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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87 views

Calculate expectation and variance

Let $(X_n)$ be a sequence of independent RVs which are uniformly distributed on $[0,1]$ interval. For $0<x\le 1$ we define $$N(x):=\inf\{n:X_1+\dots+X_n\ge x\}.$$ Show that $$\mathbb{P}(N(x)\ge ...
4
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2answers
75 views

Typo on Wikipedia's entry on Hoeffding's inequality?

In Wikipedia's entry on Hoeffding's inequality, they state that if $\overline{X} := \frac 1 n \sum_{i=1}^n X_i$, then $$ P(\overline{X}-E[\overline{X}] \ge t) \le \exp (-2n^2 t^2)$$ if we assume for ...
3
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1answer
69 views

Remembering the Portmanteau Theorem

I'm looking for a good way to remember/understand part of the Portmanteau theorem. Specifically, let $X$ be a metric space. The part of the Portmanteau theorem I'm asking about says that for a ...
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1answer
69 views

Fourier transform of Lévy process

I have the following definition of a Lévy process: An adapted process $X=(X_t)$ with $X_0=0$ a.s. is a Lévy process if it has independent increments of the past, i.e. $X_t-X_s$ is ...
2
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1answer
134 views

central limit theorem for high dimensional random walk

Consider random walk in $\mathbb{Z}^d$, $d>1$, with $x(t) = x(t-1) + \xi$, where $\xi$ has some probability distribution in $\mathbb{Z}^d$ with finite support, expectation $m = \sum_{v \in ...
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1answer
21 views

The probability with equations

Of the three independent events $E_1,E_2,E_3$,the probability that only $E_1$ occurs is $\alpha$,only $E_2$ is $\beta$ and only $E_3$ is $\gamma$. Let the probability $p$ that none of the events ...
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10answers
1k views

Is conditional probability also probability?

Some pondering leads me to the question below, which prevents me from the reckless calculation of conditional probability... As defined, conditional probability is: $$ P(A|B)=P(A\cap B)/P(B). $$ ...
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1answer
63 views

Adaptedness of random variables

Suppose we have an RCLL adapted process $(X_t)$. Moreover we are given a stopping time $T$. We define $\mathcal{F}_T=\{A\in\mathcal{F}\mid A\cap\{T\le t\}\in \mathcal{F}_t, \text{ for all }t\ge0\}$. ...
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1answer
36 views

Solving Addition Rule other way

I know the General Addition Rule for Two Events is $$P (A\cup B) = P (A) + P(B) - P(A\cap B)$$ I know the proof where ${R_1} = A\cap B, R_2 = A\cap B', R_3 = A'\cap B,$ and $R_4 = A'\cap B'.$ ...
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1answer
29 views

If $A$ and $\bar{X}$ are independent, then so are $S^2$ and $\bar{X}$ independent

I have a question about a question in the following link: Proof of the independence of the sample mean and sample variance Here $( X_{2}-\bar{X},X_{3}-\bar{X},\cdots,X_{n}-\bar{X}) =A$, $S^2$ the ...
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3answers
129 views

A Puzzle based on probability

(Based on true story) A friend of mine and myself were drinking and we wanted to decide who will pay for the next round of drink. We decided to toss a coin so as to ensure a fair chance (1/2 prob.) ...
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1answer
143 views

Find m.g.f. given $E(X^r)$ function?

"Let $X$ be a random variable with $E(X^r) = 1 / (1 + r)$, where $r = 1, 2, 3,\ldots,n$. Find the series representation for the m.g.f. of $X$, $M(t)$. Sum this series. Identify (name) the ...
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2answers
81 views

Probability $1$ item is defective out of a sample of $10$?

Question A lifetime of a brand of LEDS has an exponential density with mean 10000 hours. An LED is considered defective if its lifetime is less than 5000 hours. Assuming that the lifetimes of LEDs ...
2
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1answer
4k views

Bonferroni inequality proof

Is this proof for $P(\bigcup_{i=1}^n A_i)\le\sum_{i=1}^nP(A_i)$ correct? Pf. By induction. For $n=2$, $$P(A\cup B)=P(A)+P(B)-P(A\cap B)\le P(A)+P(B)$$ Assume that the statement is true for $n-1$, ...
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1answer
329 views

probability of maximum of two independent random variable

Suppose $X$ and $Y$ are two independant random variable with exponential distribution with paramet $\lambda=1$ and $M=$max{$X$,$Y$}. Then $P(M \ge 4)$ is equal to : Answer: 0.036 how do i come to ...
2
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2answers
124 views

If $P(X \geq a) = 1 - \frac{1}{4}a^2$, $0 \leq a \leq 2$, then what is the expectation of $X$?

Suppose for a random variable $X$ it is given $P(X \ge a)=1-\frac{1}{4}a^2$, for $0\le a\le 2$. what is the expectation of $X$? Correct answer: $\frac{4}{3}$ I have difficulty solving the problem ...
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0answers
21 views

Continuity of covariance kernels

Let $I$ be a locally compact Hausdorff (LCH) topological space. Let $c : I \times I \to \mathbb R$ be a covariance kernel, that is, a symmetric, nonnegative-definite function. Does it follow that $c$ ...
5
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1answer
206 views

Is there a characterization of the shift-invariant ergodic measures?

Consider probability measures $\mu$ on the space $\{0,1\}^\mathbb{N}$ that are shift-invariant with respect to the left-shift map. Is there a nice characterization of the ergodic shift-invariant ...
2
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2answers
297 views

What does $E[XY]$ mean?

Let's say I have two random variables, $X$ and $Y$. $X$ is the value of a fair die, $Y$ is the result of a coin flip, with heads being 1 and tails being 0. $E[X] = \sum_{k=1}^{6}{\frac{k}{6}} = ...
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1answer
142 views

Shop customers Poisson process

People arrive at some shop as Poisson process of rate $N$. There are $N$ corridors in the shop and each customer chooses one at random, independently of the other customers. Now let $X_t^N$ be the ...
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1answer
484 views

Poisson process arrivals of buses

I would like to check if my calculations are correct: We have two buses, arrivals of bus (A) form a Poisson process of rate $1$ bus per hour and arrivals of bus (B) form an indep. Poisson process of ...
3
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1answer
37 views

How to show that $n^{-r} \sum_{j=1}^n (X_j - \mu) \rightarrow 0$ in probability

I need your help to prove the following statement. Let $X_1, \cdots X_n$ be stochasticaly independent, identically distributed random variables. Assume they have a finite expected value $\mu$ and a ...
2
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1answer
456 views

Expected value of sample median given the sample mean.

Let $Y$ denote the median and let $\bar{X}$ denote the mean of a random sample of size $n=2k+1$ from a distribution that is $N(\mu,\sigma^2)$. How can I compute $E(Y|\bar{X}=\bar{x})$? Intuitively, ...
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1answer
29 views

Resulting pdf of multiple trials

If I have a pdf, $f_0(x)$, from which I sample $n$ times, what is the resulting pdf, $f_n(x)$, which tells me the relative likelihood to take a given value at least once? For example, if there's ...
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0answers
39 views

Is the expected value $\mu$ in the WLLN a random variable?

Am I right in thinking that the weak law of large numbers, when stating that $$\bar{X_n} \to \mu$$ in probability convergence is stating that the sequence of random variables $\{X_n\}$ tends to the ...
2
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3answers
64 views

Probability computation, tossing two dice

I have some ideas on how to solve the problem, but simulations do not support my analytical results :) Toss two dice and sum their value and write it down: Denote by $X_n$ the result at $n$-th toss. ...
2
votes
1answer
74 views

Why the two probability results are the same?

Suppose there are 3 red balls and 2 white balls in a bag. We want to pick out 2 balls without replacement. What's the probability of the 1st and 2nd balls are both red? Solution 1: Use the ...
5
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2answers
175 views

What is probability? [closed]

I tried to understand the most fundamental foundation of the mathematical definition of probability in the most natural/human way. (At first, I thought I may have found a proper understanding like ...
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2answers
83 views

Random interarrival times (poisson process)

In their monograph "Queues", Cox and Smith state (paraphrased - this is p5): In interval $(t, \Delta t)$ the probability of no arrivals in a completely random process is $1 - \alpha \Delta t + ...
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1answer
10k views

Gamma Distribution out of sum of exponential random variables

I have a sequence $T_1,T_2,\ldots$ of independent exponential random variables with paramter $\lambda$. I take the sum $S=\sum_{i=1}^n T_i$ and now I would like to calculate the probability density ...
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1answer
46 views

$Q$-matrices and Markov Chain properties

I would like to start with the definition of a $Q$-matrix on a countable set $I$. A $Q$-matrix is a matrix $Q=(q_{ij}:i,j\in I)$ satisfying the following conditions: (i) $0\le-q_{ii}<\infty\forall ...
1
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2answers
73 views

CLT in case the sum of variances grows as log N

Consider the following (bounded) random variables $X_i$, which take outcome $1$ with probability $C/i$ and outcome $0$ with probability $1 - C/i$, where $C$ is a positive constant. Then one has, ...
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4answers
67 views

Reference request for stochastic process and applications

I am looking for a text book that will cover the following topics I hope someone could suggest me a good text book that will provide me a good guidance regarding the following; Generating functions, ...
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1answer
80 views

Expected value and variance of random variable

Suppose I pick an integer independently and uniformly at random from $[1,n]$ and I repeat this experiment $t$ times. Let $a_i$ be the number of times I pick number $i$, and let $a_{max}$ be the ...
3
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1answer
81 views

A problem on limit superior

Let $A_n$ be the square $[(x,y) : |x|\leq 1, |y|\leq 1]$ rotated through the angle $2\pi n\theta$. I need to find the geometric description of $lim sup_n A_n$ when $\theta$ irrational. I understand ...
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0answers
33 views

limit of limit superior w.r.t truncated set

Let $\Theta\subseteq\mathbb{R}^d$ is open set and $(\cal X, \cal A)$ be a measurable space . For every $\theta\in\Theta$, suppose that $P_\theta$ is a probability measure on $(\cal X, \cal A)$. ...
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1answer
561 views

First step analysis on random walk

Let us consider random walk on integers {0,1,...,N} where $P(N,N)=1$,$P(0,1)=1$, $P(N,N-1)=0$ and all other connections have probability $\frac{1}{2}$. Using first step analysis, compute $p_{00}$ for ...
2
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1answer
43 views

Martingale and mean squared error

In preparation for a course I am doing later in the semester I have been trying to brush up on my knowledge about martingales. But I am struggling with the following problem: Let ...
1
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1answer
53 views

Integrability of the supremum of a sum of Birkhoff averages

Let $(X, \mathcal{A},\mu)$ be a probability space and $T:X\rightarrow X$ an ergodic transformation. The Birkhoff averages of a function $\phi:X \rightarrow \mathbb{R}$ are defined by $$ ...
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2answers
75 views

iid random variables random walk

In preparation for a course I am doing later in the semester I have been trying to brush up on some probability theory. But I am struggling with the following problem: Let $\{Y_n,n\geq1\}$ be ...
0
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0answers
46 views

uncorrelated vs independent in exponential family

If $X$ and $Y$ are jointly Gaussian (the joint measure admits a Gaussian density), we know that $X$ and $Y$ are independent if and only if they are uncorrelated. If this true for all distributions in ...
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0answers
30 views

Charateristic function evaluation

I have a signal given by the following equation: $$y_k = X_k S_0 + \sum_{l=0 \& l\neq k}^{N-1}S_{l-k}+n_k$$ where $X_k$ are independent and identically distributed random variables. $n_k$ is a ...
2
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1answer
31 views

Variance of sum of square differences corrupted by noise

I've been trying to extend a proof I have for a univariate function $f(x)$ to a multivariate function $f(x,y)$ but the result I get looks weird. I'm wondering whether there is a mistake in any step ...
3
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1answer
79 views

The branching process $\mu^{-n}Z_n = \mu^{-n}\sum_{k=1}^{Z_n{-1}}X_{n,k}$ is a martingale

Let $\{X_{n,k} : n,k \geq 1\}$ be a collection of i.i.d. $\mathbb{Z}_+$-random variables with finite variance $\sigma^2 > 0$ and mean $\mu > 0$. Define $(Z_n)_{n\geq 0}$ recursively by ...
0
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1answer
56 views

Probability qual problem about Polya's criterion (I guess)

Suppose $\mu$ is non negative, $\sigma$-finite measure on $(0,\infty)$ so that $$c:=\int_0^\infty x\mu (dx)\in(0,\infty)$$ Let $$\phi(u):=\exp\left(\int(e^{iux}-1)\mu (dx)\right)$$ Prove that there ...
0
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2answers
58 views

Third Moment of a Sum of Normal and Gamma

I just ran into the next problem: The random variables $X$ and $Y$ are independent, where $X \sim Normal(1,1)$ and $Y \sim Gamma(\lambda,p)$ with $E(Y) = 1$ and $Var(Y) = 1/2$ How do we find ...
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1answer
51 views

simple question on POKER HAND probiblity {full house}

So I have a question about the probability of having a full house in a poker hands (5 cards), from a standard deck of cards (52). The solution is ...
3
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1answer
64 views

Independent distributions determined by projections

Let $X_1 \sim \mathbb P_1, \dots, X_n \sim \mathbb P_n$ be independent random variables. Let $b_1, \dots b_n$ be a basis of $\mathbb R^n$. Question: Once I know the distributions of the random ...
2
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2answers
62 views

LLN question: convergence of sequence average

This is a question relation to the law of large numbers. Let $(X_n:N\in\mathbb{N})$ is as sequence of independent random variables such that $\mathbb{E}(X_n) = \mu$ and $\mathbb{E}(X_n^4) \leq M$ for ...
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1answer
106 views

Relation between convergence in distribution and weak convergence

If $(X_n: n\in \mathbb{N}), X$ are a sequence of random variables in $\mathbb{R}$, I wish to show that $X_n \to X$ weakly if and only if $X_n \to X$ in distribution. By 'converging weakly' I mean that ...