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].

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Expected value/distribution of a random walk that can't repeat rows or columns on a grid where each cell has a real number value?

Given that you have a matrix $A \in R^{n \times n}$, we define a random walk, $X$, of length $1 \leq k \leq n$ on $A$ as a walk where no two cells share the same row or column. An example of such a ...
kvarad's user avatar
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-3 votes
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How would I find z1 and z2? [closed]

enter image description here Hi How wouldI find z1 and z2 for e)
Math analysis 's user avatar
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0 answers
18 views

Randomly Ordering People Walking into a Room

This is a problem I thought of and am trying to figure out. Suppose there are 100 people: Person1, Person2,... Person100. A random person is selected and walks into the room. The next person to ...
konofoso's user avatar
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17 views

Show that for any $A\in\mathcal{B}([0,1])$ with $1>\mu(A)=a>0$ and any $b\in [0,a]$ so that $\mu(B)=b$.

Let $([0,1], \mathcal{B}([0,1]), \mu)$ be a measure space with Lebesgue measure. Show that for any $A\in\mathcal{B}([0,1])$ with $1>\mu(A)=a>0$ and any $b\in [0,a]$ so that $\mu(B)=b$. My idea ...
Hermi's user avatar
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-1 votes
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distribution of response variables in Maximum Likelihood Estimation [closed]

i've seen the MLE for a supervised model be $L(w)=\prod_{i}p(y_i|x_i; w)$, my question is about the distribution $p(y|x; w)$ if it is a conditional distribution or just the distribution of y with x ...
Tests Coll's user avatar
-2 votes
0 answers
13 views

Probability integral transform: What does it mean to apply the cdf of X to X? [closed]

I have just learned the probability integral transform and I find it hard to understand conceptually. I don't understand what $F_X(X)$ means. It's the probability that X is less or equal to itself? ...
Aziz Benbachir's user avatar
-1 votes
0 answers
50 views

classical theory of probability

Wikipedia definition: As stated in Laplace's Théorie analytique des probabilités, The probability of an event is the ratio of the number of cases favorable to it, to the number of all cases possible ...
Sam's user avatar
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3 votes
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The vertices of a pentagram are five random points on a circle. Conjecture: The probability that the pentagram contains the circle's centre is $3/8$.

The vertices of a pentagram are five uniformly random points on a circle. Is the following conjecture true: The probability that the pentagram contains the circle's centre is $\frac38$. (The ...
Dan's user avatar
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2 votes
1 answer
51 views

Modifying the Birthday Problem Paradox for Arbitrary Situations?

We learned about the birthday problem paradox: If people are in a room with randomly distributed birthdays, very few people are needed for at least two people to have the same birthday. As I ...
konofoso's user avatar
2 votes
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34 views

How to prove the exisitence of x_1?

For a sequence of probability density functions $\{f_n^i(x)\}, i=1,\ldots,N$, we define a normalized probability density function $F^*(x)$ as: $ F^*(x) = \frac{\prod_{i=1}^N (f_0^i(x))^{\frac{1}{N}}}{\...
huuu's user avatar
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3 votes
1 answer
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Density function of bivariate function implies independence

I was wondering if $$f_{X,Y}(x,y)=\frac{1}{2\pi}e^{\frac{-1}{2}(x^2+y^2)}$$ Is sufficient to show that X and Y have independent standard normal distributions? If not what else would I need to show? ...
edster101's user avatar
-4 votes
1 answer
24 views

Probability Question of Order

What is the probability of pulling the letters in the following order lullaby from tiles abllluy? Is it just 1/5040?
Donna Henson's user avatar
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0 answers
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Demonstrate a (continuous time) chain that increases by one, and then randomly returns to the origin, is transient

I have the following continuous time Markov chain $X = (X_t : t \geq 0)$ with generator matrix given by $g_{i,i+1} = \lambda_i$ for $i \geq 0$, $g_{i, 0} = \lambda_i \rho_i$ for $i > 0$, and $g_{ij}...
Featherball's user avatar
2 votes
2 answers
47 views

Is $X = \max \left\{X_1, X_2\right\}$ exponential variable given $X_1$ and $X_2$ are?

I know that if $X_1$ and $X_2$ are independent exponential variables with parameters $\lambda_1$ and $\lambda_2$ then $X = \min \{X_1, X_2\}$ is exponential, and with $\lambda = \lambda_1+\lambda_2$. ...
Danny Wen's user avatar
3 votes
0 answers
30 views

Example of finitely additive probability with no atom for all subsets of $[0,1]$

I am looking for an example of a probability measure $\mu$ on $[0,1]$ such that $\mu(A)$ is defined for ALL subsets $A\subset[0,1]$ $\mu$ is finitely additive but not $\sigma$-additive No atom: If $\...
Yi-Hsuan Lin's user avatar
-4 votes
0 answers
23 views

Probability of events involving set [closed]

From the sets {1,2,3,.... 10}, select 3 different numbers at random. What is the probability that one of the numbers is the average of the other two numbers
ZhangJin's user avatar
-3 votes
0 answers
15 views

Why does this equation hold? This is a question about ddpm and nscn. [closed]

Can someone help me why this equation holds: $\nabla _{x_{t}}logq(x_{t})=E_{q(x_{0})}[\nabla _{x_{t}}q(x_{t}|x_{0})]$?This equation is about the connection between Denoising diffusion probabilistic ...
Harry Li's user avatar
0 votes
1 answer
40 views

Which failure probability is low enough to be considered as 'will not happen'? [closed]

Like you gonna be fired/sued/shot if it happens. I use 1 / 100 million / year (global catastrophes happen on Earth every 100 million years so it is likely no one will care about me) But maybe there is ...
Sergey Alaev's user avatar
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The compactness of Radon probability measure

Let X be a compact subspace of $\mathbb{R}^n$. Denote by $\mathcal{R}$ the space of Radon measures on X, and $\mathcal{P}$ the space of Radon probability measures on X. In my book, it says that $\...
Lilileaf's user avatar
2 votes
1 answer
42 views

Exercise 7.14 First course in probability

Exercise 7.14 First course in probability : An urn has $m$ black balls. At each stage, a black ball is removed and a new ball that is black with probability $p$ and white with probability $1 - p$ is ...
MathematicsBeginner's user avatar
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Show that $ \mu(A_{i_1}\cap A_{i_2}\cap\dots \cap A_{i_k})>c^k-\epsilon $

Let $((0,1], \mathcal{B},\mu)$ be a measure space. Let $\{A_i\}_{i\ge 1}$ be a sequence of Borel measurable sets so that $\mu(A_i)\ge c$ for all $i\ge 1$ and some universal constants $c\in (0,1)$. ...
H.Y Duan's user avatar
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1 answer
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Give an example of $P_m$ and $f_n$ so that $\lim_{m\to\infty}\lim_{n\to\infty}\int f_n dP_m\neq \int fdP$

Let $(X,\mathcal{F})$ be a measurable space. Let $\{P_m\}_{m=1}^\infty$ be a sequence of probability measures so that $P_m\to P$ weakly as $m\to\infty$. Let $\{f_n\}$ be a sequence of continuous ...
H.Y Duan's user avatar
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0 answers
34 views

How to prove this inequality (model quantization)?

Some conditions and definitions which may not be of use : $\pi_j^{t|s}:=P_{Y_t|\{X_t,X_s^{replay}\}}(y_t=j|x_t)$ stand for the prediction probability of the t-th task data $x_{t}$ on the j-th class, ...
Shunsheng Lee's user avatar
-1 votes
0 answers
22 views

Prove approximate equality

Now, I know $\mathbb{E}\left\{\log_{2}\left(1+\frac{X}{Y}\right)\right\}\approx\log_{2}\left(1+\frac{\mathbb{E}\left\{X\right\}}{\mathbb{E}\left\{Y\right\}}\right) \tag{1}$ if $X$ and $Y$are ...
KC Lau's user avatar
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1 answer
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If $X$ is sub-Gaussian random variable with variance proxy $\sigma^2$, how to show that $E\{ \exp( t X^2) \} \leq (1 - 2 t \sigma^2)^{-1/2}$?

If $X$ is sub-Gaussian random variable with variance proxy $\sigma^2$, i.e., $E(X) = 0$ and $E\{ \exp(s X) \} \leq \exp( \frac{\sigma^2 s^2}{2} )$ for $\forall s \in \mathbb{R}$, then how to show that ...
Zifeng Zhang's user avatar
2 votes
0 answers
57 views

Is there a way to calculate this probability cleanly without brute force?

Let $L$ and $a$ be positive integer constants. I will flip a coin $L$ times. Let $n_1, \ldots, n_s$ be the count of consecutive runs of the same flip. For example, if $L = 5$ and I flipped $HHTHT$, ...
Vincent Lin's user avatar
4 votes
0 answers
65 views

Solving the Secretary Problem using Simple Math

In our computer coding class, we learned about the Secretary Problem (https://en.wikipedia.org/wiki/Secretary_problem). The goal of this problem is to find out the "optimal cutoff" for an ...
konofoso's user avatar
0 votes
1 answer
27 views

Two cards are drawn from an ordinary poker deck of 52 playing cards one by one with replacement. find Probability of getting the same number or letter

Suppose 2 cards are drawn from an ordinary poker deck of 52 playing cards one by one with replacement. Find the probability of getting the same number or letter. attempt: There are $4$ type of cards. ...
math404's user avatar
  • 447
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0 answers
24 views

2X has same distribution as sum of two iid copies of X under smooth transformation

Suppose $f:[0,+\infty)\rightarrow [0,+\infty)$ is a given strictly increasing smooth function starting from $f(0)=0$. I want to figure out under what conditions with respect to $f$ can we find ...
Fangyi Chen's user avatar
0 votes
1 answer
35 views

Prove independence of variables & find distribution

I’m struggling with the following question $$ X = A\cos\theta+B\sin\theta$$ $$Y= B\cos\theta - A\sin\theta$$ Where $A,B\sim N(0,1)$ (A and B are independent) and $\theta$ is a constant Show that $X$ ...
edster101's user avatar
1 vote
3 answers
62 views

Random variable obtained from discrete and continuous Random variables

A random variable $Z$ is obtained as follows: Given X $\sim U(0, 1)$ and $Y|(X=x) \sim Bern(x)$. If $Y = 1, Z = X$. Otherwise, the experiment is repeated until $(X, Y)$ with $Y = 1$ is obtained. It is ...
SK25's user avatar
  • 23
0 votes
1 answer
30 views

Finding average tries to get a positive result while knowing probability of getting a positive result in n tries

so if I have a test that returs either a positive or a negative result but the result depends on the number of tries, is there a way to know how many tries on avarage would it take to get a positive ...
Limofeus's user avatar
1 vote
1 answer
37 views

Memoryless property of the geometric distribution

Proving that for a geometric random variable with probability mass function $(1 - \pi)^{x - 1}\pi$ for $x \in \mathbb{N}$ and $0$ otherwise the memorylessness property $\mathbb{P}(X > s + t \; | \; ...
Riccardo Iorio's user avatar
1 vote
0 answers
20 views

How to compute probability of event E happening and having K occurences of E within the last M trials

When sampled within a time of N trial successive events, I know the probability of success P of an event E. This probability P of event E is constant and doesn’t change with each sampling trial. I ...
Guy B's user avatar
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0 votes
0 answers
46 views

I solved a problem but I'm not sure where I am wrong. - game of craps

I'm a student studying probability and got a problem while solving the following: The game of craps is played as follows: a player rolls two dice. If the sum of the dice is either a 2, 3, or 12, the ...
squid__'s user avatar
  • 55
2 votes
1 answer
24 views

Derive the 0-1 law from a functional equation of a martingale limit

I am trying to understand a proof given by Jabbour-Hattab in the paper "Martingales and Large Deviations for Binary Search Trees". In this paper, we consider the limit of a non-negative ...
CampFire's user avatar
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0 answers
53 views

Probability that two paths don't intersect

This question is inspired by this other question but it could be simpler. I think also that by using an answer here and the inclusion-exclusion principle we can get a solution for the linked question. ...
Fabius Wiesner's user avatar
-2 votes
0 answers
32 views

a limit of an integral containing a probability with a $0 \infty$ product [closed]

I'd like to compute thie limit, where p(x) is a probability $\lim_{{y \to 0}}\frac{1}{y } \int_{0}^{y} p(x) \, dx$ $p(x)\in [0,1]$. So the function inside the integral is bounded but not continuous ...
dan_k's user avatar
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0 votes
2 answers
27 views

Defining majorization of vectors using doubly stochastic matrix

On Wikipedia, it says that given two non-increasing vectors $a,b\in\mathbb{R}^d$, $a$ majorizes $b$ if and only if there exist a doubly stochastic matrix $D\in\mathbb{R}^{d\times d}$ such that $b=Da$. ...
Sagi Buchbinder Shadur's user avatar
0 votes
0 answers
37 views

Karhunen–Loève theorem proof

I was looking at Karhunen–Loève theorem on Wikipedia. It consider a square-integrable random process $X_t$ indexed over the closed interval $[a,b]$. In the proof an orthonormal basis is taken in $L^2([...
Carlo C's user avatar
  • 25
-1 votes
2 answers
27 views

Strange consequence of linear combination of normal distribution

I have trouble with a consequence of this theorem Let $X_1, ..., X_n$ be independent normal distributions with expected value $\mu _i$ and variance $\sigma _i ^2$. Then the random variable $Y = \sum ...
selenio34's user avatar
  • 156
0 votes
0 answers
20 views

a limit of an integral containing a probability [closed]

I'd like to compute thie limit, where p(x) is a probability $\lim_{{y \to 0}}\frac{1}{y } \int_{0}^{y} p(x) \, dx$
dan_k's user avatar
  • 1
1 vote
1 answer
77 views

Why does it look like the probability exceeding 1? How do you solve this problem?

This problem comes from a swedish probability and statistics book and reads as follows: "A company manufactures a type of appliance with a one-year warranty. Due to different uses of the devices, ...
lill_m8's user avatar
  • 11
3 votes
1 answer
59 views

How to Determine Independence of Events Using Probability

Let's get a deck of cards. a) Are the events "the drawn card is a heart" and "the drawn card is an ace" independent? b) Suppose we have removed the following cards from the deck: ...
Need_MathHelp's user avatar
1 vote
1 answer
46 views

Form of joint distribution of Markov model

Consider a time series model: $$ \begin{align} \vec x_t &= \mathbf A \vec x_{t-1} + \vec w,\ \text{where we have i.i.d.}\ \vec w \sim N(\vec 0, \mathbf Q) \\ \vec x_0 &\sim N(\vec \mu_0, \...
Rui's user avatar
  • 13
1 vote
0 answers
54 views

What number appears most often in an $n \times n$ multiplication table and How many times does it appear?

Recently, I have considered a problem: $$\sum_{k=1}^{n^2}\frac{a_{n,k}}{n^2}X_{k} \stackrel{a.s.}{\longrightarrow} ?$$ where $$a_{n,k} = \#\left\{(i,j) \mid ij = k, i, j \in [n]\right\}$$ and ${X_{k}}...
Defender's user avatar
0 votes
1 answer
21 views

Can Chernoff Bound Theorem be applied to functions of independent random variables

Let $X_1, X_2, \ldots X_n$ be independent random variables, can we apply the Chernoff Bound Theorem to bound $E\left[\sum_i \Theta( \widehat{X_{i}}))\right]$ if $\Theta( \widehat{X_{i}}) : \widehat{...
ephemeral's user avatar
  • 227
-1 votes
0 answers
20 views

Need some further explanation on this question. [closed]

An instructor gives her class a set of 10 problems with the information that the final exam will consist of a random selection of 5 of them. If a student has figured out how to do 7 of the problems, ...
Abhishek Singh's user avatar
1 vote
1 answer
70 views

$E[X|X^2+Y^2] = 0$ when $X$ and $Y$ are independent standard normals.

I need to show that $E[X|X^2+Y^2] = 0$ when $X$ and $Y$ are independent standard normals. $E[X|X^2+Y^2] = \int \frac{f(x, x^2+y^2)}{f(x^2+y^2)} \frac{1}{\sqrt{2\pi}}e^{\frac{x^2}{-2}}dx$. We want to ...
Adam's user avatar
  • 181
0 votes
1 answer
19 views

Setting sampling probability when sparsifying a non-negative weighted graph

Given a set of $mn$ non-negative edges, with what probability should one keep every edge $w_{ij}$ if we want $\sim pmn$ non-zero weights in our sparsified and every edge is sampled with a probability ...
meowcaroons's user avatar

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