I am trying to do the following question from Liggett. Let $A$ be a linear operator define on the space of continuous functions $C(E)$ for a compact set $E$. Define $A$ by $A=T-I$ where $T$ is a ...
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0answers
13 views

Finding where the tail starts for a probability distribution, from its generating function

Suppose we generate "random strings" over an $m$-letter alphabet, and look for the first occurrence of $k$ consecutive identical digits. I was with some effort able to find that the random variable ...
2
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2answers
27 views

How to prove $P(|X_n-X_m|>\epsilon)\leq P(|X_n-X|>\epsilon/2)+P(|X_m-X|>\epsilon/2)$?

Consider random variables $X, X_1, X_2, ...$ in a probability space $(\Omega, \mathcal F, P)$ such that $X_n\stackrel{p}{\rightarrow} X$. Let $n, m \in \mathbb N$. How can I prove that for some ...
3
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1answer
44 views

Which inequalities are there with stochastic integration?

Which inequalities can I use with stochastic integration? For example, with the standard lebesgue integral we have $$\left|\int_\Omega f(x) dx\right| \le M |\Omega|$$ (where $M$ is the maximum of ...
7
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2answers
493 views

Does strong law of large numbers hold for an average of this triangular array containing “almost i.i.d.” sequences?

If a sequence of $n$ i.i.d. positive random variables $X_1,\ldots,X_n$ has the expectation $\mathbb{E}[X_1]=\mu<\infty$, then it is known that the strong law of large numbers (SLLN) holds: ...
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0answers
25 views

Expected value and distribution of a random walk (continuous time, discrete state space)

I'm having trouble with a rather simple calculation: Let $(X_t)_{t\geq0}$ be a simple random walk in continuous time on the integer grid $\mathbb Z^d$. Let $\mathbb P_x$ and $\mathbb E_x$ denote ...
1
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1answer
15 views

Lévy Process existence of the expectation of the supremum of the past process.

Given a Lévy Process $X_{t}$ in $\mathbb{R}^{d}$, with $X_{t}^{*}:=\sup_{s\in[0,t]}|X_{s}|$. I want to show, that for $t>0$ with $E[|X_{t}|]<\infty$ for $t>0$, then ...
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0answers
16 views

To estimate the probability that a diffusion reaches a certain value

I have a diffusion process define by the following equation: \begin{equation} dX_t=X_t[\beta(N-X_t)-\alpha]dt+\sqrt{X_t(\beta(N-X_t)+\alpha}) { }dB_t \end{equation} and I proved that the solution ...
0
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1answer
23 views

Variance of Signum Function of Two Random Variables

Let $ X $ and $Y$ be two random variables with means $\mu_X$ and $\mu_Y$ respectively, as well as variances $\sigma_X$ and $\sigma_Y$ (all of which exist). I am interested in computing the following ...
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1answer
28 views

Find the joint probability density function of Max and Min

This is the problem 1.2.13 of Karlin's book An introduction to stochastic modeling: Let X and Y be independent random variables each with the uniform probability density function ...
2
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1answer
24 views

Conditional expectation of a random walk given that it is positive

Let $\{\xi_k\}$ is a sequence of iid random variables with $E(\xi_1)=0$ and $E(\xi_1)^2=\sigma^2<\infty$. Define the random walk $Y_n=\sum_{k=1}^n \xi_k$. Is it necessarily true that the ...
1
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1answer
167 views

Polya's urn model - limit distribution

Let an urn contain w white and b black balls. Draw a ball randomly from the urn and return it together with another ball of the same color. Let $b_n$ be the number of black balls and $w_n$ the number ...
2
votes
2answers
29 views

Convergence rate of mean and standard deviation.

I have a random variable simulator with Normal distribution $(\mu,\sigma^2)$. I repeatedly conduction simulation. Each time, the simulation gives $N$ numbers $x_1,x_2,\ldots,x_N$. I use the $N$ ...
0
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0answers
12 views

renewal process and jumps(question)

Let $N_t$ be a renewal process and $T_n$ the jumps with $T_n=X_1+...+X_n$.$X_1,..,X_n$ where $X_i$'s are independent random variables identically distributed law $F_X$. Let $A_t:=t-T_{N_t-1}$ and $x ...
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0answers
21 views

How to compute the Lebesgue-Stieltjes measure for given intervals

Let u be a Lebesgue-Stieltjes measure on the Borel σ-algebra. Let Fu be the associated function such that u([a,b)) = Fu(b)−Fu(a). Calculate a) u([a,b]); in terms of the function Fu. b) u((a,b)); in ...
0
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1answer
33 views

Prove the following integral is asymptotically zero

I have to solve the following exercise. I would appreciate to get a hint for it. Suppose $(\Omega, \mathcal{F}, \mu)$ be a measure space. Let $f$ be an integrable function. Show ...
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1answer
12 views

Independence: norm v.s. direction of a standard multivariate normal vector

Suppose that $v\sim N(0,\sigma^2 I_n)$ and with $||\cdot||$ denoting the Euclidean norm, define $$ u=v/||v||\quad\text{and}\quad w=||v||. $$ I've been told that $u$ and $w$ are independent and I see ...
1
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1answer
30 views

Showing that $\mu$ is a measure when continuous from above

Statment Let $\mu$ be a set function defined on a $\sigma$ -algebra. Show that $\mu$ is a measure given that $\mu \geq 0$, $\mu(\emptyset)=0$, $\mu$ is continuous from above and countably additive. ...
0
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1answer
35 views

Poisson probability of an event A before event B

I'm trying to calculate the probability of two poisson processes events happening one before the other, with two different $\lambda$s. The way I see it, I can word it as the probability of event $A$ ...
0
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0answers
10 views

Proof of substitution rule for conditional expectation

Let $v: \mathbb{R}^2 \to \mathbb{R}$ be a function and $X, Y$ random variables. It holds $$ \mathbb{E}[v(X,Y)|Y=y]=\mathbb{E}[v(X,y)|Y=y], \ y\in R(Y). $$ What would be a way to start the proof? I ...
0
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1answer
23 views

Finding $P(C)$ with Bayes's Theorem

We have two events $C$ and $D$ such that $0<P(D)<1$ and a $P(C|D)=P(C|D^{c}) = \frac{1}{3}$. I am wondering if it is possible to calculate $P(C)$ from only this information. I've tried using ...
1
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1answer
13 views

Reformulation of SLLN with continuous nondecreasing process as time

Given a filtered probability space $(\Omega,\mathcal{F}_{t},P)$ and a continuous nondecreasing process $U_{t}$ with $U_{0}=0$ and $U_{t}\rightarrow \infty$ $P-a.s.$ as $t$ goes to $\infty$. Given a ...
7
votes
1answer
538 views

Girsanov: Change of drift, that depends on the process

Known: If I am looking at an SDE like: $dX_t = b(t,\omega) dt + dW_t$ with $W_t$ a Brownian motion under a measure $P$. I know that I can change the drift by using Girsanov to $dX_t = ...
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0answers
13 views

Finding the autocorrelation of $X(t)$ and $Y(t)$ from the autocorrelation and pseudocorrelation of $Z(t) = X(t)+i Y(t)$

Consider $Z(t) = X(t) + iY(t)$, $i$ being imaginary. Knowing that $$ r(t_1,t_2) = e^{i(t_1 - t_2) - (t_1+t_2)^{2}} \quad\quad\text{and}\quad\quad \mathrm{pseudo}-r(t_1, t_2) = 0 $$ how can one ...
0
votes
1answer
41 views

Given three independent events $A,B,C$, is $I_A+2I_B$ independent of $I_C$?

Let $(\Omega,\mathcal{F},P)$ be a probability space and $A,B,C\in \mathcal{F}$ are independent. Is $I_A+2I_B$ independent of $I_C$? $I_A,I_B,I_C$ are indicator random variables. I started by ...
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votes
1answer
119 views

Distribution in Polya's Urn / existence of mgf / Stolz–Cesàro alternative / dominated convergence theorem

I know this has been asked elsewhere, but I think the values or random variables are different or something. From Williams' Probability with Martingales: I proved that $M_n$ is a $\sigma(B_1, ...
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3answers
386 views

Why does randomness exhibit a pattern in the long run?

!!! Layman here so please avoid complex math and answers. Random (usually pseudorandom) events are usually characterized along these lines: Each outcome in a trial experiment must be i.i.d.; i.e. ...
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1answer
31 views

What is the probability of unions of intersections? [on hold]

Suppose we have two unions of (possibly overlapping) events. Let me denote the unions as: $$A = IE_A^1 \cup \dots \cup IE_A^{k_A}$$ $$B = IE_B^1 \cup \dots \cup IE_B^{k_B}$$ Each $IE_X^y$ is a ...
-1
votes
1answer
31 views

How do I transform this Probability in weak law of large number form?

Let $X_{1},\dots$ be a sequence of independent random variables. Suppose, for $k=1,2,\dots$ $$P\left(X_{2k-1}=1\right)=P\left(X_{2k-1}=-1\right)=\frac{1}{2}$$ and the probability density function of ...
1
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2answers
54 views

Show that a random variable $T_x$ is uniformly distributed given that $T$ is uniformly distributed?

We have a lifetime $T$, which is uniformly distributed over $(0,b)$. We then introduce a new r.v., $T_x=T-x$, which is defined on $0<x<b$. We want to show that given $T>x$, the variable ...
5
votes
2answers
49 views

Is Probability really consistent with our world?

Say we have 6 unbiased coins, We toss 5 coins and get 5 heads. Then what is the probable outcome of the sixth toss? Mathematically every new and discrete event should be independent of the results of ...
0
votes
1answer
41 views

Conditional entropy and independent conditioning variables

Let $X,Y,Z,Y',Z'$ be random variables where $Y\sim Y', Z\sim Z'$, $Y$ and $Z$ are independent, while $Y'$ and $Z'$ are, in the sense that we have $p(X,Y,Z)=p(X|Y,Z)p(Y)p(Z)$ ...
2
votes
1answer
412 views

Characteristic function of vector-valued random variables

I just begins my self-study on Brownian motion. I got stuck on the part about random-vector and characteristic function. Here are my questions: I'm not quite get about how characteristic function of ...
0
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1answer
24 views

How to prove $E\|Y'\|\leq E\|Y'-Y''\|,$ where $Y$ is a random matrix in $\mathbb{R}^{n\times n}$, and $Y'$, $Y''$ are independent copies of $Y$?

How to prove $$E\|Y'\|\leq E\|Y'-Y''\|,$$ where $Y$ is a random matrix in $\mathbb{R}^{n\times n}$, and $Y'$, $Y''$ are independent copies of $Y$; $\|\cdot\|$ denotes the $l_2$ operator norm;$E$ ...
1
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2answers
53 views

The intersection of all events in a sequence has probability $\lim \limits _{k \to \infty} P(A_k)$

If a sequence $A_1, A_2, A_3, \dots$ of events is decreasing, show that the intersection of all events in the sequence has probability: $\lim \limits _{k \to \infty} P(A_k)$. I suck at proofs so I am ...
3
votes
3answers
86 views

Conceptual Statistics. Define for this problem, population, Samples and Estimators, and when is Normal Dist?

Students in Stanford are supposed to spend on average 3 hours of time per week for every credit hour they take. Last year, 263 randomly selected seniors were contacted and asked how much total time ...
5
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1answer
50 views

How to calculate the shortest interval, for $P ( X ≤ 1 . 645) = 0 . 95$?

The problem statement said: Based on the fact that $\Phi(1 . 645) = 0 . 95$ find an interval in which $X$ will fall with $95\%$ probability. Therefore: Since $P ( X ≤ 1 . 645) = 0 . 95, ( -∞ , ...
3
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0answers
55 views

If $X_n\nearrow X$ then $E(X_n)\rightarrow E(X)$?

Let $(X_n)$ be an increasing sequence of real valued integrable rvs on a probability space $(\Omega,\mathcal{F},P)$, such that $(X_n)$ converges ae to some rv $X$. Is it true that $E(X_n)\rightarrow ...
0
votes
2answers
30 views

Fourier transform problem with symmetric matrix. Related to Gaussian?

Hi everyone I encountered a problem that looks simple enough but I have no idea where to start. Find Fourier transform of $e^{-\langle Ax,x\rangle}$ when $A$ is a positive definite symmetric $n ...
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0answers
32 views

Are there different definitions of a continuous time Markov chain, condition on a finite or infinite number of earlier values?

My book defines a continuous time Markov chain like this. Let $\{X_t\}, t \in T$ be a stochastic process on $(\Omega, \mathcal{A}, P)$, with a countable state space $S$. The process is a Markov ...
1
vote
1answer
65 views

If $dX_{t} = X_{t}\,dt + \,dB_{t}$, why does $e^{- t}dX_{t} = e^{-t} X_{t} \,dt + e^{-t} \,dB_{t}$?

I'm taking a course in stochastic differential equations, and in order to solve $dX_{t} = X_{t}\,dt + \,dB_{t}$, the book gives a hint: to multiply both sides of this equation by $e^{-t}$. (But, as ...
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vote
2answers
34 views

If $X_n\geq 0,~X_n\rightarrow X$ ae and $E(X_n)\leq c,$ then $E(X)\leq c.$

Let $(X_n)$ be a sequence of positive valued rvs on a probability space $(\Omega,\mathcal{F},P),$ such that $(X_n)$ converges ae to a rv $X.$ If $E(X_n)\leq c<+\infty$ for all $n$, then $X$ is ...
-1
votes
0answers
39 views

show that $Y_1$ is unbiased for $\theta$ and find its variance [on hold]

Let $X_1,\ldots,X_n \stackrel {\text{iid}} {\sim} \text{$P_0$}(θ)$ $$Y_1= \frac {X_1+3X_2+5X_5} {9} $$ $$ Y_2= \sum_{i} X_i$$ Show that $Y_1$ is unbiased for $\theta$ and find its variance. Show ...
1
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1answer
25 views
0
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0answers
30 views

Probability Mass Function of a Sentence

We have a sentence: Some dogs are brown. We choose one letter (out of the 16) at random. Let Y be the length of the whole word containing the letter. How can I find the probability mass function of Y? ...