Questions tagged [probability]

For basic questions about probability and the questions associated with the calculation of probability, expected value, variance, standard deviation, or similar statistical quantities. For questions about the theoretical footing of probability (especially using [tag:measure-theory]), ask under [tag:probability-theory] instead. For questions about specific probability distributions, use [tag:probability-distributions] instead.

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Number of questions to attend to pass a multiple choice test

There are 100 multiple choice questions in a test, with 4 options each. The probability of choosing a right answer for a question is 0.25 Each right answer gets 4 marks and each wrong answer gets -1 ...
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What is the probability of finding 2 in set of primes?

Let $$S=\{p :p\in\mathbb{P}\}$$ Be the set of primes. So $2$ is also a member of it right? $$P[\text{ finding 2 in $S$}]=\frac{1}{\infty}=0 ??$$ Does that mean that probability of finding $2$ from the ...
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Is there any difference between the two limits in $(1)$ and $(2)$ as above?

I read a a paper about statistics and machine learning ("High-dimensional Asymptotics of Langevin Dynamics in Spiked Matrix Models"). Assume that $\{(Y_0^i, \sqrt{N}U^i)\}_{1\le i\le N}$ are ...
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You and your friend are each dealt two cards: hers face up and yours face down. Which of the following scenarios are you more likely to have a pair?

I am self-teaching probability, could someone please help me determine if I've solved this problem correctly, as I don't have the answers to these exercises. The following question is taken from ...
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Finding a value to prove a probability mass function.

Ive been given the probability mass function: Pr(X = x) = A(6.75x - x^2) (equation 1) with x ∈ {1,2,3,4,5} and I need to go about finding the value for A so that the probability mass function ...
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Gibbs sampling from Multinomial distribution

I need to use Gibbs sampling to sample from $(X_1, X_2, \ldots, X_n)$ that is distributed according to Mult $\left( N, 1/n, \ldots, 1/n\right)$. For this I need to compute the conditional pmf ...
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Why is it impossible to solve the heat diffusion/conduction equation as a function of time in 2D and 3D situations?

The current literature provides solutions for heat diffusion/conduct PDE only for a 1D geometric space plus time dimensions. Why there are no 2D and 3D spatial dimensions plus time. Are mathematical ...
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CLT and unknown probability of success

Suppose that I distribute flyers on the street and people, when approached, take them with the same probability $p$. On average, I need 5 hours to distribute 2500 flyers and with confidence 0.9 it is ...
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Why are Some Probability Distributions "Used More Often" for Certain Things?

I have had this question for a long time - Why are Some Probability Distributions "Used More Often" for Certain Things? I understand that for certain problems, the type of probability ...
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Find $P(W_1 > 1 \cap W_2 < 1)$ where $W_t$ is a standard Brownian motion.

Let $W_t$ be a standard Brownian motion. Find $P(W_1 > 1 \cap W_2 < 1)$. I tried to do something on my own, but because I'm new to Brownian motion, I'm stuck at the beginning of the calculations....
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Expected number of rounds to reach an intersection size

Consider the set $U=\{1,..,n\}$, and $A=\phi$ and $B=\phi$ are two empty sets to begin. Perform the following step. Choose $m$ elements from $U$ uniformly at random (with replacement) and add to $A$. ...
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a symmetrical die is thrown, then a coin is thrown as many times as the die indicates .

Exercices : a symmetrical die is thrown, then a coin is thrown as many times as the die indicates points. let $X$ be the number of Tails obtained . Determine $E(X)$ My attempts : First it's clearly ...
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flip a coin, if head, Z = X, if tail, Z = Y. X, Y if normal distribution (0, 1). what is the probability density function of Z? [closed]

I am trying to solve the following probability problem: Suppose $X\sim N(0, 1)$ and $Y\sim N(0, 4)$. A fair coin is flipped independently from $X$ and $Y$ and define $Z = X$ if if head comes up and ...
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What is the probability that Camilla and Cameron are paired with each other?

Textbook problem: A teacher with a math class of 20 students randomly pairs the students to take a test. What is the probability that Camilla and Cameron, two students in the class, are paired with ...
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How to proof Lip-constant weakens when size of function class approaches to infinity?

Taken from paper "A Universal Law of Robustness via isoperimetry" by Bubeck and Sellke. Theorem 3. Let $\mathcal{F}$ be a class of functions from $\mathbb{R}^{d} \rightarrow \mathbb{R}$ and ...
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What is the probability that no one waits for remainder? [closed]

The problem is: There are 20 people in a queue. 10 of them have 10 euro and the others have 20 euro. Everyone wants to buy a ticket that costs 10 euro. What is the probability that no one waits for ...
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What is the criterium for matrices of real numbers to be a variance-covariance matrix for some random variables

Variance-Covariance matrix of random variables $X_1, X_2, \dots, X_n$ is $\Sigma \in R^{n\times n}$, where $[\Sigma]_{i, j} = Cov(X_i, X_j)$. I'm curious if most of matrices can be the variance-...
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Proving convergence of $\frac{1}{n}W_n^\top W_n$ where $W_n\sim \mathrm{Mul}(n, \frac{1}{n}1_n)$

I want to prove $$\frac{1}{n}W_n^\top W_n \to2 \qquad \text{in probability.} $$ $W_n$ is a $n\times1$ vector st $W_n\sim \mathrm{Multinomial}(n, \frac{1}{n}1_n)$. I tried to find an upper bound of the ...
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If $X$ and $Y$ are independent, is it true that $\mathbb P (X+Y>x | X+Y>0) \geq \mathbb P (Y>x | Y>0)$ for every $x>0$?

$Y$ is a random variable with symmetric distribution around 0 and both random variables are continuous with existing density It seems true, because the random variable $X$ should just bring further ...
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Markov umbrellas

"I have $4$ umbrellas randomly distributed between my house and my office. Each day I go from my house to the office and back. If it's raining, i will take an umbrella in my way to the other ...
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How to deal with overflows while using either Welford's or Youngs-Cramper algorithms for variance?

I'm trying to use either of the mentioned algorithms to calculate sample variance. However I ran into this problem -- Welford's algorithm uses for accumulations a sum of differences of elements and ...
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Bound on entropy of n variables

Let $X_1,X_2,\dots,X_n$ be arbitrary discrete random variables. Prove that $$H(X_1^n)\leq\frac{1}{n-1}\sum_{i=1}^nH(X_1^{i-1},X_{i+1}^n),$$ where for $j\geq i$, $X_i^j$ denotes the block of random ...
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Finding the generating function of the random variable given by $P(X=m)=\frac{1}{m(m+1)}$

Finding the generating function of the random variable given by $P(X=m)=\frac{1}{m(m+1)}$. So I've tried using the definition: $G_X(s)=\sum_{n=0}^{\infty}s^nP(X=m)=\sum_{n=0}^{\infty}s^n\frac{1}{n(n+1)...
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Markov Fields as Markov Chains

I am currently looking through the literature on a topic that has so far appeared only in Georgii's Gibbs measures, chapter 10. The question really is, how do I construct a Markov Chain from a given ...
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how to find expected number of claim [closed]

An automobile insurer studied the number of claims filed by a policyholder in a given year. The probability distribution function was as follows: Number of claims $(x)$ - Probability $p(x)$ $0.72$ $...
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Can anyone please assist me with this question using inverse Laplace [closed]

[** Inverse of Laplace**][1] [1]: https://i.stack.imgur.com/l7c1Z.jpg Inverse Laplace
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Expected number of self-avoiding walks of a given length in a random graph

Let us consider the following random locally finite (i.e., every vertex has a finite degree) graph: $\mathbb{Z}^d$ ($d\geq2$) is the set of vertices and any two vertices $v$ and $w$ have an edge ...
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Finding stationary distribution of random process

Suppose we are given $x_t, \bar{x_t}, t\in \mathbb{Z_+}$ independent 2-states $\{0, 1\}$ Markov chains with positive transition probabilities. Initial states are $x_0 = 0; \bar{x}_0 = 1$. For which ...
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Scaling of moments of normal distribution/Brownian incremenrs

Let $B_t$ be Brownian motion for $t\ge0$ so that $B_t-B_s \sim N(0,t-s)$. There are a few formulae out there that show how the $p$-th moment of $E[(B_t-B_s)^p]$ is calculated. My question is: how do ...
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A Sequence of real-valued random variables is Cauchy almost surely and convergent in probability

Let $\{X_n\}_{n\geq0}$ be a sequence of real-valued random variables such that $$\text{Pr}(\{\omega: \{X_n(\omega)\}_{n=1} \text{ is Cauchy } \})=1$$ and for all $\epsilon>0$, one has $$\lim_{n \to\...
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Properties of Continuous Time Markov Chains

In school, I only learned about about "Discrete Time Markov Chains" - in Discrete Time Markov Chains, transitions can only happen at fixed time points. For example, suppose there are three ...
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Transformation of the histogram

I have two classes in my problem: 'missing' and 'present'. These are the pixel intensities of some pictures. The histogram of their values you can see below. I would like to find some transformation ...
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The probability of getting non-integer solution of SLE if $b = \text{rand}(0, 1)$

Consider a system of linear equations $Ax = b$, where $A \in \mathbb{Z^n}$, $\text{det}(A) \neq0$. We assign a random vector of 0, 1: $b = \text{rand}(0, 1) \in \mathbb{N}$, uniformly distributed, ...
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Checking for independence in simple Bayessian network

Having the simple model, for example $P(W\mid R)= 0.8$ Is probability of wet grass given rain and $P(W\mid S)= 0.6$ Is probability of wet grass given sprinkler, It is correct, that the $P(W \mid S,R)$,...
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3 votes
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527 views

$\sum_{A\in2^\Omega}P(A)=2^{|\Omega|-1}$ for probability space $(\Omega,2^\Omega,P)$ with finite $\Omega$

I'm looking for a combinatorial argument to complete a proof (below) of the following: Claim: If $(\Omega,2^\Omega,P)$ is a probability space with finite $\Omega,$ then $\sum_{A\in2^\Omega}P(A)=2^{|\...
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A circle and the probability as a function of time

I am trying to solve the following problem.Supposing that we have a circle of radius R which at t=0 does not contain any sphere as we had set a border so that no sphere was allowed to enter in it.At t=...
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Let $X$ be an exponential random variable and $\Bbb{P}(X \in A) < \Bbb{P}(X \in B)$. Is $\Bbb{P}(aX \in A) < \Bbb{P}(aX \in B)$ for $a > 0$?

Let $X$ be an exponential random variable (say with mean 1) and $\Bbb{P}(X \in A) < \Bbb{P}(X \in B)$ for two events $A, B \subset\Bbb{R}$. Is $\Bbb{P}(aX \in A) < \Bbb{P}(aX \in B)$ for $a > ...
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How objectively do we evaluate likelihood of two very rare events? [closed]

I've been pondering the concept of objectiveness to evaluate likelihood and predictability of extremely rare events. I'd like to demonstrate my questions over trivial hypothethical examples below. ...
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What is the distribution of this function of a table of random coin-flip variables

I'm working on a problem in the domain of cellular automatons and I have managed to get some results conditional on the asymptotic behavior of the following function of a table of $0$'s and $1$'s: For ...
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Expected number of fixed points in a permutation by brute force

I'm trying to compute the expected value of fixed points in a permutation without using the linearity of expectation. I tried the following: $$\sum_{i=1}^{n} i\frac{(n-i)!\binom{n}{i}}{n!}$$ However, ...
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1 vote
1 answer
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Explanation about this probabilistic proof that leads to $log_RN$

I am reading a proof based on (probabilistic) analysis on how a specific data structure performs for the case of search miss. The base assumption is that the probability that each of the $N$ keys in a ...
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2 votes
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How to find the probability of drawing a certain two cards from a pack

Could I please ask for help on the last part of this question: Two cards are drawn without replacement form a pack of playing cards. Calculate the probability: a) That both cards are aces b) that one ...
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What is the variance of the error term for the regression of the sample mean of $x$ on the sample mean of $y$ that is heteroskedastic?

I am having difficulty finding an appropriate expression for the variance of $\bar{e}_{k}$ in $$ \bar{y}_{k} = \tilde{\beta} \bar{x}_{k} + \bar{e}_{k} $$ where $i = 1, \dots, n $ individuals from a ...
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Probability of Invalid document in a large data set

I'am auditing a very large data set of documents. A document can be Valid or Invalid. Checking a document is computationally ...
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4 votes
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Multiplication of two random matrices over a finite field

Consider a matrix $\mathrm{X}$ sampled uniformly at random from the set of all rank $r$ matrices over $\mathbb{F}_q^{m \times n}$ and a matrix $\mathrm{Y}$ sampled uniformly at random from the set of ...
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There are $3$ red and $5$ black balls in bag$A$and $2$ red and $3$ black balls in bag $B$ . One ball is drawn from bag $A$ and two from bag $B$ .

There are $3$ red and $5$ black balls in bag$A$and $2$ red and $3$ black balls in bag $B$ . One ball is drawn from bag $A$ and two from bag $B$ . Find the probability that out of the three balls ...
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For the probability triple $(\Omega, \mathcal{F}, \Bbb{P})$, a random variable $X$, and a function $g$, is $g(X)$ automatically measurable?

For the probability triple $(\Omega, \mathcal{F}, \Bbb{P})$, a random variable $X: \Omega \to D$, and an arbitrary function $g: D \to E$, is $g \circ X$ also measurable and thus a random variable? I ...
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There are $3$ urns $A,B$ and $C$. Urn $A$ contains $4$ red balls and $3$ black balls. Urn $B$ contains $5$ red balls and $4$ black balls. Urn $C$

There are $3$ urns $A,B$ and $C$. Urn $A$ contains $4$ red balls and $3$ black balls. Urn $B$ contains $5$ red balls and $4$ black balls. Urn $C$ contains $4$ red balls and $4$ black balls. One ball ...
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3 votes
2 answers
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Finding the expected value of $\dfrac{X}{Y}$

Below is a problem I did. I believe I did it correctly and I am hoping that somebody here can either confirm that I did it right or tell me where I went wrong. Problem: Let $X$ be a random variable ...
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$\mathbb{P}\left(X\geqslant\frac{2\alpha}{\lambda}\right)\leqslant \left(\frac{2}{e}\right)^{\alpha}.$

Using $$\mathbb{P}(X\geqslant x)\leqslant e^{-tx}M_{X}(t),\text{ }t\geqslant0,$$ show that in the particular case that $X\overset{\underset{d}{}}{=}\Gamma(\alpha,\lambda)$, $$\mathbb{P}\left(X\...
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