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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Relation between expectation and sample points

Suppose that the expectation of a random variable $X$ is $5$. Which of the following statements is true? There is a sample point at which $X$ has the value $5$. There is a sample point at which $X$ ...
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39 views

Calculate the result of combining three multivariate Gaussian distributions

A Bayesian derivation of the Kalman Filter was provided by Ho and Lee (1964); this paper is available as a free pdf here. As part of their derivation, they substituted three multivariate Gaussian ...
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11 views

Methodology for probability dist of function or random variables

Is there a methodology that allows us to derive a distribution of functions of random variables? How would someone approach this problem? What are the key ingredients? For example in many electrical ...
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1answer
22 views

Probability Of 4 Decks of cards

I have a question about probability and I would just like to make sure im correct. This is the question: 4 standard decks, if we randomly select 100 cards without replacement find the probability of ...
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2answers
41 views

Probability density function of the negative of a random variable (Exercise 4.1.2 from Grimmett and Stirzaker)

Find the density function of $Y = a X$, where $a > 0$, in terms of the density function of $X$. Show that the continuous random variables $X$ and $-X$ have the same distribution function if and ...
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0answers
17 views

The definition of completion of one measure with respect to a family

The following is taken from the book of Revuz and Yor more or less verbatim. If $(E,\mathcal E)$ is a measure space carrying probability measure $\mu$, the completion $\mathcal E^\mu$ is the ...
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20 views

Probability that $t^2 + 2\sqrt{x}t + y = 0 $

X and Y are independent X~Geom(P) Y~Exp($\lambda $) Compute the probability that $t^2 + 2\sqrt{x}t + y = 0 $ Steps I've taken so far: Found where the determinant of the quadratic is $\geq 0$. ...
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1answer
38 views

Probability Space and Random Variable [closed]

Let $\Omega = [0, \infty)$ and define $X$ on $\Omega$ by $X(w) = \frac{1}{1+w^2} $. For each of the following intervals, $I$, find the event $(x \in I) = \{w: X(w) \in I\}$. For $I = \left[{1\over ...
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2answers
25 views

How to solve this using probability theory?

In 7 story building 3 persons got on an empty elevator on the first floor. Each of them can get out at any floor independent of each others starting from the 2nd floor. What is the probability that ...
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0answers
22 views

Compose pdf with the random variable itself and conditional probability.

When I discussed with my classmates, we was wondering whether the following statement is correct. Given $(X_k)_k$ be a discrete real-valued stochastic process and assume $p_{X_k}$ be the strictly ...
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1answer
30 views

Simplify $E(\max(X_1+Y_1, X_2+Y_2))$ when $X_1, Y_1, X_2$, and $Y_2$ are exponentially distributed

The time until A arrives is exponentially distributed with rate $\lambda_1$, and the time until B arrives is exponentially distributed with rate $\lambda_2$. Once they arrive, they will spend ...
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59 views

Pos properties.

Given $n\in\mathbb{N}$, and $f:\mathbb{N}^*\rightarrow \mathbb{N}$, let define $Pos$ as: $$Pos(f)(n)= |\{x \leq n, f(x)=f(n)\}|$$ When given $n\in\mathbb{N}$, this function gives the 'position' of ...
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1answer
22 views

Expectation of random variable and measure transform theorem

Let $(\Omega,B,\mu)$ be a probability space where $\Omega$ is $[0,1]$, $\mu$ the Lebesgue measure, $B$ the Borel $\sigma$-algebra of $[0,1]$ and $f(w)=1-w$ be a random variable. Let $\phi:\mathbb ...
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1answer
52 views

Ergodicity of stochastic process

If one can show that the process converges to a stationary process in probability, does it mean that the process is ergodic?
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2answers
59 views

How do you define a sample space with rigor?

I was reading First Course on Probability by Sheldon Ross and I came across a problem which went like this: "A customer visiting the suit department of a certain store will purchase a suit with ...
2
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1answer
31 views

Marginal independence v.s. joint independence

Suppose that $X$ is independent with $Y$ and is also independent with $Z$. No further assumption is made about the joint distribution of $Y$ and $Z$. Does it follow that $X$ is independent with ...
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4answers
76 views

Are $XZ$ and $YZ$ independent if $X$, $Y$ and $Z$ are all independent?

Let $X$ and $Y$ be independent random variables. Let $Z$ be a random variable such that $Z$ and $X$ are independent, $Z$ and $Y$ are independent. Are random variables $XZ$ and $YZ$ independent?
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29 views

Expectation of random variable $f(w) = w$.

Expectation of random variable $f(w) = 1-w$. Is this right? $Fx(x) = \mu_f$ $ \left\{\begin{matrix} \mu(\emptyset) = 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ for \ x < 0 \\ ...
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1answer
40 views

Does this joint weak convergence result hold?

Let $(X_n)_{n\geq1}$ be a sequence of random variables weakly converging to $X$ ($X_n\Rightarrow X$) as $n\rightarrow \infty$. I am trying to determine the most general conditions on a function $f$ ...
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1answer
22 views

Finding density function of a conditioned function

Let $(X, Y)$ be a point taken uniformly at random on the unit square $[−1, 1]^2$. Let X|E be X conditioned on the event $E$ = ${\{X + Y \ge 0\}}$. Find the p.d.f. of X|E. Now, the ...
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0answers
20 views

Almost sure limit of a sequence

I have a random variable $X$, which is uniformly distributed on $[0,1]$. The underlying probability space is as usual $(\Omega,\mathcal{F},P)$. For every $t$ with $0\leq t < 1$, I define a random ...
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1answer
20 views

Literature on transformed Gaussian matrices

I am considering real $n$-by-$m$ matrices of the following type: $$ M=SM^\prime,\\ M^\prime_{ij} \overset{\text{iid}} \sim N(0,1). $$ Here, $S$ is a fixed $n$-by-$n$ matrix and the entries of ...
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1answer
24 views

Law of large numbers for a sequence of independent random variables with different means

I have an independent sequence of random variables $(X_i)_{i\geq 1}$ such that $E[X_i] = \mu_i > 0$ and $E[X_i^2] < \infty$. I know that $\frac{1}{n}\sum_{i=1}^n\mu_i \to \mu < \infty$ as ...
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0answers
7 views

Can the entropy of a system of multiple stochastic processes be defined?

I'm trying to measure the entropy of BINGO cards. (waiting for laughter to die down.) Normally one would think of this as 75 choose 24, but in reality it is 15 choose 5, four times, and 15 choose 4, ...
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0answers
41 views

Sample space of infinite coin tosses experiment

In one of the online statistics/probability web sites http://www.statlect.com/asymptotic-theory/mean-square-convergencee, the following definition is given for the mean - square convergence of the ...
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0answers
22 views

Second moment method

Suppose for a sequence of random variables $(X_n)_n$we know $E(X_n)\rightarrow c$, $c$ a constant, and $\sum_{n=1}^\infty \text{Var}(X_n)<\infty$. Then $X_n\rightarrow c$ a.s is the claim. By ...
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1answer
23 views

(Uniform) continuity of push-forward operator

I am wondering about extension of the the answer given here. Namely, suppose $U$, $V$ are Polish spaces and $F:U→V$ is uniformly continuous. Does this mean that the push-forward operator $F_*: ...
2
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2answers
81 views

If X is independent to Y and Z, does it imply that X is independent to YZ ?

After years of mathematics, I am struggling with this simple question. If we have 3 r.v. $X,Y,Z$ and we have $X$ independent to $Y$ and to $Z$, then do we have that $X$ is also independent to $YZ$ ? ...
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1answer
40 views

What is the expected value of the number of random letters needed to produce my string.

Consider the string $p$ with length = $n$. Suppose we have the alphabet $\sum = (a_{1}..a_{k})$ and empty string $s$. All symbols have equal probability of choosing ($p(a_{1})=...=p(a_{k})$). We ...
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26 views

Elements of a $\sigma$-algebra

I have a probability space $(\Omega,\mathcal{F},P)$ on which a uniformly distributed random variable $X$ lives. That is, $P\{0\leq X \leq a\} = a \wedge 1$ for $ a\geq 0$. Then, I fix a $t > 0$ ...
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1answer
25 views

Example for a non-ergodic stationary process

Let $(X_n)_{n \in \mathbb{N}}$ be a (strictly) stationary process and let $T$ denote the left-shift on $\mathbb{R}^\mathbb{N}$, i.e. $T((x_n)_{n \in \mathbb{N}}) = (x_{n + 1})_{n \in \mathbb{N}}$. ...
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23 views

Poisson process deviations

How can one prove the following inequalities for a standard Poisson process $\mathbf{N}(t)$ ? $\mathbb{P}\bigg[\bigg|\frac{\mathbf{N}(\lambda)}{\lambda}-1 \bigg| > \varepsilon\bigg] \leq ...
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1answer
23 views

Expanding $E[N^2]-E[N]$

I'm trying to prove $$\sum_{i=0}^{\infty}iP[N>i]=\frac{1}{2}(E[N^2]-E[N])$$ by expanding both the RHS and LHS and showing that they are equal. The first thing I did was multiply both sides by $2$ ...
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44 views

Recommendations: Any Good books to study Path-Integration from 0 again?

I was researching and talking with some friends about I want to start from zero studying path integral, this question, and they recommended I start by studying "Quantum Mechanics and Path Integrals". ...
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2answers
37 views

What is the domain and range of the sum of two random variables?

Let $(\Omega, \mathcal{F}, P)$ be a probability space s.t. $\Omega = \{0,1\}$. Let $X_1: \Omega \rightarrow \{0,1\}$, $X_2: \Omega \rightarrow \{0,1\}$ be two random variables over $\Omega$ (i.e., ...
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1answer
33 views

Probability question using no-memory property of exponential distribution

A customer must be served first by server 1, then by server 2, and finally by server 3. The amount of time required for service by server i is an exponential random variable with rate µi , for ...
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1answer
20 views

Understanding the proof of Johnson-Lindenstrauss (JL) lemma

I have some questions about proof of Johnson-Lindenstrauss (JL) lemma. I appreciate any responses in advance. It is stated in the following paper: An elementary proof of JL lemma we argued: "Hence ...
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1answer
55 views

Attempting to show $P(|S_n| <1)$ for a martingale $(S_n)$

Now, I am stuck on the last part of the question. I managed to find the solutions, but I don't udnerstand them completely. What I don't understand is: How they got that indicator function, and why ...
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1answer
19 views

If $(Y_n)$ is an adapted sequence of random variables, with $EY_T = 0$, show $EY_n = 0$ for all $n$

Let $(Y_n)$ be an adapted sequence of integrable ranodm variables, with the property that for every boudned stopping time $T$ we have $EY_T = 0$. I am asked to show that $EY_n = 0$ for every natural ...
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1answer
57 views

Can I assume that random variables with exponential distribution are positive?

Let $(Y_n)$ be i.i.d random variables following exponential distribution with parameter $1$. Let $X_n=\min(Y_1,\dotsc, Y_n)$ Prove that $ X_n \xrightarrow{P} 0$ It's easy to prove that ...
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1answer
49 views

Determine $E\sum_0^\infty X_n1_{(T=n)}$

$X_T = \sum_0^\infty X_n 1_{(T=n)}$ where $T$ is a stopping time and $(X_n)$ is a martingale. Show that if $T$ is bounded then $EX_T = EX_0$: $T \leq N$, and then consider $X_T = X_{T\wedge N} = ...
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2answers
53 views

Consider all possible orderings of the numbers from $1$ to $10$.

Consider all possible orderings of the numbers from $1$ to $10$. For such an ordering, a number is lucky if it appears in the same position as in the usual order. Assuming all orderings have the same ...
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15 views

Theorem of Daniell-Stone (uniqueness when assuming compactness)

Theorem of Daniell-Stone. Let $L$ be a $\sigma$-continuous abstract integral on a Stone lattice V of real-valued functions on $\Omega$ and let $\mathcal{A}(V)$ denote the set of all $V$-open ...
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1answer
34 views

Understanding Bayes' theorem with uniform distribution

Station $X$ begin to transmit a message in $[0,20]$ with uniform distribution, and $Y$ also want to transmit a message in $[6,14]$ with uniform distribution. Assume that transmission takes $2$ ...
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1answer
23 views

Prove that $P(E \cap F^c) = P(E) - P(E \cap F)$

I'm trying to prove the equality $P(E \cap F^c) = P(E) - P(E \cap F)$, but I feel as if I'm going in circles after I wrote one of the probability axioms $P(E \cap F^c) = P(E) + P(F^c)-P(E \cup F^c)$. ...
2
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1answer
30 views

Almost sure and $\mathcal{L}^1$ convergence of $Y_n=(\cos a)^{-n}\cos(a(Z_1+\cdots+Z_n))$ with $(Z_n)$ i.i.d. Bernoulli

Let $(X_i)$ be i.i.d. with $P(X_i= a)=P(X_i = -a)=\frac{1}{2}$, for some $a$ such that $2a \notin\pi\mathbb Z$. Let $$Y_n=\frac1{\cos^n(a)}\cos\left(\sum_{i=1}^n X_i\right).$$ Check whether $(Y_n)$ ...
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1answer
34 views

Poisson: $P(N(4)-N(2)=5|N(4)=8)$

Let {N(t), t ≥ 0} be a Poisson process of rate 2. Determine: a) $P(N(4)-N(2)=5|N(4)=8)$ Attempt: The conditional poisson distribution is uniformly distributed between the interval [0,4]. Therefore, ...
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0answers
41 views

This is a Markov Chain?

Consider two irreducible ergodic Markov chains with the same state space $\{0, 1, . . . , N\}$, with transition matrices $P$ and $Q$ and respective stationary distributions $\pi$ and $\rho$. We ...
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4answers
56 views

Expected number of rounds

Assume that we have a bucket containing $n$ different balls. At each round we choose at random a subset of the remaining balls, and remove them from the bucket. We keep doing this until the bucket ...
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55 views

If $(X, Y)$ and $(X', Y')$ have the same distribution, then so do $\mathbb{E}(Y|\sigma(X))$ and $\mathbb{E}(Y'|\sigma(X'))$

I am having trouble figuring out the proof for the following question, maybe it's because I am not familiar with the regular conditional distributions. Let $X, X', Y,Y'$ be real random variables ...