Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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1answer
18 views

verifying stopping times…

Let $m$ be a natural number, $$g_m:=\sup\left\{ {n\leq m: S_n\leq 0}\right\}$$ and $$d_m:=\inf\left\{ {n\geq m: S_n \leq 0}\right\}$$ I have to check if they are are stopping times. It's still a new ...
1
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0answers
17 views

Finding joint distribution function?

Let $U$ and $V$ be two independent uniform (0,1) random variables and let \begin{eqnarray*} R &=&\sqrt{\frac{U^{2}+V^{2}}{2}},\\ H &=&\frac{2U}{U+V}. \end{eqnarray*} I found the ...
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0answers
21 views

Is $\exp(-2\sin^2t)$ a characteristic function?

Is $\exp(-2\sin^2t)$ the characteristic function of some random variable?
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1answer
25 views

Probability Question for two head coin [on hold]

I have tried to find the solution of this question but i am not able to get the correct solution for this... It will be very beneficial for me if someone provides me the complete solution to this ...
0
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1answer
44 views

Calculation of Conditional Expectation

I have problems with the following exercise: Let $\Omega=[-\frac{1}{3},\frac{1}{3}]$, $\mathcal{F}=\mathcal{B}(\Omega)$ the Borel-$\sigma$-algebra on $\Omega$ and P the Lebesgue-measure. ...
0
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1answer
32 views

Finding a probability density function of a function of three dependent random variables

I have three random variables that are functions of another three random variables by pairs, say: $U=fc(X,Y)$, $V=fc(Y,Z)$ and $W=fc(X,Z)$, with $X$, $Y$ and $Z$ being independent random variables ...
0
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1answer
13 views

In an irreducible, aperiodic, null-recurrent Markov chain holds $\sum_n p_{ij}^{(n)} = \infty$

My lecture notes state the following theorem: Theorem 2. Let $(X_n)$ be an irreducible, aperiodic, null-recurrent Markov chain. Then $$\forall i,j \in S : p_{ij}^{(n)}\to 0 \text{ and } \sum_n ...
2
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1answer
23 views

Basic question about conditional expectation

Consider $X$ and $Y$ tow random variables $\mathcal F_2$-measurable where $\mathcal F_1$ and $\mathcal F_2$ are two $\sigma$-algebras such that $\mathcal F_1 \subseteq \mathcal F_2 $. Can we always ...
0
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0answers
6 views

spin-off of Choosing the correct subsequence of events s.t. sum of probabilities of events diverge [on hold]

Spin-off from here: Choosing the correct subsequence of events s.t. sum of probabilities of events diverge 1 Does m have to be 2? 2 Is it correct to say that for $(A_{nm+i})_{n\in\mathbb{N}}, m\in ...
0
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1answer
19 views

Choosing the correct subsequence of events s.t. sum of probabilities of events diverge

Here is the problem. I tried choosing $B_n = A_{mn}$ since it is an independent sequence for $m \geq 2$, but I am not quite sure how to guarantee that $\sum_{n=1}^{\infty} P(A_{mn}) = \infty$. Is ...
0
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0answers
52 views

Upper bound on the covariance of two gamma processes?

Given two binary gamma processes, $X = \Gamma(t; \gamma_1, \lambda_1)$ and $Y = \Gamma(t; \gamma_2, \lambda_2)$, what is their maximum covariance? Applying this answer, it would seem that it is the ...
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0answers
24 views

Billingsley “Probability and Measure” on constructing $\sigma$-fields

i'm starting to read, very slowly, Patrick Billingsley's "Probability and Measure". in chapter 1 "Probability", section 2 "Probability Measures", there's an optional section "Constructing ...
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0answers
27 views

Bounding the size of consecutive sums of independent Bernoullis

Let $\{X_i\}$ be a sequence of independent Bernoulli random variables that take the values 1 and 0 each with probability 1/2. Is the following statement true? For any $\epsilon > 0$, there exists ...
0
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0answers
16 views

Hypercontractivity of Markov Operator

I have been reading a paper by Ahlswede and Gacs on hypercontractivity of Markov operator (see here 1) and its application to information theory. To be honest, I could not fully understand the ...
0
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3answers
24 views

Median Value + Mode for Hybrid Functions of a Continuous Probability Density Function

To find the median: should I set the integral to 0.5.... but because there are two functions that are non-zero, I am unaware of a method to find the median. To find the mode: would I need to ...
1
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1answer
13 views

Bound on variance of random process when signal is known

I am reading this paper (link to a Nature paper, may not be accessible) and I encountered the following. I have very little experience in probability theory and I could not find much helpful in ...
1
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1answer
22 views

Square Integrable local martingale or locally square integrable martingale?

I have a question about martingales. What is the difference between "locally square integrable martingale" and "square integrable local martingale"? In particular, which set does $M_{loc}^2$ ...
0
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1answer
32 views

Probability of order statistics with numerous conditions

Let $Y_i$ be the i-th order statistic of a continuous random variable $Y$ and let $z_{k-1}\leq z_k$ for all $k$. Let $1\leq j\leq n-1$. How can I evaluate or rewrite $$Pr(Y_{n-j-1}\leq ...
3
votes
1answer
43 views

Convergence of discrete random variables, show $\frac{S_n}{\sqrt{n}}\to0$ a.s.

Let $X_n$ be a sequence of independent discrete real random variables, with discrete density $$p_{X_n}(x):=\Pr(X_n=x)= \cases{ 1-\frac1{n^2} & \text{if } x= 0\cr \frac1{2n^2} & \text{if ...
1
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2answers
29 views

Can we show that $E\|X-Y\|^2 \leq E\|X-Z\|^2 + E\|Z-Y\|^2$

Let $X,Y,Z$ be some random elements on some Hilbert space $(H,\langle\cdot,\cdot\rangle)$. Can we show that $$E\|X-Y\|^2 \leq E\|X-Z\|^2 + E\|Z-Y\|^2$$ I can clearly see that $$E\|X-Y\|^2 \leq ...
0
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0answers
12 views

Limiting distribution of loss random variable?

I'm going to try to make the notation not actuarial-specific, but for those with a background in actuarial science, this relates to exam MLC. Suppose I have random variables $X_{i} \geq 0$ such that ...
1
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1answer
48 views

Sum of random variable

Considering two continuous random variables $X$ and $Y$ with $d.f \; F_X, F_Y$ I want to fin the distribution and distribution function of the sum $Z=X+Y$. \begin{align} P\{Z \leq z\} &= P\{X+Y ...
0
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1answer
21 views

Poisson Process Suitable Scenarios

I have a couple of doubts about if these scenarios are suitable to be modeled as a Poisson process. I will like to have your views and arguments why. Packets are lost due to packet overflow in the ...
1
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1answer
53 views

A sequence of nonconstant i.i.d. random variables converges with probability zero

Proove: $X_{n} iid, X_{n}$ not constant a.s. $\iff P(X_{n}$ $converges)=0$ My idea for "$\Rightarrow$": $X_{n}$ not constant a.s. $\iff \forall$ c $\in \mathbb{R}$, $\varepsilon$ > 0: ...
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0answers
23 views

Probability of occurrence of two events simultaneously

I have a question with probability of occurrence of two events simultaneously. I have a probability histogram for some events occurring individually. Is it possible to predict the simultaneous ...
0
votes
2answers
27 views

Calculating Probabilities for a cumulative distribution function within a given inequality

Given that K = 1/36, I require some help understanding (b) • Pr(1/2 ≤ X ≤ 1) Is re-written as such: Pr(X ≤ 1) - Pr(X < 1/2) I do not understand why! Is it because Pr(X ≤ 1) is solved as ...
1
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1answer
18 views

Calculating Probabilities using a cumulative distribution function

For (b) Pr(X greater than or equal to 2) = ? The textbook says as such but I am confused: Pr(X greater than or equal to 2) = 1 - pr(X less than 2) I do not understand why they re-write the ...
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0answers
31 views

Determining the Cramer-Rao lower bound

Let $X = (X_1,\dots,X_n)$ be a vector of iid variables from the smooth density $f(x,\theta_0), \theta_0 \in \Theta \subset \mathbb{R}$. Let $L(\theta)$ be the likelihood and $I(\theta)$ the ...
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0answers
11 views

Product of Gaussian random variable with hermition of another independent gaussian random variable. [on hold]

If X∼ CN(0,1) and Y ∼ CN(0,1). X and Y are vectors independent of one another. How to find the E[(X†)Y]. What will be the probability density funtion of Z, If Z = (X†).Y ?
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41 views

Values of $\mu$ for which $S_n=e^{\sum_{i=1}^n X_i}$, is a martingale ($X_i ~ \mathcal{N}(\mu,1)$) [on hold]

Let $(X_n)_{n\geq 1}$ a sequence of $\mathcal{N}(\mu,1)$ $\mathcal{F}$-adapted and $S_n=e^{\sum_{i=1}^n X_i}$. I have to write the conditions for which S_n is a martingale, then I have to show that ...
1
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1answer
31 views

Poisson process has independent and stationary increments

Being $N_t$ a Poisson process, defined as $$N_t:=\sum_{n\geq 1} n \mathbb{1}_{[T_n,T_{n+1}[}(t)$$ where $T_n$ are sums of independent exponential random variables, how can I prove it has stationary ...
2
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1answer
17 views

Why $E[X|\mathcal{G}]=X$ if $X$ is $\mathcal{G}$-measurable?

If $X$ is a $\mathcal{G}$-measurable random variable, why $E[X|\mathcal{G}] = X$? I know the intuition (basicly we're conditioning on the same informations on which $X$ is defined, $\sigma(X)$, we ...
0
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2answers
27 views

I'm not sure if I'm supposed to use a Poisson distribution or Conditional Probability (or both) to answer this question

I have a question that I'm trying to solve. I have the answer but I don't know how they arrived at the answer so I can't compare my work and see where I went wrong. The number of injury claims per ...
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2answers
25 views

Riemann Integral in Probability With Martingales by Williams

"Note on the Riemann integral" in chapter 5 of Probability with Martingales by Williams reads: If, for example, f is a non-negative Riemann integrable function on $([0,1],\mathcal{B}[0,1],Leb)$ ...
0
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2answers
11 views

Expected Profit for Binomial Variable

Part (a) I am familiar with: (a) P(batch is rejected) = P(X greater than or equal to 3) and n = 15 and p(defective) = 0.1 This gives me the correct answer of 0.1841 I am stuck at part 2! I have ...
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0answers
23 views

Will the branching process go extinct with probability 1?

I am trying check whether the branching process goes extinct with probability one. Single Type Branching Process with Pk = (1/2n)(n/k), for k = 0,.....,n with n > 2. Assuming, i can be able to ...
0
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2answers
30 views

The relation of $P(X=x+1)$ and $P(X=x)$ in binomial distribution

If I substitute the values to the binomial probability theory, it appears as such $${n \choose x+1} p^{x+1} (1-p)^{n-x-1}$$ I don't know how to move on... What am I doing wrong, or are you ...
0
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1answer
39 views

How to apply the strong Markov property in this case?

I'm trying to understand the following proof: Theorem: Let $(X_n)$ be an irreducible $(\alpha, \mathbf p)$-Markov chain with a finite state space $S$. Then $(X_n)$ is positive recurrent. ...
0
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0answers
21 views

Let $X_t$ be a Brownian motion find $X_2>2, x_1>x_2,$ and $x_t<4$ for all $2\leq t\leq 3 $ [on hold]

Let $X_t$ be a Brownian motion find $X_2>2, x_1>x_2,$ and $x_t<4$ for all $2\leq t\leq 3 $ Can you help me with tips and bibliography... I don't understand very good the topic, and I can't ...
0
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1answer
31 views

Transformation of random variable

I want to prove the following: $$\text{Let F be a distribution function of any random variable $\\$ and G(x) the quantile function (or inverse) of } \frac 1 {1-F(x)}$$ $$\text{Then, for a standard ...
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1answer
20 views

What are possible theories/models to describe social influence based on frequency?

I am working on social influence based on frequency, meaning if a person repeatedly appears at a location many times then he/she is more influential at that location. In other words, he/she can ...
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0answers
25 views
+50

How to get closed form solutions to stopped martingale problems?

Way back when, I took a course in stochastic processes in college. I remember being frustrated by the plethora of abstract proofs without much in the way of how to use them to get actual results. It ...
2
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2answers
37 views

How to maximize pay with repeated toss of coin

repeated toss a coin and you can stop anytime and payoff is just #times you got head divided by total number of throws, how do you maximize your pay. Does anyone have a clever strategy for this? ...
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3answers
66 views

Transition Matrix of M/M/1 Queue

We know that for an M/M/1 queue the state space is $S=\{0,1,2,... \}$. Further the probability to go from state $i$ to $i+1$ is $\lambda$ for all $i$ in $S$. Moreover, to go from $i$ to $i-1$ is the ...
1
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1answer
53 views

Confusion with real numbers and random variables; Integration and Independence in Williams

In David Williams' Probability with Martingales, $\exists$ this exercise. Let $X_n$ be iid RVs with the same continuous dist function. Let $E_1 = \Omega$ and for $n \geq 2, E_n = (X_n > X_m ...
2
votes
1answer
62 views

“Fair” game in Williams

In David Williams' Probability with Martingales, $\exists$ this exercise. What's fair about a fair game? Let $X_i$, i = 1, 2, ... be indp RVs s.t. $X_i = i^2 - 1$ with prob $1/i^2$ and $-1$ with ...
0
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1answer
25 views

Probability of highest common factor in Williams

In David Williams' Probability with Martingales, $\exists$ this exercise. Let s > 1 and let $\zeta(s) = \sum_{n=1}^{\infty} {n^{-s}}$. Let X and Y be independent $\mathbb{N}$-valued random variables ...
2
votes
2answers
38 views

Let $X$ be the value of the first die and $Y$ the sum of the two dice. Find $E(X / Y)$ [closed]

Consider an experiment of rolling two dice. Let $X$ be the value of the first die and $Y$ the sum of the two dice. Find $E(X / Y)$, ie, obtain the value of $E(x/y) (y)$ for all $y$ Good evening, I ...
0
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0answers
50 views

Mean and variance of a stochastic process

Let \begin{equation} \begin{array}{l} y_1(t)=e^{-\kappa_1 t}y_1(0)+\displaystyle\int_0^t\kappa_1 e^{\kappa_1(s-t)}\theta_1ds +\sigma_1\displaystyle\int_0^te^{\kappa_1(s-t)}\sqrt{y_1(s)}dZ_1(s),\\ ...
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0answers
12 views

probability distributions [closed]

![ Question 1. 1. Using the probability distribution table, what is the value of P(X = 2 or X = 0)? X 0 1 2 3 4 5 P 0.3 0.05 0.1 0.15 0.15 0.25 P(X = 2 or X = 0) = _____ (Points : 1) ...