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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An airline company annually gives $5000$ vouchers to its clients. In the previous year, $67 $% of the clients that received a voucher redeemed it.

(a) Let X be number of clients that will redeem the voucher this year. Suggest a distribution for X. State any assumptions that you need to make. (b) If the average cost of each voucher is $ $125$, ...
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6 views

Almost sure convergence of $\hat{\sigma^2}$

Let $Y \sim N(X\beta,\sigma^2I)$ where $Rank(X_{n\times p})=p \leq n$. The least square estimate of $\sigma^2$ is $\hat{\sigma^2}=\frac{Y'(I-P)Y}{n-p}$ where $P=X(X'X)^{-1}X'$ is the projection matrix ...
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1answer
12 views

Prove that $\sigma$-algebras $A_1,\ldots,A_n$ are independent if and only if $A_i$ is independent of each $A_1,\ldots,A_{i-1}$, for all $i=2,\ldots,n$

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space and $\mathcal{A}_1,\ldots,\mathcal{A}_n\subseteq 2^\Omega$ be $\sigma$-algebras. How can we show, that ...
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0answers
12 views

Distribution/law of a random variable after conditioning on an event

I have a probability triple $(\Omega,\mathcal{B},P)$ and a random variable $X:(\Omega,\mathcal{B},P)\to(\mathbb R,\mathcal{R})$ with distribution $\mu_X := P \circ X^{-1}$. If I condition on an event ...
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1answer
47 views

If X,Y and Z are independent, are X and YZ independent?

If yes: I know that f(X) and g(Y) are independent if X and Y are independent and f and g are "measurable".* If that is to be used, is g(Y) = YZ measurable? If not, how else to approach this? If ...
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2answers
16 views

let x and y be uniformly distributed independent random variables on [0 ,1].the probability that the distance between x and y is less than 1/2 is?

I have a question about probability: let x and y be uniformly distributed independent random variables on [0 ,1].the probability that the distance between x and y is less than 1/2 is? can someone ...
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1answer
17 views

Computation of Conditional Expectation using Measures

Here's a definition of conditional expectation of $X$ found on p 363 of this book: Let $(\Omega, \mathcal{F}, P)$ be a probability space, and let $X$ be in integrable random variable. If $B \in ...
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1answer
28 views

Is it possible to assign probability to a set $X$ with $|X|>2^{\aleph_0}$?

Is it possible to assign probability to a set $X$ with cardinality $|X| > 2^{\aleph_0}$? Example would be a set $|X| = 2^{2^{\aleph_0}}$.
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1answer
10 views

how do I parametrise a stochastic matrix

I have a matrix $\mathbf{t}$ that maps one $d$ dimensional probability distribution to another $\mathbf{t}^T x = q$, i.e. with $\sum\limits_i t_{ij} x_i = q_j$ and $\sum\limits_j t_{ij} = 1$ $\forall$ ...
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1answer
18 views

If $(B_t)_{t\ge 0}$ is a Brownian motion and $\tau$ is a stopping time, then the stopped process $(B_{\min(\tau,t)})_{t\ge 0}$ is integrable

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$. By definition $B_t$ is normally distributed with mean $0$ and variance $t$. Now, let ...
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0answers
11 views

Optional stopping/sampling for right-continuous supermartingales

Let $\mathbb{F}$ be a filtration $(X_t)_{t\ge 0}$ be a right-continuous $\mathbb{F}$-supermartingale $\sigma,\tau$ be bounded $\mathbb{F}$-stopping times with $\sigma\le \tau$ and ...
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0answers
29 views

Assume a die is rolled repeatedly. Find the markov matrix $P$ for the random variable of the time until the next $6$.

Assume a die is rolled repeatedly. Find the markov/transition matrix $P$ for the random variable $X_r$ = the time until the next six at time $r$. My solution was: For $i,j \geq 0$, $P$ is given ...
3
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1answer
31 views

Variation processes and strong solutions of stochastic differential equations

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $\mathbb{F}$ be a filtration on $(\Omega,\mathcal{A})$ $\tau$ be a $\mathbb{F}$-stopping time An $\mathbb{F}$-adapted, ...
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0answers
19 views

Power Spectral Density Approximation

Let $X_t$ be a zero-mean, stationary random process. Let $X_f$ be the Fourier transform of $X_t$; $X_f$ is also a random process, but as a function of $f$. Let us denote the power spectral density ...
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0answers
31 views

Deriving density of a function of a random variable

Let $(\Omega, \mathcal{F}, P)$ be a probability space and $X: \Omega \to \mathbb{R}$ a continuous random variable. Let $Y:\mathbb{R} \to \mathbb{R}$ be Borel-measurable. Finally, let $f_X: ...
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3answers
30 views

The non-uniform probability of sums from the throw of multiple dice

I'm reading The Drunkards Walk by Leonard Mlodinow. In the book, the author writes: From a throw of three dice, a sum of 9 and 10 can be constructed in an equal combinations. However, the outcome ...
2
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1answer
19 views

independence of random objects when forming product spaces

Suppose we have two probability spaces $(\Omega_1, \mathscr{F}_1, \{\mathcal{F}^1_t\},\mathbb{P})$ and $(\Omega_2, \mathscr{F}_2, \{\mathcal{F}^2_t\},\mathbb{P}_2)$, if we take product space $$\Omega ...
0
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1answer
25 views

independence of random variable

Suppose we have $2$ Independent random variables $X$ AND $Y$. Let $f(X)$ and $g(Y)$ are functions of those $2$ random variables. 1.) my question can we say that the functions $g(X)$ AND $f(Y)$ are ...
2
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1answer
41 views

Doob decomposition of $|S_n|$ where $S_n$ is simple random walk.

Let $X_n$, $n\geqslant 1$ be iid Rademacher random variables, i.e. $X_1$ takes values $\pm 1$ each with probability $\frac12$. Define $S_0=0$ and $S_n=\sum_{i=0}^n X_i$, and $\mathcal F_n = ...
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2answers
22 views

Probability theory combinatoric problem

A total of $n$ bar magnets are placed end to end in a line with random independent orientations. Adjacent ends with equal polarities repel each other, and adjacent ends with opposite polarities ...
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0answers
27 views

How would you understand the notation $\operatorname E(\zeta \mid x)$.

Let ($\Omega=X\times Y, 2^{\Omega}, \operatorname{P})$ be a discrete probability space, so $\Omega$ consists of pairs $(x,y)$. Let $\zeta$ be a random variable $\Omega\rightarrow\mathbb{R}$ on that ...
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1answer
34 views

If $(X_t,t\in I)$ is a process with values in $(E,\mathcal{E})$, are $\sigma(X_t,t\in I)$ and $\sigma(X)=X^{-1}(\mathcal{E}^{\otimes I})$ equal?

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $E$ be a Polish space and $\mathcal{E}$ be the Borel $\sigma$-algebra on $E$ $I$ be an index set $X_t$ be a random variable on ...
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2answers
22 views

Probability of two strings being equal

Given a matrix $A\in F_2^{n\times m}$, (let $m< n$ and $A$ has full column rank) what is the probability under the distribution ( $y,y'$ uniformly random in $\{0,1\}^m$), such that $Ay=y'$? I am ...
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1answer
14 views

What's the probability, and how to choose the right formula?

Question 1: Toss a coin 4 times. Let $A$ denote the event that a head is obtained on the first toss, and let $B$ denote the event that a head is obtained on the fourth toss. Is $A \cap B$ empty? ...
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5answers
363 views

Is there an alternative intuition for solving the probability of having one ace card in every bridge player's hand?

I am trying to get to know probability a little better since it's a weak point for me and I was wondering what other ways there were to intuitively think about the problem of finding the probability ...
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1answer
38 views

What is the probability of two random line segments crossing in a unit square?

For the purposes of this question a random line segment is defined by connecting two random points inside the unit square, where a random point is found by generating two random numbers between 0 and ...
3
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2answers
50 views

Markov Chains - Strong Markov Property

I'm struck with an exercise. I tried, but the results don't seem to fit to those proposed. Exercise: Two players play the following game. The one who begins draws two cards from a deck of 40 cards ...
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2answers
39 views

Exchanging expectation and limits

Exchanging expectation and limits I have a stochastic process, ${b_t} \, (t=0, 1, 2, \ldots)$, which follows a random walk. Specifically, ${b_0} = 0$ and for $t$ greater than zero, $\displaystyle ...
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1answer
34 views

Let X and Y be a random variables with $E(X) = 5$, $Var(X) = 30$, $E(Y ) = -􀀀5$, $Var(Y ) = 10$ and $Cov(X, Y ) = 7$

(a) Find $E(2X-3Y+1)$. (b) Find $E((X-2Y)^2)$. (c) Find $Var(3X-Y+pi)$ First I found $E(X^2)$ and $E(Y^2)$ using the given values for (a) I have $2E(X)-3E(Y)+1$ for (b) I come up with: ...
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1answer
32 views

Marginal Distributions from Joint Distribution

Here's a seemingly common proof for the formula of a marginal distribution using a bivariate joint distribution, for which I'm not clear on each step: Setup: Let $(\Omega, \mathcal{F}, P)$ be a ...
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1answer
25 views

Law of Iterated Expectation with Probability?

I'm trying to follow a proof of the following proposition (source) Let X and Y be two independent random variables and denote by $F_X(x)$ and $F_Y(y)$ their distribution functions. Let $$Z=X+Y$$ ...
3
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5answers
286 views

Intuition behind Chebyshev's inequality

Is there any intuition behind Chebyshev's inequality or is that only pure mathematics? What strikes me is that any random variable (whatever distribution it has) applies to that. $$ ...
0
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1answer
22 views

Laplace-Stieltjes :Functions of independent random variables

I am reading a book about stochastic modelling and I came across something and I couldn't really figure it out. First question would be are Probabilty Generating Functions (PGF) only for discrete ...
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1answer
42 views

Probability of winning consecutively [on hold]

India and USA play $7$ football matches. No match ends in a draw. Both the countries are of same strength. Find the probability that India wins at least $3$ consecutive matches.
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1answer
25 views

How do we know the joint probability distribution measure is valid?

Let $X,Y$ be $\mathbb{R}$-valued random variables on $(\Omega, \mathcal{F})$. Then $(X,Y) : \Omega \to \mathbb{R}^2$ induces a joint probability distribution measure $\mu_{X,Y}: \mathcal{B} \otimes ...
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1answer
24 views

Convolution with one of the variables is mixed and the other continuous

Suppose $X$ and $Y$ are independent random variables with CDF $F$ and $G$ and nonnegative support. If $X$ has a point mass $p$ at $0$ and otherwise some "density" $f$ (that is, ...
3
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3answers
29 views

Simulation of interacting Ornstein-Uhlenbeck processes

I would like to simulate the following system of interacting OU processes on $[0,T]$: $$dX_t^1=(X_t^2-X_t^1)\,dt+\sigma_1 \,dW_t^1,\quad X_0^1=x_1$$ $$dX_t^2=(X_t^1-X_t^2)\,dt+\sigma_2 ...
3
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1answer
33 views

Joint Distribution Implies Independence…?

Consider the measurable space $(\mathbb{R}, \mathcal{B})$ and a probability space $(\Omega, \mathcal{F}, P)$. Define a finite sequence of random variables $X_1,\ldots,X_n: \Omega \to \mathbb{R}$. ...
0
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1answer
13 views

Equivalence of definitions of Gaussian Measure

Wikipedia's article on Gaussian measures notes this as the definition of Gaussian measures: $\gamma_{\mu, \sigma^{2}}^{n} (A) := \frac{1}{\sqrt{2 \pi \sigma^{2}}^{n}} \int_{A} \exp \left( - \frac{1}{2 ...
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0answers
20 views

Local Martingales in a Finite Time Horizon setting

I apologise if this question has been answered somewhere else. Consider the following definition. Let $T \in [0, \infty), d \in \mathbb{N}$, let $(\Omega, \mathcal{F}, P, (\mathbb{F}_t)_{t \in [0, ...
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1answer
38 views

Trouble finding the expected value of a random variable

Suppose that we have a procedure A that we run once and it returns as a result either success or ...
0
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0answers
15 views

How to deal with set-valued(set in $\Re^n$) random variables?

I'm trying to attack a problem where the random variable are sets i.e set-valued random variable. Suppose $S = \{X_1, X_2,\cdots,X_n\}$ is a set of sets($X_i$) and $f(X_i)$ is the probability ...
2
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1answer
27 views

Is a martingale with bounded variance therefore bounded in $L^2$?

If a martingale $W_n$ has bounded variance, does this mean that $W_n$ is automatically bounded in $L^2$? I feel like this ought to be obvious but I don't see how to prove it and I haven't been able to ...
1
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1answer
28 views

A conjecture on the connection between the difference of two independent Poisson random variables and their parameters.

Let $X$ and $Y$ be two independent poisson random variables with parameters $\mu$ and $\lambda$, respectively. Assuming that $\mu\geq\lambda$ , is it true that $P\left(X=Y-k\right)$ is decreasing in ...
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2answers
61 views

Is a limit of measure of a sequence of sets equal to measure of limit of the sequence of sets?

I'm sitting at the same question desk as this: Limit of the measure of the converging sequence of sets. Actually, I can't prove it neither. PA6OTA gave a hint to show there is subsequence $A_{n_k}$ ...
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1answer
23 views

Find probability mass function from text [on hold]

$5$ persons (each independent of the other) when in a good mood it opens the tap with probability $\frac{1}{2}$ or in a bad mood with probability $\frac{1}{2}$. When that person is in a good mood it ...
1
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1answer
22 views

Normally Distributed = Absolute Continuity?

Let $(\Omega, \mathcal{F}, P)$ be a probability space. A random variable $X: \Omega \to \mathbb{R}$ is said to have the standard normal distribution if it has the density $f:\mathbb{R} \to ...
0
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2answers
37 views

How to find E(Y) given that the random variable X is exponentially distributed with lambda equal to 0.5?

Random variable $X$ is exponentially distributed with the parameter $\lambda$ equal to $0.5$. Define also $Y = 1 - 2X$ Find $E(Y)$ , Var(Y) and the moment generating function of Y. I have $f_x(X)= ...
1
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2answers
37 views

How is the definition of joint probability not circular?

According to the chain rule: $$ P(A,B) = P(A \cap B) = P(A \mid B)P(B) $$ Yet, according to wikipedia, the Kolmogorov definition of conditional probability is (axiomatically) as follows: $$ P(A ...
2
votes
1answer
46 views

Convergence of a sequence

Let $X$ be a random variable with a distribution function such that $n^t P(|X|>n) \to 0$ as $n \to \infty$, for some $t>0$. Then, I know that for any $\epsilon>0$, there exists $n_0\in ...