Questions tagged [probability]
For questions about probability. independence, total probability and conditional probability. For questions about the theoretical footing of probability use [tag:probability-theory]. For questions about specific probability distributions, use [tag:probability-distributions].
104,259
questions
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Bayes' theorem and card colors
This is an expansion/generalization of a previous question I've asked here. Some of the simplifications I made in the original question turned out to be too simplifying, so I'm trying again. The most ...
- 33
0
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0
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5
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Probability of getting a correct Bit
I have a probability problem that goes like this: I want to sent a bit across a channel that has a certain error rate. The probability of getting a bit wrong is $0.3$, and so to increase the chances ...
- 305
0
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0
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9
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Martingales and stopping rules [closed]
\textbf{Question:}
Let (X_1, X_2, \ldots, X_n) be a sequence of independent and identically distributed (iid) uniform variables on the interval ([-2, 2]), and define (S_n = X_1 + X_2 + \ldots + X_n). ...
- 1
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1
answer
14
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Find marginal distribution of X and Y and the expectation of XY
This is an exercise that confuse me
$$ let \ \ x,y \in \{0,1,2......\}$$
$$\mathbb{P}(X=x,Y=y)=\frac{e^{-2}}{x!(y-x)!},x=\{0,1,...\}, y=\{x,x+1,...\}$$
$$ otherwise \ \mathbb{P}(X=x,Y=y) = 0$$
In the ...
- 33
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0
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12
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Logarithm of probability limit question
Let $X_1, X_2, ...$ be a sequence of i.i.d random vectors taking values in $\mathbb{R^n}$ such that $L(\lambda):= \log \mathbb{E} [e^{\langle \lambda, X_1 \rangle}] < \infty$ for all $\lambda \in \...
- 25
1
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2
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41
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Expected value of a continuous random variable must be strictly bounded
Given a probability space $(\Omega, \mathcal{F}, P)$ and a continuous random variable $X: \Omega \to I$ where $I$ is an interval of $\mathbb{R}$. I'm trying to show that the expected value $$E[X] = \...
- 406
1
vote
3
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53
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How to prove/check a random generator is a unifrom random distribution?
If I have access to a random integer generator that produces values in the range from 0 to N-1, how can I verify whether this generator truly produces a uniform random distribution?
Simply analyzing ...
- 1,145
1
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0
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22
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Probability of finding the argmax over inner products
Let $U(\mathbb{S}^{d-1})$ be the uniform probability over the unit sphere. Given $x_1,\dots,x_d, a,b\in \mathbb{R}^d$, I'm looking at the following probability:
$$P(\mathcal{X},a,b) = \text{Pr}_{y\sim ...
- 853
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0
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30
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Conditional version of Borel Cantelli's lemma
Let's say we consider a discrete time random walk on a continous space $W_n= \sum_{i=1}^{n}X_i$ where $X_i$ are i.i.d. random variables. We define the event $A_n = \{W_n \geq \sqrt{n}\}$ and we ...
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Question regarding independent (Bernoulli) trials: rolling a die 10 times and defining success as rolling a 1 or a 6.
I just want to make sure that the scenario I'm describing is indeed a Bernoulli trial type of experiment. Here is the set up and query regarding the set up:
A six-sided die is rolled 10 times. What is ...
-3
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0
answers
21
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How many times a particular number shows up in an infinite sequence [closed]
A program using Python to calculate how many times a number appears in an n sequence
0
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0
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13
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Local Polynomial Regression as Linear Smoother
I am interested in Local Polynomial Regression. I was reading the Local Polynomial Regression Paper (Fan,1992), where we want to minimize
\begin{equation}
\sum_{j=1}^n\left\{Y_j-a-b\left(x-X_j\right)\...
3
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55
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Balls are pulled and distributed randomly into the boxes. What is the probability that in no boxes there will be balls of the same color?
Full Question: The pool of 100 balls contains 20 pink balls, 10 yellow balls, 20 purple balls, 15 indigo balls and 35 green balls. We randomly pull 10 balls out of 100 and randomly distribute them ...
- 31
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21
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The number of states of (Stochastic) Probabilistic Finite Automata that accepts string that is divisible by 3 and 5 (two distinct primes)?
My question is inspired by The number of states in DFA that accepts strings with length that is divisible by $3$ or $5$. 's conclusion, that DFA requires full a*b number of states. So - I am trying to ...
- 1,265
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0
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21
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Proof of Marcinkiewicz-Zygmund strong law of large numbers
Marcinkiewicz-Zygmund strong law of large numbers:
Let $X_1,X_2,···$ be i.i.d. with $E|X_1|^p< \infty$ for some $0< p <2$. Then
$$
\begin{cases}
\frac{S_n - nEX}{n^{1/p}} \to 0 \text{ a.s.} &...
- 1
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votes
1
answer
29
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Estimating of unknown parameters using the method of moments
Let $x_1, x_2, ..., x_n$ be the results of $n$ independent repeated observations on random value $\xi$, which density function:
$f(x, \theta) = pf_1(x,\theta)+(1-p)f_2(x, \theta)$,
where $f_1(x, \...
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1
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0
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33
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How to calculate probabilities in a Brownian motion?
I have a Brownian motion, $X(t)$.
Further, $T_a$ $=$ $inf($$t:$ $X(t) = a$$)$
And $M(t)$ $=$ $max($$X(s) : $ $0 ≤ s ≤ t$$)$.
I am trying to prove that:
$P(T_a \le t) =$ $2$$\int_a^\infty \frac{1}{\...
- 191
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0
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49
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Uniform distribution always exists on a closed convex compact set?
Let $X$ be a topological space with a weak order $\succsim$.
$C\subset X$ is a closed convex compact set.
What additional topological assumptions do we need to ensure that $C$ always has uniform ...
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47
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Is there a probability problem about "probability theory and its applications”
William Feller's relatively old book "Probability Theory and Its Applications",Chapter 3,Section 3.1:
Galton's rank order test.Suppose that a quantity (such as the height of) is measured on ...
- 1
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1
answer
29
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Selecting a marble from two boxes where the marbles are either red or blue
Problem:
A box contains $3$ blue and $2$ red marbles which another box contains $2$
blue and $5$ red marbles. A marble is drawn at random from one of the boxes to be
blue. What is the probability that ...
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29
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What is the probability of having 1 boy [closed]
What is the probability Family of 3 children will have 1 boy and 2 girls
0
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2
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55
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Number of Coin Flips Before Some Sequence Happens?
The Negative Binomial Distribution (https://en.wikipedia.org/wiki/Negative_binomial_distribution) describes the probability of getting $k$ successes before $r$ failures. For example, on a single 6 ...
- 2,358
2
votes
1
answer
551
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Chebyshev's Inequality Improvement
I'm having trouble improving the upper bound on Markov/Chebyshev's inequality in this particular example:
We have IID random variables where each $\mathbb{E}[|X_i|^2]<\infty$. Show that $$\lim_{n\...
- 304
1
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0
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39
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Given $n$ uniform random variables, probability no two differ by less than $x$
Given $n$ random variables uniformly distributed from $0$ to $1$, what is the probability that no two differ by less than some $x$?
My initial thought was that for $n$ random variables there are $n \...
0
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0
answers
28
views
Bound from below the probability to go beyond a certain point in a zero mean random walk
Consider a random walk on a continous space, call it $W_n = \sum_{i=1}^{n}X_i$ where the $X_i$ are i.i.d. such that $\mathbb{E}(X_i) = 0$ and with finite variance. I would like to bound from below the ...
- 764
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0
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25
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Chebyshev's inequality for sum of dependent random variables
Let $X_1, X_2, ...$ be a sequence of square integrable RV's with mean $\mu$ and covariance:
$$
\textrm{Cov}(X_i,X_j) =
\begin{cases}
\sigma^2 & \text{for } i = j \\
\rho & \text{for } |i - ...
- 25
2
votes
1
answer
57
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How to prove a result about the accumulation point / cluster point in Brownian Motion?
I have a standard Brownian motion, $B(t)$, $t\ge0$.
I am trying to prove the following result:
Every $t > 0$ is an accumulation point (or cluster point) of $(s: B(s) = B(t))$ from the right, with ...
- 191
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0
answers
26
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Deviation with respect of the Mode, part 2 - Weighted linear combination
Deviation with respect of the Mode, part 2 - Weighted linear combination
I a previous question I made, I wonder about How the Variance will behave if is defined in respect to the Mode "$\nu$"...
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answers
18
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Computing $E[\|ABB^TA^T\|_F^2]$ for Gaussian $A,B$
Suppose $A$ and $B$ are $n\times n$ matrices with IID standard normal entries.
What is the value of the following expression as the function of $n$?
$$E[\|ABB^TA^T\|_F^2]$$
The first few values are ...
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49
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How can I pick a point at random in the plane?
I am preparing an experiment for a math presentation that I will show to middle schoolers on December 12th. My idea is the following: I would like to find a squared box and draw inside it a circle and ...
- 1
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1
answer
30
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About upper asymptotic density.
Let $A$ be a subset of $\mathbb{N}$ and $(p_i)_i$ a strictly increasing sequence of positive integers. We define
$\overline{d}(A|(p_i)_i):=\limsup_n \frac{\left| A \cap \left\{ p_1,\cdots , p_n \right\...
- 11
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1
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23
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Given bivariate bernoulli with an integral as a parameter prove that are marginally identically distributed and correlation is positive
So this is a question from a past exam. The joint density function is
\begin{equation}
\begin{aligned}
P\left(X_1=x_1, X_2=x_2\right) & =I_{\{0,1\}}\left(x_1\right) I_{\{0,1\}}\left(x_2\right) \...
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votes
1
answer
41
views
Picking random point many times on segment
Let $AB$ be a segment of length $1$.
Let $P_1$ be a randomly chosen point on $AB$.
Let $P_2$ be a randomly chosen point on $AP_1$.
Let $P_3$ be a randomly chosen point on $AP_2$.
...
Let $P_{10}$ be a ...
- 11
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votes
2
answers
72
views
How to detect if a moving point has passed through the connecting line between two other fixed points?
I have a moving point (Tag) and two fixed points (anchors). I have the distance from the anchors to tag at each time. I also have the position of the anchors and the last position of the tag. I ...
- 3
1
vote
1
answer
46
views
Central limit theorem for inhomogeneous Poisson process
I consider $N_t$ an inhomogeneous Poisson process, especially $N_t$ follows a Poisson law with parameter $\int_{0}^{t}\lambda(s)ds$.
We assume that $\frac{1}{t}\int_{0}^{t}\lambda(s)ds\to \sigma^{2}$ ...
- 1,116
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0
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25
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How do I solve the average number of attempts to succeed when the probability increases with each failure?
Our base probability of success begins at .03 and each failed attempt increases the probability by .002 so that the second attempt has a probability of .032, the third .034, and so on. (The trials ...
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58
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Recursive Expectation Question
Suppose we draw some k ~ Unif(0, 1). Then, we will draw some $u_1$ ~ Unif(0, 1). If $u_1 < k,$ we stop. Else, we will draw $u_2$ ~ Unif(0, $u_1$). We will continue drawing until $u_n < k.$ What ...
0
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1
answer
25
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What is the difference in my reasoning for sampling with replacement and the correct reasoning?
I am currently reading Blitzen and Hwang's Introduction to Probability book, and am on Theorem 1.4.7, which states that when you are sampling n objects and making k choices from them one at a time ...
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1
answer
61
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probability distribution of total number of cells with 1/2 probability of dividing.
There is a cell that has a 1/2 probability of dividing into two daughter cells (the parent cell disappears) and a 1/2 probability of stop dividing. And each daughter cell has a 1/2 probability of ...
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1
vote
1
answer
46
views
How can I check that the following process is a martingale?
Let $(\Omega, \mathcal{F}, (\mathcal{F}_t), \Bbb{P})$ be a filtered probability space. Net $N$ be a poisson process with parameter $\lambda>0$. Let $h$ be a bounded measurable function and define $...
- 2,008
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votes
0
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17
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Why call the states of a Markov chain positive recurrent and null recurrent?
If a Markov chain is recurrent, its states will all be visited an infinite number of times on average if the chain is run forever. This name makes sense, the event of visiting the state keeps ...
- 6,581
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1
answer
41
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Taking the second derivative of the moment generating function.
I am following along from a proof of the central limit theorem using moment generating function from this lecture.
At one stage, we are taking derivatives of the moment generating function of an r.v. $...
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votes
0
answers
21
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Probability question (standard - 12)
A matrix is choosen randomly form the set of matrix of order 3 elements of matries are 0, 1, 2 or 3. It is found that the matrix is diagonal matrix. Find the prob- ability that the matrix is non-...
2
votes
1
answer
39
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Brownian motion and logarithm inside averaging
We consider Brownian motion as a process
$$ V(0)=0, \qquad \overline{V(x)}=0, \qquad \overline{(V(x)-V(y))^2}= 2 |x-y|. $$
I do not know how to prove the statement:
$$ \overline{\log\left(\int \mathrm{...
- 23
0
votes
0
answers
31
views
The smallest d which almost ensures every polygon will contain expected number of samples +- d
Let's uniformly sample $n$ times from the unit square. Given $m$ polygons contained within the unit square with areas $A_i$, where $m = kn$, and where the polygons overlap arbitrarily, what is the ...
- 23
2
votes
1
answer
95
views
Probability that every polygon contains almost exactly the expected number of samples
Let's uniformly sample $n$ times from the unit square. Define a polygon contained within the unit square with area $A$. Surely as $n \to \infty$, the probability that the polygon contains at least $\...
- 23
0
votes
0
answers
32
views
Proof birth and death process
I need some help understanding the following proof:
I don't really see how the first inequality is derived. I guess the authors are conditioning on whether $X_0^0 = X_0^p$ or $X_0^0 \neq X_0^p$, but ...
- 83
2
votes
0
answers
42
views
Stochastic process vs. random function
On the wiki page on stochastic processes, it says that they can be interpreted as random elements of a function space, which makes complete sense to me. However, it goes on to say that this ...
- 196
0
votes
0
answers
23
views
Find the density function of a random vector $w = (X_{(3)}, X_{(2)}, X_{(1)})$
Let a random vector $v = (X_1, X_2, X_3)$ be given, the components of which are independent and uniformly distributed on the segment $[1, 5]$. Find the density function of a random vector $w = (X_{(3)}...
- 119
2
votes
1
answer
46
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Understanding the Likelihood Principle - What is the Big Deal here?
I am reading about the Likelihood Principle (https://en.wikipedia.org/wiki/Likelihood_principle).
In short, I think the Likelihood Principle discusses the phenomena that if two experiments have the ...
- 2,358