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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7 views

Constants Which Make Random Variables Independent [on hold]

Let $X_1,X_2,...,X_n$ be i.i.d absolutely continuous random variables with density function $\theta \exp(-\theta x)$, $x>0,\theta>0$. Describe all combinations of real valued constants ...
2
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1answer
20 views

Intuition of the distribution of the minimum of exponential random variables

Let $X,Y$ be two independent random variables with exponential distribution with parameters $a$ and $b$ respectively. It is known (see e.g. here) that $Z:=\min\{X,Y\}$ is exponential distributed with ...
0
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0answers
30 views

On the linear combination of $\pm 1$ random variables

Let $X_1,\dots, X_n$ be i.i.d symmetric $\pm 1$ random variables, i.e. $X_j$ takes values in $\{-1,1\}$ with $$\mathbb{P}(X_j=1)=\mathbb{P}(X_j=-1)=\frac{1}{2}.$$ Let $a_1,\dots,a_n\in\mathbb{R}$, ...
0
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1answer
39 views

Probability - draw balls

Assume there are 100 balls in the box, 50 are white and 50 are black. What is the probability that I draw 9 balls in which at most 4 are white (without replacement)? $(p(9 black)+p(1 white + 8 black) ...
2
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1answer
21 views

Understanding derangement.

From the inclusion-exclusion principle we get that out of $N$ objects with one label each, there is a probability of $$\sum_{k=1}^N (-1)^{k+1}\frac{1}{k!}$$ that a random assignment of the $N$ labels ...
3
votes
1answer
31 views

How does the posterior of a dirac prior look like?

Edit for the Moderators: Should this question migrate to stats.stackexchange? I have a very basic question concerning updating from a prior to a posterior in bayesian statistics. Setting: I ...
2
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0answers
19 views

How to compute the characteristic function

Let $X_n$ be a sequence of identically distributed, independent random variables and let the distribution of $X_n$ be $\mu_{X_n}(\{1\}) = a$ and $\mu_{X_n}(\{-1\}) = 1-a$ for 0 < a < 1. ...
0
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0answers
11 views

How to approach deriving a folder distribution's pdf from original pdf?

Let's say I have an Erlang-distributed random variable $x$, and now I'm only taking the samples of $x$ for which it holds that $x>T$, where $T$ is some constant. The probability $P[x>T]$ can be ...
4
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0answers
32 views

Stochastic domination

Suppose we have two probability measures on a space $X$, $\mu$ and $\nu$, such that $\nu$ stochastically dominates $\mu$, i.e.there exist a coupling of $\mu$ and $\nu$ on the product space $X \times ...
2
votes
3answers
30 views

What is the probability that Person A will be chosen last every time$?$

There are $14$ people in a pool of subjects. People are selected at random one at a time. All people are chosen and then Person A.(Person A is chosen last every time This trial occurs $3$ ...
0
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0answers
42 views

Why do we always consider real-valued $f$ in the Itō formula to find an expression for $f(t,X_t)$

The Itō formula (see Da Prato, Theorem 4.32) yields an expression for $f(t,X_t)$ where $${\rm d}X_t=\phi\;{\rm d}t+\Phi\;{\rm d}W_t\;,\;\;\;X_0=\xi\;.\tag 1$$ Even when $X$ takes values in a Hilbert ...
1
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1answer
25 views

Prob: Observation coming from a specific continuous distribution

I have a two-staged random process: First, I draw a type, which can be either $\tilde f$ or $\tilde g$. The (unconditional) probability of drawing the former type is $P_F$. Then, these two types ...
1
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1answer
68 views

Sum of probabilities is infinite

I'm stucked solving this problem: Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of i.i.d. random variables with exponential distribution and $\lambda=1$. Show that ...
0
votes
2answers
37 views

Find the density function from a joint density function

I try to solve the following task and I don't know what the correct way to do is. Let $p\in(0,1)$ and $(X,Y)$ be a pair of random variables with distribution density function ...
1
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0answers
42 views

For a finite state irreducible aperiodic MC, show that $P^{d^2}$ has all coordinates positive.

Suppose $X_n$ is an irreducible aperiodic finite state MC, with $P$ being the transition matrix. Then we know that $P^n$ has all positive entries for some $n\in\mathbb N$. If the state space $S$ of ...
0
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1answer
56 views

I am not given figures to answer this question. Whats the right approach?

Z is a random variable defined as the sum of N independent Bernoulli trials where the probability of each Bernoulli taking the value 1 is given by p. The number of Bernoulli trials N is itself a ...
1
vote
1answer
29 views

Why does the given condition imply the following random variables are not independent?

Let $Y ∼ U[0, 1]$ be uniformly distributed in the interval $[0, 1]$. Define the random variables $X_1, X_2$ as $$X_1 = sin(2πY )$$ $$X_2 = cos(2πY )$$ Why does the fact that $X_1^2 + X_2^2 = 1$ imply ...
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0answers
21 views

How to calculate weighted negative log-likelihood? [on hold]

I'm calculating the negative log-likelihood for a bunch of tasks. The tasks appear at a certain time point and I want to give to the newer tasks a higher weight. $$ NLL = p_{1} * p_{2} *... * p_{n} = ...
0
votes
0answers
6 views

Finding the norm of estimation error asymptotically

Let $\theta \in \mathbb{R}^p$ be such that it has uniform distribution on the set of standard unit vectors $\{\tau e_1,\ldots,\tau e_p\}$, for $\tau=\sqrt{(2-\varepsilon)\log p}, \varepsilon>0$. ...
1
vote
1answer
13 views

Give necessary and sufficient conditions so the sum of random variables converges almost surely

$\{X_k\}_{k}$ are independent random variables on the probability space $(\Omega, \mathcal F, P)$ and $X_k$ has gamma density $f_k(x)$ where $f_k(x)=\dfrac{x^{a_x-1}e^{-x}}{\Gamma(a_k)}$ where $x,a_k ...
0
votes
1answer
31 views

How to deal with a conditional binomial question involving coin flips?

Suppose that we have a sequence of fair coin flips. At each round, it is either Heads, which we denote by $H$, or tails, which we denote as $T$. Now, at the end of $N$ flips, which is assumed to be a ...
1
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1answer
19 views

Number of balls in a slot

Let $N$ balls are distributed among $r$ cells at random, each cell being free to receive any number of balls. Calculate the probability that a particular cell contains k balls ...
0
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1answer
32 views

Expected win in a lottery

In a lottery $1000$ tickets are sold and the cost of a ticket is if $10$dollars. The lottery offers a first prize of if $1,000\,$ dollars, two second prizes of $500\,$ ...
2
votes
1answer
30 views

Theoretical interpretation of simulating from a distribution

Suppose there is a random variable $X$ with marginal density $p_X$. However only the conditional densities $\{p_{X\mid\Theta}(\cdot\mid\theta):\theta \in \mathbf{T}\}$ are known directly, where ...
0
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1answer
11 views

Limes superior and random variables

I want to show the following: Let $X_1,X_2\dots$ be i.i.d. random variables. Let $\text{E}[|X|^p]=\infty$ for $p>0$. Show that $$P(\limsup\limits_{n\to\infty }\{|X_n|\geq n^{1/p}\})=1$$ What I ...
0
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0answers
17 views

measure-theoretic probability, (sets of) null events and non-zero probability

Assuming a well-defined probability space $ (\Bbb{R},\mathscr{B},\Bbb{P}_X) $, where $\mathscr{B}$ is the Borel $\sigma$-field, and for a random variable $X$ having a continuous probability density ...
2
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0answers
52 views
+50

Can we apply an Itō formula to find an expression for $f(t,X_t)$, if $f$ is taking values in a Hilbert space?

Let $U$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space ...
2
votes
2answers
48 views

Showing that this is a martingale.(4.13 in Øksendals SDE)

This is an exercise from Øksendals stochastic differential equations, where I get stuck. It is exercise number 4.13.(I simplified the notation a bit.) I have that X is an Itô-process where: ...
0
votes
1answer
49 views

Infinite probability density?

I've read that for a "[..]random variable strongly "localized" around a single value", the probability density function (PDF) could be: $p(x)=\frac {1}{2\epsilon}$, with $\epsilon \to 0$, and ...
1
vote
1answer
23 views

Let $X\sim\mathcal N(0,A)$ , where $A\sim Exp(1)$. How do I recover the joint distribution for $Z=(X,A)$?

Unfortunately there is no recipe for computing the joint distribution, just the other way around (from the joint distr. to the marginal ones). Would appreciate any help to find an Ansatz for this ...
3
votes
0answers
36 views

Hitting Times for Brownian Motion - Levy Process?

Let $X$ be a Brownian motion and let $$H_a = \inf\{ s \ge 0 \mid X_s = a \} \;\ \text{and} \;\ S_a = \inf\{ s \ge 0 \mid X_s > a \}.$$ Now, I've shown that $H_a$ and $S_a$ are equal almost surely ...
0
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2answers
35 views

Find dependent event when two dice are thrown simultaneously.

Let two fair six-faced dice $A$ and $B$ be thrown simultaneously. If $E_1$ is the event that die $A$ shows up four, $E_2$ is the event that die $B$ shows up two and $E_3$ is the event that the sum of ...
0
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0answers
27 views

Waiting Time Distribution

Let X be a random variable which denotes the amount of time spent in a state(say state 'I') before changing state. As X is a random variable it must have a Probability space/sample space and a sigma ...
1
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0answers
52 views
+50

Show $P\left[A<Z \mid \mathcal{G} \right]=e^{-A}$ for $Z$ standard exponential and $A$ nonnegative $\mathcal G$-measurable

I have a question about exponential distribution and conditional probability. Let $(\Omega, \mathcal{F}, P)$ be a probability space and $\mathcal{G}$ be a sub $\sigma$ algebra of $\mathcal{F}$. Let ...
0
votes
1answer
28 views

square-root rule of time

I tried to test the square-root-rule of time for quantiles of a normal distribution. So i created with the statiscal programming language R two variables a<-rnorm(100,mean=2,sd=1) ...
1
vote
2answers
31 views

Probability Question - Moment Generating Function

$$ f(x) = \begin{cases} xe^{-x}, & \text{x ≥ 0} \\ 0, & \text{elsewhere} \end{cases}$$ Q: Find the Moment Generating Function of X. Hi, I was trying to solve this question by putting the ...
2
votes
3answers
64 views

Picking two random points on a disk

I try to solve the following: Pick two arbitrary points $M$ and $N$ independently on a disk $\{(x,y)\in\mathbb{R}^2:x^2+y^2 \leq 1\}$ that is unformily inside. Let $P$ be the distance between those ...
2
votes
0answers
32 views

About the random $\pm 1$ matrices

I was reading the paper "On the probability that a random $\pm 1$ matrix is singular". In the paper the author defined the following notations: $M_n$: a random $n\times n$ matrix with i.i.d entries ...
2
votes
3answers
54 views

What is a sample of a random variable?

I've tried to learn probability, one way or another, many times in the last 50 years, and finally settled on the Kolmogorov approach, where a random variable isn't described as something like "a roll ...
0
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0answers
8 views

Proposition in Daniel Kuhn et al paper “Primal and dual linear decision rules in stochastic and robust optimization”

Paper link:http://www.optimization-online.org/DB_FILE/2009/02/2218.pdf In this paper in page number 8 the autors make the following proposition: Let $\mathbb{P}$ be a probability measure on ...
2
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1answer
13 views

Every collection of measures on a compact space is uniformly tight

I am looking for a proper statement of the sentence in the title and its proof. First, let me give some context. I have a covariance stationary time series, $X$. The autocovariance function of $X$ is ...
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votes
0answers
19 views

Function of two continuous random variables. find CDF [closed]

[\begin{array}{l}{\rm{Let X be a continuous random variable with uniform distribution on }}\left[ {0,1} \right].{\rm{ }}\\{\rm{Let Y be a continuous random variable with uniform distribution on ...
2
votes
0answers
19 views

Almost sure convergence and limes superior

I'm trying to prove the following exercises and I don't know if my attempts are correct. A sequence of real random variables $(X_n)$ almost surely converges to $X$ if and only if for every $\epsilon ...
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votes
1answer
39 views

Let X be a continuous random variable with pdf… [closed]

a.) Let X be a continuous random variable with pdf $f_x(t) = \exp[-t-e^{-t}]$ for all t in the reals. Find $F_X(x)$ My solution is $$F_X(x)= P(X \le x) = ...
0
votes
1answer
13 views

Continuous and discrete random variables defined on the same probability space?

I am confused on the definition of continuous/discrete random variables defined on the same probability space. Consider the random variables $X,Y$ defined on the same probability space $(\Omega, ...
0
votes
0answers
13 views

Properties of Kernel Integral inner Product of Gaussian Process

Can anyone give any reference / suggest how to get the rigorous mathematical properties of the following : $$ Y=\int_{a}^{b} K_{X} (t) \ f(t) \ dt $$ where $$f \sim GP (\mu(\cdot), R ...
0
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0answers
10 views

Connect the MGF of the Survivor, Cumulative and Mass disttributions

Assume that $X$ has a known distribution $P_X$, with a generating function $\hat P_X$. What relationship links $\hat P_X$ with the MGF of X's CDF ($\hat C_X$) and SDF ($\hat S_X$). Would that ...
0
votes
2answers
49 views

Prove $P(X=k \mid X+Y=n) = \frac1{n+1}$

Let $f_X(k) = f_Y(k)= p(1-p)^k~$ for all $k = 0,1,2,\ldots$ for some $0 < p < 1$. Show that for any $n \ge 0$ $$P(X=k \mid X+Y=n) = \frac1{n+1}$$ for any $0 \le k \le n$. What is confusing ...
0
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0answers
9 views

Deriving the asymptotic properties of two-stage estimators

Suppose $Y_n = g(X_1, \cdots, X_n)$ is a statistic and that $\sqrt{n}(Y_n -\theta) \stackrel{d}{\to} N(0,V)$, where $\theta$ is a constant. Define $Z_{i,n} = f(X_i,Y_n)$, where $f$ is a continuously ...
0
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0answers
30 views

Dependence/Independence Problem in probability [closed]

Suppose a student is taught by $N$ teachers. The pdf of the marks that the student gets from $i$-th teacher is $p_{m_i}(x)$ and we assume that all $p_{m_i}(x)$'s are i.i.d i.e. ...