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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Generating stochastic processes from distributions of random variables

A stochastic process is a sequence of random variables $\{Y_t : t=0,\pm 1,\pm 2,\pm 3,\pm 4,\dots \}$. How is this determined by the set of distributions of all finite collections of $Y$'s? I do not ...
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190 views

Joint distribution of an infinite collection of random variables?

Let's say we have a countable collection of random variables $X_1, X_2, ...$, in $(\Omega, \mathscr{F}, \mathbb{P})$ Can we define a joint distribution function for all of them ie $$F_{X_1,X_2, ...
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43 views

Probability distribution of the distance between two random, uniformly distributed points in the unit ball

Take two vectors, $p1$ and $p2$, that are uniformly distributed within the unit ball. Let $d = \|p1-p2\|$. What probability distribution do we have for $\frac{1}{d^2}$? Let us consider the ...
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65 views

Characterization of renewal processes, and examples with exactly one of stationary or independent increments

A continuous-time process $\{N(t) : t\in[0,\infty)\}$ is a counting process if it satisfies $N(t)\geqslant0$ a.s. (nonnegative) $\mathbb P(N(t)\in\mathbb N\cup\{0\}) = 1$ (integer-valued) If ...
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18 views

Uniform randomly select elements to two sets and compare them after $2$ selections

I have a matrix $A_{5 \times 5}$. Let $a_{3,3}$ is center element of the matrix. Uniform randomly select other elements (expect $a_{3,3}$, each element only choose $1$time), into two set $X,Y$ and ...
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2answers
51 views

Compute $\mathbf E[B_s^4B_t^2-2B_sB_t^5+B^6_s]$

Let $\mathbf B=\{B_t\}_{t\ge0}$ a continuous Brownian motion, what is then $\large\mathbf E[B_s^4B_t^2-2B_sB_t^5+B^6_s]$, for $t\ge s$ ? How can I factorize the expression in the parenthesis, If ...
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1answer
42 views

Rotationally invariant random variable implies uniformly distributed?

Let $X = (X_{1}, \ldots, X_{n})$ be a random vector such that $\|X\|_{2} = 1$ almost surely and $UX$ is equal in distribution to $X$ for all orthogonal matrices $U$ (so $X$ is rotationally invariant). ...
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27 views

Approximating a probability distribution by the moments

In a version of Levy's theorem we know that if we have a sequence of characteristic functions $\phi_n$, such that $\phi_n\rightarrow \phi$ pointwise, then $\phi$ is the characteristic function of a ...
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1answer
116 views

When Superposition of Two Renewal Processes is another Renewal Process?

When superposition of two renewal processes is another renewal process? If you merge (superpose) two Poisson processes with parameters $\lambda_1$ and $\lambda_2$, the outcome is another Poisson ...
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248 views

Expected Value of Square Root of Poisson Random Variable

Find the expected value of $\sqrt{K}$ where $K$ is a random variable according to Poisson distribution with parameter $\lambda$. I don't know how to calculate the following sum: $E[\sqrt{K}]= ...
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60 views

A stationary distribution of Markov chain

For a irreducible finite Markov chain, I know that the definition of a stationary distribution is as follows: $\pi P = \pi$, where $P$ is a transition matrix and $\pi$ is a stationary distribution ...
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1answer
32 views

Lemma about probability space P (From Grimmett and Stirzaker)

This is from Grimmett and Stirzaker, Chapter 1, page 7. Lemma. Let $A_{1},A_{2},...$ be an increasing sequence of events, so that $A_{1}\subseteq{A_{2}}\subseteq{A_{3}}\subseteq{...}$, and write A ...
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43 views

Correlation Coefficient as Cosine

I've read that the correlation coefficient between two random variables may be viewed as the cosine as the angle between them, but I can't find any solid explanation. To be concrete, let $X$ and $Y$ ...
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1answer
370 views

Sheldon Ross vs My TA, what answer is wrong?

I have the solution of this problem, 1) The game of Clue involves 6 suspects, 6 weapons, and 9 rooms. One of each is chosen randomly and the object of the game is to guess the chosen three. In one ...
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2answers
44 views

Suppose $\{X_n\}$ is uncorrelated sequence,meaning $Cov(X_i,X_j)=0, i\not= j$

Suppose $\{X_n\}$ is uncorrelated sequence,meaning $$Cov(X_i,X_j)=0, i\not= j$$ If there exists a constat $c>0$ such that $Var(X_n)\leq c$ for all $n\geq 1$, then for any $\alpha > \frac{1}{2}$ ...
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78 views

Permutations of passwords without a specific substring

I've been a little bit stuck on a problem for my discrete math class. You need to set a password as a string which is a permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Your birth day is ...
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38 views

Calculate sample size given Conditional Probability and level of significance

On the basis of a pilot study, it was found that the predation probability for dark-coloured moths on a dark background is 0.10, in contrast to 0.90 on a light background. What should be the sample ...
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2answers
27 views

Calculate marginal densities

Let $f(u,v)=\mathbb{1}_{\{0\le v \le 2u\}} \cdot \mathbb{1}_{\{0\le u \le 1\}}$ How can I calculate marginal densities? I know $f(u)=\int_{-\infty}^{\infty} f(u,v)dv$ and ...
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1answer
35 views

Why is $\mathcal{E}(X)=\mathcal{E}(X,X^*)$?

According to a course about $\sigma$-agebras in infinite dimensional space they said that it is easy to see that : $$\mathcal{E}(X)=\mathcal{E}(X,X^*)$$ where: $X$ is separable real Banach space. ...
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35 views

Example of a $\sigma$-additive function over a $\sigma$-algebra

While working in measure theory I've decided to study each theorem using a concrete example of concepts I deal with. In this case, I want proof that: If ϕ is σ-aditive function on a σ-field, ...
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1answer
64 views

Counterexample to infinite sum of expectations = sum of infinite expectations?

I am looking for the counterexample to the statement $\sum E[X_n] = E[\sum X_n]$ given that $\sum E[X_n]$ and $\sum X_n$ exist. This would be also a counterexample to the dominated convergence theorem ...
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1answer
70 views

Laplace functional of sum of independent uniformly distributed random variables

I'm doing some of the exercises in Cinlar's "Probability and Stochastics" to better understand the material. This exercise (VI.1.17) is taken from page 247: Fix an integer $n \geq 1$. Let ...
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64 views

Suppose $X$ and $Y$ have joint density $f(c,y)=6xy^2$ for $x,y\in(0,1)$. What is $P(X+Y<1)$?

My attempt: $$f(x,y)=6xy^2$$ $$P(X+Y<1)=\int_{0}^1\int_{0}^{1-y} 6xy^2dxdy$$ $$=\frac{1}{10}$$ However, the answer was supposed to be $\frac{3}{5}$, and the bounds on the integral were supposed ...
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1answer
50 views

Calculate the variance of the joint probability. {Actuarial Problem, Exam P. Help!}

I'm studing for exam P, and I found the following problem: I don't know how to solve it, and I will really apreciate if someone know how to do it, and learn from it. Thanks comunity again.
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1answer
60 views

How to Compute Var($4X-Y$) on this case?

Let $X$ be the number of $1$'s and $Y$ be the number of $2$'s that occur in $n$ rolls of a fair die. $(a)$ Compute Cov($X; Y$ ). Solution are given in other post: Compute Cov(X,Y) while X is ...
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15 views

Suppose that $X_n$ are integer valued, and $X_n$ converges in distribution to $X$.

How can we show that $m+\frac{1}{2}$ is a continuity point of the distribution function of $X$, if $m$ is an integer? Do I need to show that $X$ is also integer valued?
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1answer
29 views

Extrema of a $\sigma$-additive function are attained.

I've written down a proof for the following theorem which appears in Loeve's book of probability (page 86). I believe I'm missing something because Loeve's proof is much more complicated but I can't ...
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2answers
55 views

$P(X\in B)=1$ when $X_n$ converges in distribution to $X$ for open and closed $B$

Suppose that $X_n \rightarrow X$ in distribution. If $B$ is a closed Borel subset of $\mathbb{R}$, and $P(X_n \in B) = 1$ for all $n$, then (a) Prove that $P(X \in B) = 1$. (b) Show that, however, ...
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12 views

Auto-correlation of random process (integration)

Consider the following problem: Suppose we know that $f_X(x)=1$, now I want to calculate the autocorrelation: $E[\underbrace{X(t)X(t+\tau)}_{g(t)}]=\int_0^1 g(t)f_X(x)dx=\int_0^1 ...
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46 views

Uniqueness of Limit in Convergence in Distribution

If $X_n \rightarrow X$ in distribution and $X_n \rightarrow Y$ in distribution , show that X and Y have the same distribution. My Approach: Suppose the distribution of X and Y are not same, i.e., ...
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18 views

finding the likelihood function for a sample

I need some help with finding the likelihood (or log-likelihood) function for this problem: A sample $x_1,...x_n$is generated as follows: for $i=1...n$ 5-bit binary string $w \in \{s_0,...s_{31}\}$ is ...
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0answers
70 views

Sum of two families of uniform integrable random variables

Suppose $\{X_n\}$ and $\{Y_n\}$ are two families of u.i(uniform integrable) random variables defined on the same probability space. Is $\{X_n+Y_n\}$ u.i? Proof Given $$\mathbb{E}[|X_n|\,I_{|X_n|\geq ...
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1answer
47 views

Independence and change of measure

Let $(\Omega, \mathscr F)$ be a finite probability space and $\mu$ be a probability measure on $\Omega$. Consider a sequence of random variable $\xi_n$ that are independent and identically ...
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50 views

I need a help to understand a paper about Balls and Bins and bins max load

I must declare that I have asked a similar question before but I did not get any answer. I do need a concrete example and formula allowing me to determine maximum number of balls in a bin with ...
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1answer
27 views

Fisher information for a single sampling of an exponential distribution

I am viewing an example of finding the Fisher information for a single sampling from an exponential distribution where: $$P(x|\theta) = \frac{1}{\theta}e^{-\frac{x}{\theta}}$$ The score $S$ is ...
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1answer
34 views

Equivalent measures and independence

Suppose that $\mu$ and $\lambda$ are two equivalent measures in a probability space $(\Omega, \mathscr F)$. Suppose that random variables $\xi_1,..,\xi_n$ in $(\Omega, \mathscr F)$ are independent ...
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1answer
51 views

Prove that $ n \sum\limits_{k=n}^\infty \frac{1}{k^2 \log k} \to0$

I am wondering if $\displaystyle \lim_{n \to \infty} n \sum_{k=n}^\infty \frac{1}{k^2 \log k} = 0$, and how I would go about proving the result if the limit is correct. I thought of using an integral ...
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1answer
93 views

Random vector $(X,Y)$ is uniformly distributed on the disk. Find the joint distribution of $R=\sqrt{X^2+Y^2}$ and $\theta =\arctan (Y/X)$

Random vector $(X,Y)$ is uniformly distributed on the disk $D_r$ defined by $$D_r=\{(x,y)\in \mathbb R^2\mid x^2+y^2\leq r\}.$$ Find the joint distribution of $R=\sqrt{X^2+Y^2}$ and ...
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48 views

Understanding the Normal Distribution?

If a sample is normal with observations independent and identically distributed: $\mu|\sigma^2 \propto N(\beta \,,\,\sigma^2/\, n_0)$ How can I show that $\mu\,|\,x_1,x_2,....x_n\,,\,\sigma^2 \sim ...
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2answers
80 views

Prove $E((X+Y)^p)\leq 2^p (E(X^p)+E(Y^p))$ for nonnegative random variables $X,Y$ and $p\ge0$

Suppose $X \geq 0$ and $Y \geq 0$ are random variables and that $p\geq 0$ Prove $$E((X+Y)^p)\leq 2^p (E(X^p)+E(Y^p))$$ Proof Since $(X+Y)^p \leq (2 \> \max\{X,Y\})^p=2^p \> \max ...
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2answers
58 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 ...
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1answer
84 views

Calculate the density function of $Y=\frac{1}{X}-X$ where $X\sim U[0,1]$

I know that : $$f_X(x)=\cases{1 & $x\in [0,1]$\\0 & $x\notin[0,1]$}$$ Then: $$P(Y\leq y)=P(\frac{1}{X}-X\leq y)=P(X\leq\frac{1}{2}(\sqrt{y^2+4}-y))$$ as $$\frac{1}{x}-x=y\rightarrow ...
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32 views

rate of convergence central limit theorem

Let $X_{1},X_{2,\ldots }$ be i.i.d. with finite second moment random variables. With mean $\mu$ and $\sigma^{2}$ Then the classical CLT states: $\sqrt{n}(\overline{X_{n}}-\mu)$ convergences to ...
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43 views

Central Limit Theorem: Lindeberg condition application

Let $0 < a_1 < a_2 < · · ·$ be fixed real numbers and let $\{X_k\}_{k \geq 1}$ be a sequence of i.i.d. random variables with zero mean and unit variance. Let $T_n = \sum_{i=1}^{n} a_iX_i$. ...
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1answer
69 views

Expectation of supremum of a submartingale

I have a probability space $(\mathbb{P}, \Omega, \mathcal{F})$ and in this space, I have a submartingale $(X_n)_n$ with the following two properties: $\inf_n X_n < 0$ $\mathbb{E}[X_0] \geq 0$ ...
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25 views

Lebesgue measure vs Lebesgue-Stieltjes measure

Reading advanced probability theory book I've come across Lebesgue-Stieltjes measure. Could someone explain what is the difference between it and "standard" Lebesgue measure on $\mathbb{R}$? Thank ...
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1answer
46 views

$(g(X_t))_{t \geq 0}$ is continuous in probability if $g$ is uniformly continuous and $(X_t)_{t \geq 0}$ continuous in probability

How to prove this fact? $(X_{t})_{t \ge 0}$ is continuous stochastic process(It means that $\lim_{s \to t}P(\left|X_{t}-X_{s} \right|>\epsilon)=0)$) and function $g:\mathbb{R} \rightarrow ...
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1answer
55 views

Probability of winning the election

Suppose you do a poll to one hundred people. 52% of the people say that will vote for candidate A and 48% of the people say that will vote for candidate B. What is the probability that candidate B ...
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1answer
65 views

Limit of random variables indipendent from a fixed sigma algebra.

Let $(X_n)_{ n \in \mathbb{N}}$ be a sequence of random variables that converges (in probability) to a random variable $X$. Let's suppose that each $X_n$ is independent from a fixed sigma-algebra $F$, ...
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17 views

Independent increment property

$X=(X_k, k \geq 0)$ is defined recursively by $X_0=1$ and $X_{k+1}=(1+X_k)U_k$ where $U_0,U_1,\ldots$ are independent random variables each uniformly distriubted on $[0,1]$. Determine if the process ...