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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 (probability-theory) instead. For questions about specific probability distributions, please use (probability-distributions).

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probability of a non-central xhi-squared RV to be smaller than a chi-squared RV

Given $\alpha > 0$ and $\beta > 0$, I am looking for a bound on the probability $$\text{Pr}\left( (\alpha + X)^2 \leq \beta Y^2 \right)$$ where $X$ and $Y$ are distributed $\mathcal{N}(0,1)$. ...
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Expected value of a function of multiple random variables

I was wondering what is the definition of the expected value of a function of two or more random variables? And how does one show it is consistent? So if you have a random variable $z = g(x,y)$ ...
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Variance of Product of Ind. Variables

Whats wrong with my approach to answer the following question? The number of customers arriving to a fast food restaurant between 7 am and 9 am has the Poisson distribution with mean 40. Suppose that ...
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6answers
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1 in 4, two chances, only one needs to hit

Currently in disagreement with a friend over the probability of a specific situation. My friend believes that if you have a 1 in 4 chance of something happening, getting two opportunities at it makes ...
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can a Markov chain have a periodic and transient state?

I want to say that in a Markov chain it is not possible for there to exist a state that is both transient and periodic. Here are the definitions I am working with. Let $P$ denote the transition ...
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Weak convergence of $(n^{-1}T_{k,n})_{n \geq \lceil k/2 \rceil}$

After a busy day, Santa asked $2n$ of his coworkers for dinner. There were $k$ elves and $2n-k$ gnomes. Santa and his guests sat randomly at the round table. Let $T_{k,n}$ be the number of creatures ...
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2answers
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Probability measure domain

On a measurable space $(\Omega, F)$, where $\Omega$ is a set of outcomes and $F$ is a $\sigma -$field, what exactly is the domain of a probability measure $P$? If it's a specific $\sigma -$field such ...
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1answer
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Cumulative distribution function and Brownian motion

I am having troubles with one exercise. Your help will be great! Let be $\Phi$ the C.D.F of a standard gaussian random variable (i.e. mean = 0 and variance = 1) and $(B_t)_{t\geq0}$ a Brownian motion ...
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Markov Chain generated by iterations of functions

$X_n$ is a Markov Chain on (..,-2, -1, 0, 1,..) obtained by random iterations with functions $f_1(x)=x+2$, $f_2(x)=x−1$, $f_3(x)=0$. In each iteration step we choose function to iterate with equal ...
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1answer
16 views

Flip an unfair coin

An unfair coin has an probability of heads on a single flip $p=\frac 14$, the coin is flipped n times, and the probability of getting 2 heads is the same as the probability of getting 3 heads, what is ...
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2answers
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What is the probability that a set of nine children will contain three or fewer girls?

I can't decide if it is 4/9 because there is the possibility of there being 0, 1, 2, or 3, or if it is 25% using combinations. Thanks!
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1answer
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What is the probability that I actually won the game

Question I have two blue dice, with which I play a game. If I throw a double six (i.e. if I get two six on both the dices) then I win the game. I separately throw a red die. If I get a one, then I ...
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1answer
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Sufficient condition for a Markov chain to be Aperiodic

If I want to prove that a Markov chain is aperiodic, then if I can show that $P(X_{n+1}=i\mid X_n=i)\gt 0$ $ \forall i$. Then can I say that the chain is aperiodic?
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How to accordingly adjust the % of a & b if I manually adjust c,d,e [on hold]

I have 10 different items all with a different share that totals to a 100%. I have manually adjusted some items' shares to either increasing or decreasing. What is the calculation to do so that all ...
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1answer
13 views

Convergence in mean of a series of random variables

I'm stuck on the following proof: Let $(X_1,.. X_n)$ random variables so that $E(Xi) = \mu$, $V(Xi) = \sigma^2$ final for all $1 \leq i \leq n$. It is also given that for all $i \neq j$, $Cov(Xi, Xj) ...
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Independent, exponentially distributed variables probability

The expected lifetime of a RV of type 1 is 2 years and the expected lifetime of a RV of type 2 is 1 year. Both are exponentially distributed and independent. I was therefore wondering how to calculate ...
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What is the probability two independent exponentially distributed random variables being greater than one another?

I have two exponentially distributed independent random variables A and B. The expectations of $A$ and $B$ are $2$ and $1$ respectively What is the probability that $B > A$?
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Is it true that $\lim_{n \to \infty} {(P(\forall i,j\leq n \text{ } [X_i, X_j] = e))}^{\frac{1}{n}} = P(X_1 \in Z(G))$?

Suppose $G$ is a group. $\{X_n\}_{n = 1}^{\infty}$ is a sequence of i.i.d. random elements of $G$ satisfying the condition that $$\forall H \leq G \text{ } P(X_1 \in H) = \begin{cases} \frac{1}{...
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2answers
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Reading Binomial Tables

While reading a table of cumulative binomial probabilities, if I need to find the probability of, for example, exactly 4 successful events happening and all the rest failures occurring, how would I go ...
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0answers
15 views

probability that a random variable is greater than a limit in given ordering of random variables

I am currently working on a modified version of the classic greedy algorithm for the 0/1 knapsack problem. Suppose that one has $N$ items with given weights and profits that are iid random variables ...
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1answer
25 views

Definition of ergodicity and ergodic process

I am confused by the definitions of ergodicity in wikipedia, see formal definition here which says that a measure-preserving transformation $T$ is ergodic if for every event $E$, $T^{-1}(E) = E$ ...
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2answers
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Question on two independent random variables under Poisson distribution

$X$, $Y$ are independent random variables, $X$ ~ Poiss$(λ)$, $Y$ ~ Poiss$(μ)$. How to find: a) $P( X > 0 | X+Y )$ b) $E( X | X+Y) $ ?
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1answer
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Calculate Binomial Probability for inequality

How could the binomial probability be calculated for a case of $P(X \geq 2)$ for a given value of $n$ and $p$ ?
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Let $X_1,…,X_n$ iid with density function $f_X(x) = \frac{1}{x^2} \mathbb{1}[1,\infty]$

Let $X_1,....,X_n$ iid with density function $f_X(x) = \frac{1}{x^2} \mathbb{1}[1,\infty]$. Let $M_n = max\{X_1,...X_n\}$ I want to study the law convergence of $\frac{M_n}{n}$ and of $\frac{n}{M_n}$....
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1answer
34 views

$P(T_{1}<T_{2})=\frac{\omega_{1}}{\omega_{1}+\omega_{2}}$

Supposing that $ T_{1} $ and $T_{2}$ are independent and exponentially distributed, with parameters $\omega_{1} $ and $\omega_{2}$ respectively. Then, $$P(T_{1}<T_{2})=\frac{\omega_{1}}{\omega_{1}+...
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0answers
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$\dfrac{1}{2d}\mathbb{E}N(x_{i}) \leq N(x^\prime_{i}) \leq \mathbb{E} N(x_{i})$.

1) Consider $n$ Points, $x_{1}, x_{2},...x_{n}$ distributed uniformly in $[0, 1]^d$. Term $d$ is the dimension. 2) Then, I construced a grid points $x^\prime_{1}, x^\prime_{2},...x^\prime_{n}$ that ...
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0answers
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Markov property of SDE's solution

Considering the SDE $dX_t=b(t,X_t)dt+\sigma (t,X_t)dW_t$ ($W$ is Brownian motion)  If there exists weak solution $(X,W),(\Omega ,\mathscr{F} ,P),\{\mathscr{F}_t\}$, is $X$ Markov process? I know ...
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0answers
10 views

Weibull Distribution Problem

I've been searching around various texts to find a good proof that the heavy-tailed Weibull distribution, with tail function $$\bar{F}(x)=e^{-x^\alpha}$$ is subexponential. Could anyone prove this ...
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0answers
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It is not necessarily true that if $X,Y$ are continuous then $(X,Y)$, but if they are independent?

I have a general questions. It is easy to disprove that for any X,Y continuous random variables then $(X,Y)$ is continuous. But what about if they are independent? Then I can define a new ...
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1answer
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solving random walk problem with law of total probability

A drunk man goes right with probability $P$ and left with probability $1-p$ . What's the probability he will get to $-1$ before getting to $2$ ? (The drunk man starts at 0) I tried to solve it with ...
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1answer
39 views

Integrating wrt y first and x first gives different answers.

I am solving trying to solve for the CDF of $$f(x,y)=2(x+y) , 0 \leq x \leq y \leq 1$$ I am doing it this way$$\int_{x}^{y} \bigg(\int_{0}^{x}2(x+y)dx \bigg) dy$$ And this way $$ \int_{0}^{x}\bigg(\...
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1answer
19 views

Comparing inequalities using convexity of the function

There was a question about comparing two entropies and showing that one of them is greater than or equal to other. After getting rid of the same terms for both expressions, I am left with the ...
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0answers
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Calculating the integrity of the result of a weighted voting system

I have an ensemble model, which votes over many regression systems. I give my observation to all the models and record their output. Now I have knowledge of models accuracies as follows: I know the ...
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0answers
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Countable sum of point processes

Let $(\mathbb{X},\mathcal{X})$ be a measurable space. A point process is defined as a measurable mapping $$\eta : (\Omega, \mathcal{F}) \rightarrow (\textbf{N}, \mathcal{N}),$$ where $\textbf{N}$ ...
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1answer
16 views

maximum of uniform (continuous) random variables

Let $X_1$, ..., $X_n$ be i.i.d. uniform random variables in $[0,a]$, with $a>0$. If $0<t<a$, I computed $P(\max_{1\leq i \leq n} X_i \leq a-t)$ : \begin{align*} P(\max_{1\leq i \leq n} X_i \...
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1answer
23 views

Are $P(\min(X_iY_i)-\min( X_i) \min (Y_i) \leq x)$ and $P(\min (X_iZ_i)-\min (X_i) \min (Z_i)\leq x)$ equivalent?

Imagine I have three independent random variables $X, Y, Z$ and each with positive support; Now imagine I have an iid sample of each random variable, namely $X_1,...,X_n$, $Y_1,...,Y_n$ and $Z_1,...,...
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1answer
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Compute the sum $ \sum_{n=1}^{\infty}(\frac{2}{3})^{n-1}\frac{1}{3}e^{-\frac{2n(1-t)}{t}}$

As part of a probability exam, we were required to compute the following sum: $ S=\sum_{n=1}^{\infty}(\frac{2}{3})^{n-1}\frac{1}{3}e^{-\frac{2n(1-t)}{t}}$ Our lecturer likes finding probabilistic ...
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2answers
39 views

puzzles of probabilities

A grandmother is cleaning out her garage and finds 6 items that she no longer wants to store. She has two charities that she likes to donate to, so she is going to put the 6 items into two boxes. ...
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1answer
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Probability Distributions in Statistics [on hold]

Certain tablets are packed 12 per box. If 5% of the tablets manufactured are chipped, what is the probability that a randomly selected box will be: a)be free of chipped tablets? b)have not more than ...
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On Probability & Statistics

If printers experience an error rate of 0.075 errors per page, what is the probability of finding 12 errors in a 200 page book?
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1answer
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Strange probability question found in book

I have seen one question in book and I suppose that it is not correct, question is following : If a family has three children, find the probability that two of the three children are girls. To ...
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48 views

What is the probability that the process stops?

Suppose there are $1$ white and $2$ black balls in a bag. We take two balls at a time at random from the bag. If the two balls are of different colour we throw them away but if they are of the same ...
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1answer
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A tricky but intuitive probability problem?

A bag consists of unknown numbers of white and black balls. Randomly take 2 balls out of the bag each time: If the two balls are of different colour, throw them away; If the two balls are of ...
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1answer
34 views

Use the change of variables to determine the density for a uniform distribution on $[a,b]$

Knowing that the density of a uniform random variable on $[0,1]$ is: $f_{U}=\left\{\begin{matrix} 1 & x\in [0,1]\\ 0 & x\notin[0,1] \end{matrix}\right.$ How to determine the density of a ...
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1answer
17 views

Bayesian updating with fair and unfair coins.

Given a list of coin tosses with 100,000 outcomes, suppose you know that they were generated by either a fair or a biased coin with a 51% chance of heads. How do you determine which coin it was ...
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1answer
16 views

How can I show that P is a probability measure on (\omega,sigma(X)) and how do i find the distribution of X?

I have already done problem 1, problem 2 references #1. I need help showing that P is a probability measure and how to find its distribution
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1answer
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Is this sequence almost sure convergent?

Consider the sequence of independent random variables $\{X_n\}$ such that $$\begin{align} P(X_n = 1) &= 1/n \\P(X_n = 0) &= 1 - 1/n \end{align}$$ I saw this as an example of convergent in ...
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1answer
24 views

method of moments and maximum likelihood estimators

I'm looking to find the estimate of $\mu$ for $n$ data using the method of moments and the maximum likelihood for the pdf given by $f(x) = \begin{cases} e^{-(x-\mu)}, & \text{if} \, x \geq \mu \\ ...
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0answers
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Asymptotics of expected value of maximum of normal variables

Let $X_n = \max (|Z_1|, \dots, |Z_n|) $ where $(Z_i)$ are i.i.d. standard normally distributed random variables. I'd like to show $\mathbb{E} [X_n] \sim \sqrt{2 \log n}$. I've shown already that $$\...
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0answers
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Probability, Expectation and Conditional

Two life insurance policies, each with a death benefit of 10,000 and a onetime premium of 500, are sold to a couple, one for each person. The policies will expire at the end of the tenth year. The ...