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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776 views

A point in a circle is selected at random. Calculate probability that point is closer to centre than circumference

State any assumption(s) you make Well, I decided to draw a circle with a center at the origin of a Cartesian plane. It had radius r so it's coordinates on the axes were (0, r), etc. I then drew ...
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1answer
26 views

On a real line R points a,b are randomly selected such that -2<=a<=2 and 0<=b<=3. Find the probability that | a - b | > 1

Let's say that C is the set where |a-b|>1 So I suppose you could say plot it as coordinates where the x-axis (labelled a) is from [-2,2] and the y-axis (labelled b) is from [0,3]. Now |a-b| must be ...
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1answer
21 views

Show $P(h(X)\ge a)\ge\frac{E(h(X))-a}{b-a}$

Show that, if $h:\mathbb R\to[0,b]$ and $0\le a< b$ then, $\displaystyle P(h(X)\ge a)\ge\frac{E(h(X))-a}{b-a}$ So $h$ is nonnegative and bounded. If $a=0$ then the inequality holds. because ...
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2answers
32 views

Confused by Probability Notation and Solution

Suppose $P(Z=0)=P(Z=1)=1/2$ and $Y$ ~ $N(0,1)$. $Y$ and $Z$ are independent, and $X=YZ$ The question is to find the law of $X$. I know the Law is the function $P_X$ from $B$ to $[0,1]$, where $B$ ...
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2answers
79 views

Independence of Random Variables and Distribution Functions

Let $X_1, X_2,\ldots$ be random variables on $(\Omega, \mathcal{A}, \mathbb{P})$. If $\mathbb{P}(X_1 \leq x, X_2 \leq y)=\mathbb{P}(X_1 \leq x)\mathbb{P}(X_2 \leq y)$ for all $x,y \in \mathbb{R}$. ...
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3answers
65 views

How to understand the variance formula?

How is the variance of Bernoulli distribution derived from the variance definition?
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1answer
50 views

Understanding the Poisson Memorylessness Proof

I was hoping someone can explain to me step by step the proof of Poisson memorylessness property. First i understand that , ...
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0answers
50 views

Random sampling and i.i.d.

Can you help me to clarify the following concepts by stating whether what I have written below is right or wrong? -random sampling: units are drawn from the population with a known probability of ...
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2answers
37 views

Finding covariance between profit and quality

The quality $X$ of an item is uniformly distributed on the interval $[0,1]$ and the profit $Y$ is given by $Y = X^5$. Find the covariance between $X$ and $Y$ . Can someone interpret this question ...
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42 views

What is the distribution of $Y_n$ and its convergency

Let $X_n$ be a iid sequence of Poisson random variables with parameter 1. Define $Y_0 = 1$ and $Y_n := X_nY_{n-1}$ for $n\geq 1$. How to show that $Y_n$ converges to $0$ almost surely, please? I think ...
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1answer
39 views

A discrete time Markov chain with such a transient state that $\mathbb P(T_i<\infty \ | \ X_0=i) \neq 0$

All examples of discrete time Markov chains my text provides are where $S$ is finite, and as far as I can tell, it makes all transient states have $$\mathbb P(T_i<\infty \ | \ X_0=i) = 0.$$ Are ...
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0answers
36 views

monotonicity of a complex function referring normal distribution

In my research I need to make clear the following point: Suppose that a random variable $\theta\sim N(\mu, \sigma^2)$. There are two imperfect signals about $\theta$: $X=\theta+\sigma_x\xi$ and ...
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1answer
51 views

On the gist of $\sigma(X_1,\ldots, X_n)$

As far as I understand the reason we have $\sigma(X_1,\ldots, X_n)$ all over the probability theory is that it tells us what questions are answerable by $X_1,\ldots, X_n$. Say, we run an experiment ...
2
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1answer
47 views

Estimator for Uniform Random Variables

Let $(U_n)$ be iid uniform random variables over $(0, \theta)$. Consider to estimators for $\theta$, namely, $\max\{2\bar U_n, 0\}$ and $\max\{U_n\}$. My question is how to show that both of these ...
2
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1answer
222 views

Maximizing expected profit

Suppose that a person is going to sell Fizzy Cola at a football game and must decide in advance how much to order. Suppose that he makes a gain of $m$ cents on each quart that he sells at the game but ...
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1answer
44 views

Finding conditional expectation $\mathbb{E}(X\mid (X-0.5)^2)$

I have $X$ uniformly distributed in $(0,1)$. How do I find the conditional expectation, $\mathbb{E}(X\mid (X-0.5)^2)$? My try: \begin{align}\mathbb{E}((X-0.5)^2\mid (X-0.5)^2)&=(X-0.5)^2\\ ...
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1answer
176 views

Sub sigma algebra and probability spaces — definition

I am reading this book and I am a bit lost with the definitions because they are not provided and I can't seem to find it online: Let $L_2(\Omega,A,P)$ be a probability space such that $f \in L_2$ ...
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1answer
73 views

Mean of random sum of random variable

Suppose that we have $X_1, X_2, \ldots$ is a sequence of i.i.d random variables with $E(X_i)<+\infty$ and $N$ is a random variable taking values in $\{1,2,\ldots\}$, $N$ is independent with $X_1, ...
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1answer
356 views

Using binary entropy function to approximate log(N choose K)

I am not a mathematician and struggling with the exercises while reading this book Information Theory, Inference and Learning Algorithms. The author introduced the binary entropy function at the ...
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1answer
32 views

Independence of a family with random measures having independent increments.

First of all, sry for my stupid editing. I have restored my first question and also kept my second version at the bottom, so future readers won't be too confused. Let $\mathcal{N}$ be a family of ...
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1answer
21 views

do we have $n\mathbb{P}_X([n,+\infty[)\to 0 \quad as\quad n\to +\infty$?

Let X be a random variable. I can't find a rigorous proof to show that $n\mathbb{P}_X([n,+\infty[)\to 0 \quad as\quad n\to +\infty$
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2answers
64 views

Can conditional expectation always be realized in a standard probability space?

Given any integrable random variable $X : (\Omega, \mathcal F, \mathbb P) \to \mathbb R$ (where $(\Omega, \mathcal F, \mathbb P)$ is not necessarily a standard probability space) and a ...
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1answer
89 views

What is purpose of these paragraph?

The following paragraphs are from my study notes on probability theory. It is a section within the independence discussion. But to me, they seem to appear here out of blue. I do not understand what ...
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2answers
184 views

What does $\vee$ mean in set theory?

The following proof is from Probability by Davar Khoshnevisan. There is a symbol $\vee$ in the third sentence of the proof. What does this symbol mean, please? There seems no definition about it in ...
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1answer
58 views

Independent Events or Random Variables

First recall the following definition of independent random variables. Let $(X_t)_{t \in \mathcal T}$ be a set of random variables, where $\mathcal T$ is an arbitrary index set. Then $(X_t)$ is ...
3
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0answers
91 views

Strange definition of Ergodicity

In an engineering course, a stationary process was defined to be ergodic if for all $k\in \mathbb{N}$ and for any bounded (measurable) function of $k+1$ variables we have $$\lim_{N\rightarrow ...
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1answer
75 views

Eigenvalue markov chain

I have a questions: We said that if we have a positive recurrent Markov chain, then there is a unique stationary distribution. 1.) Does this mean that if I have several positive recurrent classes, ...
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1answer
48 views

Is $\mathcal{L}^p \subset \mathcal{L}^{p-1} $?

A random variable $X$ is called integrable if $E[X] < \infty$. We say that $X \in \mathcal{L}^1$ if $E[X] < \infty$, and in general $X \in \mathcal{L}^p$ if $E[|X|^p] < \infty$. I know that ...
1
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1answer
99 views

random walk with sticky barriers

Consider a random walk on the line 1,...,d. You start at point 1. At each step you flip a coin: heads means go left, tails means go right. If you're at 1 and get a heads, just stay where you are ...
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2answers
47 views

“Show experimentally” that for large $N$, $X$ appears to be normally distributed.

I'm a bit confused about the following problem: Let $X$ be the random variable $$X = \frac{X_1+X_2+...+X_N}{\sqrt{N}}$$ where $X_k$ is the outcome from the $kth$ flip of a fair coin where heads ...
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1answer
23 views

Injections of Markov Processes

In office hours, a professor mentioned that an injective transformation of a Markov process remains Markov. Intuitively, this makes sense to me as you can "recover" the original Markov process from ...
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2answers
112 views

Probability of Either of Two Teams Winning the World Cup

I'm just trying to get my head around some basic probability theory. Say we have a World Cup sweepstake and each better has been drawn two random teams from a hat. Let's assume that I have been given ...
2
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1answer
77 views

One Corollary of the Kolmogorov Zero-One Law

Here is an application of the Kolmogorov Zero-One Law given in my textbook (a probability path by Resnick page 107-108). It states that the random variables $\limsup_n X_n$ and $\liminf_n X_n$ are ...
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1answer
36 views

drawing balls from box without replacemnt

In box we have $n$ black and $m$ white balls without replacement. Let's denote $B_k$ - number of black balls drawed in first $k$ draws $W_k$ - same for white ones Let's assume that we drawed ...
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0answers
103 views

Upper bound or approximate form for the CDF of Hypo exponential random variable

The CDF of hypo exponential random variable (sum of $n$ independent exponential random variables $X_{i} $ with different rates $\lambda_{i}$) is given by I seek for an upper bound or an approximate ...
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1answer
59 views

Limit of the measure of the converging sequence of sets

This is a question from Ash's book, "Probability and Measure Theory". Let $\mu$ be a finite measure on the $\sigma$-field $\mathcal{F}$. If $A_n\in\mathcal{F}$, $n=1,2\dots$ and $A=\lim_nA_n$, show ...
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0answers
93 views

Transient/Recurrent Markov chain

I am currently studying the concept of recurrent and transient states and was wondering about the following: Is this concept dependent on the initial distribution? Let me take this example: You can ...
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1answer
95 views

About pairwise and mutual independence of random variables

If I have some r.v. $(X_1,\dots,X_n)$ that are (mutually) independent and another r.v. $X_{n+1}$ that is (pairwise) independent from each $X_k, 1\leq k\leq n$. Does this make the vector ...
0
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1answer
37 views

number of ones with neighbours in a random binary string

Consider a sequence of i.i.d. random variables $(\xi_i)_{1 \leq i \leq L}$ such that $\xi_1 \in \{0,1\}$ and $P(\xi=1)=p$. Introduce the function $N : \{0,1\}^{L} \rightarrow \mathbb{N}$ which counts ...
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2answers
21 views

Question about independence

First of all is true that given $X,Y$ two random variables indenpendent; $(X,Y)\in D\subset \mathbb{R}^2$ then $\text{Cov}(X,Y)=0$? I tried to prove it and this is my solution: If $D=[a,b]\times ...
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1answer
101 views

Can someone please help to understand the following probability

I was reading something on communication, then I came across the following equation: $Power_{rx}=Power_{tx}*|R|^2/(1+d^2)$ where $Power_{tx}$ and $d$ can be assume to be constant, and R is the ...
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2answers
66 views

Probability Generating Functions with Three Dice

Three identical dice are thrown. The dice are fair, that is, for all three dice the probability of turning up face $j$ is $1/6$, $1 \le j \le 6$. Let $X_1,\ X_2,\ X_3$ be the independent random ...
3
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0answers
41 views

Asymptotic Bounds for the distribution of $f_n(X_n)$.

Let $\{X_n\}_{n \in \mathbb{N}}$ be a sequence of $\mathbb{R}^{k}$-valued random variables defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ converging almost surely to $X$. ...
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1answer
54 views

A question about tail $\sigma $-algebras

How do I show formally that the event $\{w\colon\, \lim_{k\rightarrow\infty} X_k(w)$ exists $\}$ is in the tail $\sigma$-algebra of the sequence $X_1, X_2,\ldots$? Intuitively is quite obvious. The ...
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1answer
78 views

Independence of Filtration

Given a sequence of random variables $\{X_{t}\}_{t=1}^{T}$ , and a filtration $\mathcal{F}_{T}$, what does it mean that $X_{t+1}$ is independent of $\mathcal{F}_{t}$? Is it simply that $X_{t+1}$ is ...
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1answer
82 views

Finding expected value of mean of an estimator

We have a set of unidimensional data, $X_1, . . . , X_n$. : The data are drawn from a uniform distribution on the interval $[a, b]$. This model has two positive real parameters, a and b, such that $0 ...
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0answers
37 views

Question on independent $\sigma$-fields

Let $(\Omega,\mathcal{A},P)$ be a probability space and $\mathcal{F}$, $\mathcal{G}\subset\mathcal{A}$ be two sub collections of sets which are closed under finite intersection. Furthermore assume ...
3
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1answer
80 views

$X_n \to X$ a.s. implies…

Let $X_n$ be independent. $ X_n \to X \, a.s.$ implies $ \sum_n P( |X_n-X|>\epsilon) < \infty$ I tried to prove as follows, which is wrong. Note that $ X_n \to X \, a.s.$ is ...
1
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1answer
87 views

Finding the pdf of an estimator

We have a set of unidimensional data, $X_1, \ldots , X_n$ drawn from the positive reals. We define a model for its distribution: The data are drawn from a uniform distribution on the interval $[0, ...
2
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1answer
48 views

Scaled integrated Brownian motion has limit

Let $B$ be a standard Brownian motion and put $$X(t)=\frac{1}{\sqrt{t}}\int_{0}^{t}f(B(s))ds,$$ where $f \in L_1(\mathbb{R}^{1})$ and $\int f(x)dx=1$. Show that $$ \lim_{t \rightarrow \infty} EX(t) ...