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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2answers
18 views
Random Variables from [0,1] - Integration Limits
I was wondering if someone could help me understand the first steps I should take for solving the next problem:
Let $U$, $V$ be random numbers chosen independently from the interval $[0, 1]$ with ...
0
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1answer
21 views
the distribution of the inverse of a standardized uniform variable
If $u$ is a standardized uniform variable, what is the mean and variance of $x=\frac{1}{u}$? What can be said about the distribution of $x$?
1
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1answer
20 views
Bottom to top explanation of the Mahanalobis distance?
I'm studying Pattern recognition and statistics and almost every book I open on the subject I bump into the concept of Mahanalobis distance. The books give sort of intuitive explanations, but still ...
0
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1answer
28 views
How does this expression arise: $\pi(10.5) = \phi (-z_{1-\alpha} + \sqrt{n} \frac{\mu_0-\mu}{2})$?
$X_i$ is $N(\mu,\sigma^2)$ distributed and the following is given $H_0: \mu \geq 12, H_a: \mu < 12$, and $\alpha=0.01$. I'm asked to calculate $\beta=P[TII]$ if in fact $\mu=10.5$
Now this is the ...
0
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1answer
24 views
Sequences of i.i.d. subgaussian RVs and uniform integrability
Consider a sequence of i.i.d. subgaussian RVs $\{a_{j}\}^{n-1}_{j = 0}$; is $\{a^2_{j}\}^{n-1}_{j = 0}$ uniformly integrable (UI)?
Intuitively it appears to be so; if we take for example $\{a_j\}$ ...
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0answers
28 views
Total set of functions in $L^2(\Omega)$
Are the sets of functions $\{e^{\int_0^T h(s)dB_s -\frac{1}{2}\int_0^T h(s)^2 ds}\}$ and $\{e^{\int_0^T h(s)dB_s}\}$ total in $L^2(\Omega)$? What is the difference? What should one use to prove weak ...
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1answer
25 views
How to calculate $\Pr[\max(X,Y)<4]$?
Suppose the joint PDF of X,Y is $f(x,y)=1/40$ and $0 < x < 5$ and $0 < y < 8$.
How to calculate $\Pr[\max(X,Y)<4]$?
1
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1answer
28 views
Finding the expectation and characteristic function of a mixed distribution.
I'm having difficulty with a practice exam question. Here's a modified version.
First, some notation. Let $E$ denote the exponential distribution, and $B$ the Bernoulli distribution. Also, given a ...
11
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0answers
49 views
Minesweeper - Chance of one-click win
I'd like to know if it's possible to calculate the odds of winning a game of Minesweeper (on easy difficulty) in a single click. This page documents a bug that occurs if you do so, and they calculate ...
1
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1answer
32 views
Convergence almost surely
Let $X_n$ and $X$ be random variables. If $X_n \to X$ almost surely, then we have that
$$ \mathbb{P}\left( \lim_{n \to \infty} X_n = X\right) = 1. $$
My question is, can we conclude that
$$ \lim_{n ...
3
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3answers
78 views
Central Limit Theorem Definition
My friend and I have a bet going about the definition of the Central Limit Theorem.
If we define an example as a number drawn at random from some probability density function where the function has a ...
1
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1answer
30 views
Counting probability question-what is the sample space in this problem?
Hi folks this is a self learn probability (counting) question from DeGroot. The question is:
Suppose that a box contains r red balls and w white balls. Suppose also that the balls are drawn out ...
2
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4answers
67 views
Why is the Expected Value different from the number needed for 50% chance of success?
An event with probability $p$ of being success is executed $\frac{1}{p}$ times. For example, if $p=5\%$, the event would then be executed $20$ times.
The Expected Value for the total number of trials ...
1
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1answer
18 views
Probability element in subset
Let be $A$ a set of naturals numbers from $1$ to $N$. Let be $B\subset A$ with $M=\operatorname{card}(B)$. Is $M/N$ the probability that finds an element belong $M$ choosing randomly any number from ...
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0answers
45 views
Central Limit Theorem: probability density function of the true mean?
A friend and I are arguing over the meaning of the Central Limit Theorem. I am stating that the normal distribution seen by taking the mean of a large number of samples is a probability density ...
0
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1answer
28 views
Is the Uniform family of Distributions dominated by the Lebesgue Measure?
The answer to this question should be fairly easy, but I can not just see it.
I want to say something like: let us consider a measure $P_{\theta}\in\mathcal{P}$ where ...
2
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1answer
56 views
What is the probability of the number 1 and number 2 employees getting the bonus at a call center?
Two weeks ago, a friend working at a call center told me about their staff bonus policy. Here I paraphrase it.
Suppose employee A answers the maximum number ($N_1$) of calls among the staff, and ...
0
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1answer
38 views
cauchy schwarz equality: difference in proving style for linear algebra and expectation version
I am interested in proving the following sub version of Cauchy Schawrz equality.
1) LA version :
If $x$ and $y$ are two real vectors and the following holds
$$<x,y> = ||x||.||y||$$ then $x$ ...
2
votes
2answers
29 views
“Expected Probability”: Withdrawing a white ball from a bucket with a random variable X = number of whites
I have a problem with the subject of "Expected Probability" (I don't really know what is the right name for it).
This is an example of a question: (I am not looking for the specific answer, just for ...
0
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0answers
10 views
Optimal stopping for random walks
Let $X_0=0, X_1, X_2,\dots, X_N$ be i.i.d. random variables, with Gaussian distribution $\cal N (0,1)$. For $k=0,\dots, N, S_k=\sum_{i=1}^k X_i$ and $Z_k= (N+1-k)S_k^2$. The goal is to get ...
1
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0answers
27 views
Odds of turning all the columns in solitaire without drawing from the pack
Windows solitaire experiment.
Start a game and see how many cards can be exposed in the columns before a draw from the pack is needed.
(Single or three card game makes no difference here)
As a time ...
1
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0answers
32 views
maximize the expected value of the logarithm of the weighted average of random variables
I'm trying to do the following.
$$\max_{m\in\mathbb{R}} \mathbb{E}\left[\log (wA + (1-w)B_m)\right],$$
where $0<w<1$ and $A, B_m > 0$ are correlated random variables. $A$ does not depend ...
1
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0answers
13 views
Conditions for the ground state of Gibbs ensemble not to be “degenerate”
I am looking at the Wikipedia article on Partition function -- As a measure. Unfortunately the article has no relevant references or reading suggestions.
I am looking for books or other resources ...
2
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1answer
56 views
Expected value: Showing $[\Bbb E(X)-\Bbb E(Y)]^2 \geq 2 \cdot \Bbb{Cov}(X,Y)$
The original question is to show that for any Random variables $X,Y$ and $0\leq p \leq 1$
$$p\Bbb V(X)+(1-p)\Bbb V(Y) + p(1-p)[\Bbb E(X)-\Bbb E(Y)]^2 \geq p^2 \Bbb V(X)+(1-p)^2 \Bbb V(Y) +2p(1-p) ...
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votes
2answers
40 views
Probability that I am not selected in any of 2000 samples?
The population contains 100 million adults, which includes myself. Simple random sampling is used to choose a sample of 1000 adults, 2000 times, independently.
I need to find the probability that I ...
0
votes
3answers
50 views
Distance between two Random Variables by comparing Cumulative Distribution Functions
Suppose $X$ and $Y$ are two random variables. Define the distance between $X$ and $Y$, $d(X, Y)$ as: $$d(X, Y) = \int_{-\infty}^{\infty}\left|\mathbb{P}(X < t) - \mathbb{P}(Y < t)\right|dt$$
...
14
votes
1answer
100 views
How long does it take a person with this “cheating” data-gathering strategy to achieve a desired result?
I have a perfectly fair coin, and my goal is to prove that it is unfair with a confidence level of 95%. In order to accomplish this, I will cheat. Whenever I fail to have enough evidence, I will ...
3
votes
2answers
80 views
Speculating on the stock exchange
Imagine you model each stock as a random walk (fractal) and also that you can buy and sell at any price. Suppose also that it 'walks' with the pace of 1.
If you buy, for example, 1000 shares of ...
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2answers
42 views
Yet another balls and boxes problem; minimum number of throws so that we have no empty boxes.
I managed to figure out how many empty boxes will be left given n amount of throws, just having a hard time figuring out the minimum number of throws necessary so that we have no empty boxes. Would it ...
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0answers
47 views
Borel-Cantelli (?) to show that a lim sup inequality of rapid decay random variables holds with prob. 1
A random variable $\xi$ in probability space $(\Omega,\mathcal{A},P)$ is said to have $c$-rapid decay if $P(\xi > k)\leq e^{-ck}$ for sufficiently large $k$. I'm given a sequence $\xi_n$ of random ...
1
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1answer
45 views
Expectation and Variance, number of balls
Question:
n boxes are ordered in a row on a table , labeled 1 to n.
In every box there's a ball. In every step Nicole chooses a ball randomly (in uniform distribution) , puts it out of the box, then ...
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1answer
20 views
Draw two or three balls from an urn with ten balls
My urn contains two black balls and eight white balls. What is the probability that I get the two black balls
a) after two draws
b) after three draws?
My approach is to draw a decision tree.
...
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0answers
15 views
find the probability ether on for one student [closed]
In a certain college, the students engage in various sports. The statistics are
60% of all students play football
50% of all students play basketball
both football and basketball of all students
...
1
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1answer
47 views
I'm gonna give probability regularization classes
There's this group of high-school level kids that failed probability and they want me to teach them so they can pass that subject, i agreed to be their teacher for this 2 weeks, however i'm not ...
1
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0answers
30 views
Jensen-like Inequality
I have the following question:
Suppose we have a function $g:\mathbb{Z}_+ \cup \{0\} \rightarrow \mathbb{R}_+$ with the property, $g(\lfloor \frac{x+y}{2} \rfloor)$ + $g(\lceil \frac{x+y}{2} \rceil) ...
0
votes
1answer
20 views
Is that Probability function only for discrete case?
Most of the books and sites define Probability function for discrete case that is they use the term as the synonym of Probability mass function.
Is that Probability function define for only ...
0
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0answers
25 views
Help with homework 4.3
The following table shows opening songs performed by a group for the 20 concerts played.
Title: Frequency:
Bucket 3
Jackstraw 3
Touch of Gray 3
Cold Rain 2
Good Times ...
1
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0answers
19 views
Convergence in distribution of a quadratic form
If $Q_n=X_nM_nX_n=\sum_{i,j=1}^n X_i m_{nij}X_j$, $X_n=(X_1,...,X_n)$ where $X_j$ are iid random variables and $M_n=(m_{nij})$ is a symmetric matrix with extending rownumber in $n\to\infty$.
Iam ...
1
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2answers
44 views
The probability that an event with exponential distribution will happen before an event with a Poisson distribution
I have two variables depicting arrival. One (lets call it $A$) has a Poisson distribution, so the probability of $n$ elements arriving in time period $\tau$ is: $P_n(\tau)=\frac{\left(\lambda ...
0
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0answers
34 views
$\operatorname{Prob}\limits_{x,y\in\mathbb{Z}_q^*}[\gcd(xy \bmod q, pq)>1: \gcd(x,pq)=\gcd(y,pq)=1]=?$
For given two distinct odd primes $p$ and $q$, how to count the probability
$$\operatorname{Prob}\limits_{x,y\in\mathbb{Z}_q^*}[\gcd(xy\bmod q, pq)>1: \gcd(x,pq)=\gcd(y,pq)=1]=\, ?$$
0
votes
2answers
25 views
Probabilities: That $A,B$ sit next to each other; That there are $3$ people are between $X$ and $Y$
1) Five persons $A,B,C,D,E$ occupy seats in a row at random. What is the probability that $A$ and $B$ sit next to each other?
2) $X$ and $Y$ stand in a line at random with $10$ other people. The ...
2
votes
1answer
24 views
Average number of heads in filtered coin toss
I have a coin that, when tossed, produces heads with probability $p \geq 0.5$ and tails with probability $1-p$.
I start a coin-tossing experiment. Whenever I get more than one tail in a row, I ...
1
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0answers
33 views
+50
Converting Expected value to integrals and differentiating
Can you suggest me how to convert the following expected value function in to an integral and differentiate it with respect to $a$.
\begin{equation*}
g \equiv E \left[ \max \left( a + x-b,0 \right) ...
0
votes
1answer
32 views
Birthday probability for three [duplicate]
There are 32 students in a class-room. What is the probability that at least 3 of them have their birthdays in the same month?
How to get the total possibilities?
1
vote
1answer
22 views
Probability, making a selection of 5 people from 10, with two married couples with restrictions
10 people. must make a committee of 5 people
So the restrictions are
1) Mr and Mrs Q can't be separated
2) Mr and Mrs P can't be in the same committee.
So how many possible committees ...
0
votes
2answers
27 views
Probability of selecting q red balls from m red balls and n blue balls
Suppose there are $m$ red balls and $n$ blue balls in an urn. We randomly choose $p:m<p<n$ balls uniformly from the urn. What is the probability that exactly $q$ red balls are chosen?
Note:- ...
3
votes
2answers
45 views
Probability: Two people get multiple choice questions
I was wondering... If two people do multiple choice test, and the questions are taken from a pool of question (the size of the pool is unknown). Both people must be given 30 questions. They both ...
0
votes
1answer
29 views
Expected Value of Dice
Given 2 fair dice are rolled until a sum of n is showing or we have rolled the dice m times, what is the expected value of X?, where X is the random variable equal to the number of rolls.
1
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0answers
26 views
Likelihood ratio test
Problem 47, pg 296 from Jun Shao: Mathematical statistics
Let $(X_1, \dots, X_n)$ be a random sample from $N(\mu, \sigma^2)$. Suppose that $\sigma^2 = \gamma \mu^2$ with unknown $\gamma >0$ and ...
0
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0answers
25 views
Exact test for 3 x 3 contingency table
I'm aware of the Fisher exact test, for determining the probability distribution for a $2\times2$ contingency table.
Is there an exact test for a $3\times3$ table? Or is there a way to combine the ...
