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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1answer
21 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
7 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
17 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
14 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
8 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 ...
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 ...
-1
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0answers
25 views

Probability of passing an exam with random % [on hold]

A student gives an exam with 4 questions, 1 right answer.He knows the answer for 20% , he knows that 30% has 2 wrong answers,and at 50% he don't know anything. What's the probability to answer at 1 ...
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
20 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 ...
2
votes
1answer
23 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 ...
0
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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
32 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 ...
1
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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
29 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
34 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
24 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 ...
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 ...
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 ...
0
votes
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 ...
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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
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 ...
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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 ...
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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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
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;$E$ ...
3
votes
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 ...
1
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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
vote
1answer
25 views

Example of a non square-integrable martingale?

Are there (simple) examples of martingales which aren't square integrable?
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 ...
3
votes
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 ...
2
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0answers
21 views

Application of Doob's optional stopping theorem to an elementary probability problem

The elementary probability problem is as follows. Let $(X_k)_{k\in\mathbb{N}}$ be a sequence of i.i.d. random variables such that $X_k \sim U(0,1)$ for each $k$. Define $\tau := \inf\{n\geq 0: ...
0
votes
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? ...
1
vote
1answer
39 views

How to show that this is a martingale?

Let $H_s$ be a predictable and bounded process. How can I show that $$M_t = \int_0^t H_s \, dW_s$$ is a martingale? Clearly since $H_s \in L^2_\text{loc} (W)$ we have that $M_t$ is a local ...
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
26 views

The “how many pieces do you have buy on average” problem, a markov problem?

I recently discovered a problem similar to this one in a book about Markov chains: Assume you can buy $n-$ different set of cards in a store, but you do not know which one you'll buy: What is the ...
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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 ...
0
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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 $p(X,Y,Z)=p(X|Y,Z)p(Y)p(Z)$ ...
1
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0answers
34 views

If $P(X_1 < X_2)$, what is $P(X_1 < X_2 \cap X_1 < X_3)$?

Say $X_i$ can have a real value in the range [1,100]. All $X_i$ are independent of each other and all values are equally likely. So then $\mathbb{P}(X_1 < X_2) = \frac{1}{2}$, right? But then, ...
0
votes
1answer
23 views

uniform Distribution on uncountable Lebesgue $0$-Sets

I know that for every measurable Set A it is possible to create a uniform Distribution on A if - A is finite - A is not a lebesgue 0-set and its not possible for infinite countable sets so I ...
0
votes
1answer
19 views

Probability of a vector of normal distribution

Given the set of vectors $\{\mathbf{g}^{1}, \ldots, \mathbf{g}^{N-1} \}$ where $\mathbf{g}^{i} \in \mathbf{R}^M$. Assume that $N \leq M$ and elements of $\mathbf{g}^{i}$ follows normal distribution, ...
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0answers
44 views

Distribution and expectation value of ceiling function of Poisson

There is Poisson random variable $X$ $$P(X=x)=\frac{\lambda^{x}}{x!}e^{-\lambda}$$ And define random variable $Z=\lceil \beta X \rceil$ ( $\beta$ is rational number which is less than 1). How can I ...
-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
vote
1answer
38 views

Expected value problem: flip $6$ fair coins before we obtain $3$ heads and $3$ tails?

How many times on average (expected value) must we flip $6$ fair coins before we obtain $3$ heads and $3$ tails? I know I need $∑ xp(x)$. I just don't know how to apply it.
0
votes
2answers
24 views

Can covariance (X,Y) be easily expressed in term of Var(X), Var(Y), E(X), and E(Y)?

Can $Cov(X,Y)$ be easily expressed in term of $Var(X), Var(Y), E(X), $ and $E(Y)$ ?
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0answers
17 views

Continuity of random variable as function of a random variable

Suppose, we are given a continuos random variable $X$ and a continuous and nondecreasing function $f$. Can it be shown that a second random variable $Y=f(X)$ is continuos on the support of $X$? What ...
0
votes
1answer
29 views

Poisson Probability with rate $\lambda (t)=-(t-4)^2+16$

The rate at which customer arrive to the bookstore is $\lambda (t)=-(t-4)^2+16 $ where $t$ measured in hours. The customers can buy a book with probability $0.5$ and they can also buy a coffee ...
4
votes
1answer
22 views

Ehrenfest Chain: stationary distribution

In the Ehrenfest Chain model: There are M balls which are divided between urn A and urn B. At each stage, if a ball is chosen, then it would be moved into a different urn. Let $X_n$ be the # of ...
3
votes
3answers
60 views

Upper bound for difference of Poisson random variables

Let $X, Y$ be random variables with Poisson$(\lambda)$ and Poisson$(2\lambda)$ distributions, respectively.Then (i) If we assume that $X, Y$ are independent, $$\mathbb{P}(X \geq Y) \leq ...