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Questions tagged [probability]

This tag is for basic questions about probability and for questions in which one wants to calculate a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using measure theory), please ask under (...

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Connecting information entropy to the prime number theorem for compression of numbers?

Prime number theorem states how often primes occur (approx. how densely they are distributed). The Shannon theorem of information entropy gives us a lower bound of how much data is at least required ...
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Conditional independence and conditional expectation for joint Gaussian vectors.

I am reading Hajek's book "Random Processes for Engineers". In example 4.5 it says: Let $X,Y,Z$ be joint Gaussian vectors. Then $X$ and $Z$ are conditionally independent to each other given $Y$ if and ...
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Probability of discovery of new values in one to n distinct element with repetition one to n

Let's say S = {1, 2, ..., n} and S2 = [1, 2, 2, 3, 3, 3, ..., n_1, ...n_n] A random function ...
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1answer
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How to find probability in case of RVs?

I have this question: There are 120 students in McDonalds, and 120 meals to serve them. 60 of the meals are Big Macs while the other 60 are Cheese Burgers. Every student, independently of the others, ...
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Probability of two shooters A and B that are shooting separately targets

Two shooters A and B are shooting two separately targets, Probabilities of target hitting are 0.3 For A and 0.6 For B. Calculate these probabilities : A) At least one shooter hits the target: My ...
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Exponential distribution convergence

Let $(X_n)_{n}∈_{\mathbb{N}_+}$ be a sequence of independent random variables such that Xn ∼ Exp{n} and let $Y_n :=\frac{1}{n} \sum_{i=1}^{n}X_{i}$ for n$\in \mathbb{N}_+$ Does the sequence ($Y_n$) ...
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How to know if a probabilistic function is increasing or decreasing without actually computing it?

Problem Setup Consider the following optimization problem \begin{equation*} \begin{aligned} & \underset{P_{a},\lambda_{a}}{\text{maximize}} & & P_{cov}(P_{a},\lambda_{a}) \\ & \text{...
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1st Yr Probability: Question about the Poisson Process in my book, what is $P(N(h) = 0)$

Background I'm reading Sheldon Ross and he gives two proofs of the same result: that given some assumptions, $N(t)$ has a Poisson distribution with mean $\lambda t$. The first proof is in chapter 4 ...
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How would I solve a classic Bayes Theorem problem using a probability tree?

For example Assume that a test for a disease gives a positive result for 2.5% of people who do not have the disease, but does not test negative if the person has the disease. What is the ...
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100 prisoners 100 boxes variant

Here is the classic 100 prisoners 100 boxes problem: The director of a prison offers 100 death row prisoners, who are numbered from 1 to 100, a last chance. A room contains a cupboard with 100 ...
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If $Pr(A)+Pr(B) > 1$ then what is the maximum and minimum value for $Pr(A\cap B)$?

If $Pr(A)+Pr(B) > 1$ then what is the maximum and minimum value for $Pr(A\cap B)$? I guess that the minimum should be when $Pr(A \cup B) = 1$ because then you have that $Pr(A \cap B) = Pr(A)+Pr(B)-...
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bound on the moment generating function

Let $X$ and $Y$ be discrete random variables. Is there a known class of joint distributions $p(x,y)$ which satisfies the following property: $$\mathbb{E}\left[ e^{\lambda X} e^{\lambda Y} \right]&...
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continuity correction of normal approximation

In a game you win 10 with probability $\frac{1}{20}$ and lose 1 with probability $\frac{19}{20}$. Approximate the probability that you lost less than $100 after the first 200 games.How will ...
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Bounding Survival Probability of an Asymmetric Random Walk by a Symmetric one

Consider two random walks that start from point $x=0$ and time $t=0$ and move either to right $x+1$ or left $x-1$: 1) Walker 1's first move is with equal probability to the right/left. However, ...
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Weird visual result for MLE of normal distribution

I am trying to visualize the log likelihood estimator for normal distribution via python but in vain. I am not sure if its a python issue or formuala issue in the code. Can you kindly have a look? ...
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1answer
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division of two PDFs

I am trying to solve the following question. I have come up a solution to this but need to know that is my approach correct in solving the question. The question is as follows A Test is 1 hour long, ...
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2answers
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When does $E[X] \leq E[Y]$ imply $E[X^2] \leq E[Y^2]$?

Assume $X, Y$ are two non-negative random variables. When does $E[X] \leq E[Y]$ imply $E[X^2] \leq E[Y^2]$? Always? I think the conclusion is not always true.
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A game of probability

An unbiased die having the numbers 1,2,3,4,5,6 is rolled 4 times. What is the probability that the minimum face value is 2? According to my reasoning the answer should be $\frac{5^4}{6^4}$ because we ...
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Determine the probability mass function, given fifty modems, one thousand customers and an independent p of 0.01

This is Problem 2.4 from Tsitsiklis, Bertsekas, Introduction to Probability, 2nd edition. My understanding is that represents disjoint outcomes. is the probability that X takes on the value 3. ...
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1answer
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How to geometrically interpret conditional expectation property tower rule??

I have a question on how to interpret conditional expectation its properties geometrically. There are two properties of conditional expectation in particular that I’m trying to interpret: Given a ...
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Why do we use the Poisson Probability Mass Function any longer?

My understanding is the Poisson Distribution is helpful to model experiments where there are many independent trials of Bernoulli experiments and each trial has a small chance of success. It is used ...
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Metric on Space of Measures [on hold]

Hi can you help me with the below exercise. I am having some problem in trying to solve it. Assume that $S:(\Omega,\zeta,Pr(\cdot))$ is a valid probability space where $\Omega $ is a set in the ...
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1answer
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Why does the binomial formula use multiplication?

This is Problem 2.2 from Tsitsiklis, Bertsekas, Introduction to Probability, 2nd edition. You go to a party with 500 guests. What is the probability that exactly one other guest has the same ...
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Lottery question-probability of pairing Mega numbers on two different tickets

Sorry to burden you with ASLQ (another stupid lottery question), but y'all are better at probability calcs than I. Last Tuesday I bought $10$ Mega Millions throws in $2$ tickets of $5$ each. The ...
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Calculating MLE for unknown parameters

Given a random variable $X$, assume it takes on the values $\{1,2,3\}$ with probabilities $Pr[X=1] = p_1$, $Pr[X=2] = 2p_1$, and $Pr[X=3] = p_2$. $p_1$ and $p_2$ are unknown parameters we must ...
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Adding a probability to a number (probability theory)

In the textbook of stochastic process, it says Suppose there is only one server. Let $L$ be the long-run average number of customers in the system. Let $L_Q$ be the average queue length in ...
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Probability that k or more of n independent Bernoulli events occurring where each $p_i$ is different

Basically what the title says I have n independent events (in reality 15) Each has some probability $p_i$ I want to know the probability that $k$ or more occur (in reality 8) I was reading ...
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The expectation of number of times to sample all balls in the black box

I got a probability problem, suppose you have a black box, and there are 4 balls in the box which have a different colour, red, blue, green and black. Each time you get a ball and put it back, how ...
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Chebyshev's Inequality Estimation

The problem says: X is the number of Heads obtained when tossing a fair coin 1000 times. Estimate the probability P(480 ≤ X ≤ 520) using Chebyshev's Inequality. I got that E(X)=1/2 and Var(X)=1/...
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A geometric probability question

Find the probability of distance of two points ,which are selected in $[0,a]$ closed interval, is less than $ka$ $k \lt 1$ What did I write : $P(A)$ = (Area measure of set $A$)/(Area measure of set $...
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Probabilistic properties that are invariant under a change of sample space.

This may be a standard question, but I couldn't find it anywhere. I am having trouble understanding why in many results in probability we can omit the actual structure $(\Omega, \mathbb{F}, \mathbb{P})...
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The proof of the weak law of large numbers in Billingsley's Probability and Measure

I am reading Billingsley's "Probability and Measure". In the last paragraph of page 5, the author said that "With the Riemann integral in the role of expected value, the usual application of Chevyshev'...
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1answer
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How to calculate a composition of random variables

I'm not able to start solving this problem, can you help me? N clients come in a shop in 1 hour. N ~ Poisson(q), but q~U(0,2). How can I calculate the distribution of clients coming in the shop in ...
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How can I calculate this conditional expectation?

I tried to solve this problem but I can't find an easy way to do it. I have n short boards and m long boards (twice the short one). I randomly choose one board at a time to fill my floor: if I choose ...
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What is the sample space for the experiment : each hour approximately 12 people come in the shop.

Each hour, there are approximately 12 peoples that come in the shop. Let $X$ the random variable that denote the number of people that come in the shop during one hour. What is the probability space $(...
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NON-martingale approach to ABRACADABRA problem

The well-known ABRACADABRA problem states (see D. Williams, "Probability with martingales", for example): a monkey is typing letters A-Z randomly and independently of each other, each letter ...
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Probability of limit

I have two closely related questions. One of them, I think, I could solve but the other I do not even really understand what is asked. Here we go: Let $\mathcal{A} = \{a_1,a_2,...\}$ be a countable ...
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Randomized Submatrix of a Sparse Matrix

I have a sparse square matrix $A$ with size $n \times n$ and number of nonzero entries $nnz$. The goal is making a sub-matrix $B$ with $s$ nonzeros which are randomly chosen from $A$. Duplicates are ...
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Multiple Raffle Risk/Reward Optimization Problem

This question ponders whether there can be a statistically backed process to determine a distribution of tickets, among a set of raffles, that would probabilistically maximize the net value likely to ...
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I need to write python code for one conditional probability question ?Need idea how i can write

Question is There are 2 identical boxes which have money in it (One has an amount twice than the other ). You may pick one box and keep the money it contains. Having chosen a box at will, but before ...
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1answer
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Conditional probability of a Royal flush

What is the probability of drawing a 10, jack, queen, king, and ace of all the same suit from 5 random cards given that one card drawn is an ace of spades? Solution Let $R$ denote the event that the ...
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1answer
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Conditional probability of a Poisson process

The question is as follows: Consider a Poisson process $\left \{ N_{t} \right \}_{t\geq 0}$ with $\lambda =3$ events per hour. What is the probability that $2$ events occur in the first hour and $...
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1answer
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Ordered Chain Rule of Probability

For the chain rule of probability the order of variables does not matter and sets are not ordered as of themselves anyway. So therefore $P(A | B) P(B) = P(A \cap B) = P(B|A) P(A)$. However, I cannot ...
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2answers
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Law of large numbers for sum

Liz is standing on the real number line at position 0. She rolls a die repeatedly. If the roll is 1 or 2, she takes one step to the right (in the positive direction). If the roll is 3, 4, 5 or 6, she ...
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1answer
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Binomial random distribution & conditional probability

I have solved following two problems related to binomial distribution probability and conditional probability, but I am unsure that both are correct; especially for c, I have written down If the ...
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conditional probability of Dirichlet multinomial model

For a Dirichlet multinomial model with m = (1, 1, 1, 1) for A = {a, b, c, d}, what is the probability of P(x = a|baaa)? I actually don't understand what does a|baaa means, I tried searching online ...
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Statistics and probability - the basics

With $x$ assuming random values if $1$, $2$, or $4$ and if you know the expected value of $x$ is equal to $2.7$. If you are only given one other piece of information $P(X=1) = 0.3$ then how can you ...
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Rate of Change in the variances in PDF

I have four data sets with probability density function (PDF)s as shown in the figure. Each of the PDFs has different mean (near to zero) and different variance. I want to derive the rate of change in ...
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1answer
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Binomial exercise with different variances? Is it possible?

Good evening, I honestly can't understand why in this excercise the variance is computed normally but then... In this one the variance is computed by dividing by n instead if multiplying by it as in ...
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Integrable random variable uniformly integrable [on hold]

Let Y be an integrable random variable. Prove that $E(Y|$ $\mathcal{G}$: $\mathcal{G}$ is a sub-$\sigma$-field of $\mathcal{G)}$ is uniformly integrable