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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1
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1answer
387 views

Normal distribution probability

just a quick question dealing with probability. The annual returns on stocks and treasury bonds over the next 12 months are uncertain. Suppose that these returns can be described by normal ...
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1answer
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Probability increases as sample size increases?

I was talking with a friend and we were discussing a math problem disguised as a social situation: If the chance that someone to accept your request to go out with them was 1% and you asked 1 person ...
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3answers
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How many right angled triangles can a circle have?

Here's what I recall of the question from CNML Grade 11, 2010/2011 Contest #3, Question 7: There are 2010 points on a circle, evenly spaced. Ford Prefect will* randomly choose three points on ...
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208 views

Asymptotic behavior of the first step in a best strategy

Consider the game described here, but for a sequence $X_1,\ldots,X_n$ of i.i.d. uniform rv's on $\lbrace 1,\ldots,n \rbrace$ (in the original game $n=6$). Using the original notation, let $a_n$ denote ...
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3answers
2k views

Best Strategy for a die game

You are allowed to roll a die up to six times. Anytime you stop, you get the dollar amount of the face value of your last roll. Question: What is the best strategy? According to my calculation, for ...
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2answers
177 views

How to select the optimum combination of numbers from a random list that add to up to a certain total (or as close to)

I'm developing a computer program, and need an algorithm to solve the following problem: How to select the optimum combination of numbers from a random list that add to up to a certain total (or as ...
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1answer
2k views

Uniform White Noise

I found a contradiction I couldn't resolve by my self. It's about a "Uniform White Noise". Let ${x}_{t}$ be a "White Noise" i.i.d. Random Process: $ \forall t \in \mathbb{R}, \ {x}_{t} \sim U[-1, ...
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6answers
31k views

Average length of the longest segment

This post is related to a previous SE post If a 1 meter rope …. concerning average length of a smallest segment. A rope of 1m is divided into three pieces by two random points. Find the average ...
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1answer
167 views

Probability of the sum of n numbers giving the same last d digits

Can anybody give me a hint how to approach the following problem, please? I actually am having a hard time stating the problem. I think an example would help you understand what the problem is. ...
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6answers
14k views

Probability of picking a random natural number

I randomly pick a natural number n. Assuming that I would have picked each number with the same probability, what was the probability for me to pick n before I did it?
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4answers
284 views

What probability distribution is this?

This image show a histogram (200 bins) of accumulated distances from a radar distance meter (very noisy). The peak around 7 meters is an object. At thought this looked kind of like a normal ...
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1answer
239 views

How to calculate likelihood to succeed knowing attempts and successful attempts amounts?

Say I have two algorithms that I don't know how they work but I know what they are meant to achieve. I tried algorithm A once and it succeeded. And tried algorithm B 100 times and succeeded 99 times. ...
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2answers
250 views

Doubt in a probability problem

Problem: Alex flip a fair coin three times. what is the probability that she gets two heads given that the first is a head? My solution is based on the argument that from the given information the ...
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32k views

If a 1 meter rope is cut at two uniformly randomly chosen points, what is the average length of the smallest piece?

If a $1$ meter rope is cut at two uniformly randomly chosen points (to give three pieces), what is the average length of the smallest piece? I got this question as a mathematical puzzle from a friend....
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2answers
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Are these transformations of the $\beta^\prime$ distribution from $\beta$ and to $F$ correct?

Motivation I have a prior on a random variable $X\sim \beta(\alpha,\beta)$ but I need to transform the variable to $Y=\frac{X}{1-X}$, for use in an analysis and I would like to know the distribution ...
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1answer
378 views

Reducing quantification to probability

I was thinking about some problem involving quantifiers (the existencial and universal quantifiers) and I noticed how it might resemble probability in a sense. They both assume a variable and its ...
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1answer
2k views

density function by the method of transformations

We have a random variable $Y$ that has a uniform distribution on the interval $[1,5]$. The cost of delay is given by $U = 2Y^2 + 3$. Use the method of transformations to derive the density function of ...
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4answers
869 views

Probability Problems

Problem: The probability that a man who is 85 years. old will die before attaining the age of 90 is $\frac13$. A,B,C,D are four person who are 85 years old. what is the probability that A will die ...
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2answers
137 views

Doubts on a probability problem

Problem: From the deck of the 52 cards,cards are drawn randomly without replacement.What is the probability of drawing a king of hearts at the third attempt? If it was drawn at the 15th attempt, what ...
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1answer
10k views

martingale and filtration

As I understand, martingale is a stochastic process (i.e., a sequence of random variables) such that the conditional expected value of an observation at some time $t$, given all the observations up to ...
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3answers
3k views

binomial random variable

Let $Y$ be a binomial random variable with $n$ trials and probability of success given by $p$. Show that $n-Y$ is a binomial random variable with $n$ trials and probability of success given by $1-p$. ...
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1answer
3k views

statistics - moment-generating function

Let $Y_1,\dots,Y_n$ be independent and identically distributed random variables such that for $0 < p < 1$, $P(Y_i = 1) = p$ and $P(Y_i = 0) = q = 1-p$. A. Find the moment-generating functions ...
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0answers
440 views

An application of the Optional Sampling Theorem

let $S(k), k\geq 0$ a discrete random process. Suppose $S(N)$ is with probability one either 100 or 0 and that $S(0)=50$. Suppose further there is at least a sixty percent probability that the price ...
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1answer
357 views

Negativity in a CIR model discretized by Ito-Taylor expansion

Let $X = (X_t: t \in [0,T])$ be a stochastic process satisfying a CIR model $$ dX_t = \beta (X_t - \gamma) dt + \sigma\sqrt{X_t} dB_t, $$ where $B_t$ is a standard Brownian motion, $\beta$ is a ...
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1answer
529 views

distribution of iid sequence of integrable random variables

I came across an interesting problem in Jacod's probability book. But have no idea how to approach it. Should I approach it using induction? Any ideas? Let $X_1, X_2, \cdots$ be an infinite sequence ...
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1answer
12k views

Find an unbiased estimate for λs (Poisson distribution)

So I have this problem to solve... Let X denote the number of paint defects found in a square yard section of a car body painted by a robot. These data are obtained: 8, 5, 0, 10, 0, 3, 1, 12, 2, 7, ...
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1answer
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Doubts on Mutually exclusive and Independent events

Problem: In a school competition,the probability of hitting the target bu Dick is $\frac{1}{2}$,by Betty is $\frac{1}{3}$ and by Joe is $\frac{3}{5}$.If all of them fire independently,calculate the ...
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3answers
1k views

Density of a function of random variables

I recently learned about finding density functions of functions of random variables using the cumulative distribution function. For example, computing the density function for the difference of two ...
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1answer
3k views

Statistical Inference Question

From Statistical Inference Second Edition (George Casella, Roger L. Berger) "My telephone rings 12 times each week, the calls being randomly distributed among the 7 days. What is the probability that ...
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2answers
3k views

A continuous analogue of the binomial distribution

For any positive integer $N$, the binomial$(N!,p)$ distribution has the following property: for any $1 \leq n \leq N$, there exist i.i.d. random variables $X_1,\ldots,X_n$ such that $X_1 + \cdots + ...
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1answer
209 views

Understanding birthday attack probabilities

I have two sets $M$ and $H$. $M$ is an arbitrary string of length $k$ and $H$ is an string of length $p$. Both are constructed from a charset of length $r$. And $p<k$. Hash function $f(m)=h$. I ...
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1answer
647 views

binomial-Poisson/beta hierarchy

X|N,P ~ Binomial(N, P) N ~ Poisson(11) P ~ Beta(2,3) What is the moment generating function for X?
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1answer
512 views

Question about Brownian motion

Let $\{B(t), t \in \mathbb{R} \}$ be a two sided brownian motion defined as $$ B(t) = \begin{cases} B_1(t),\quad t >0 \\ 0, \quad t = 0 \\ B_2(-t), \quad t < 0 \end{cases} $$ where $B_1$ ...
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1answer
163 views

Risk process in Insurance

Let $R(t) = u + ct - \sum_{k=1}^{X(t)}Z_{k}, t\geq 0 $, be a risk process, where $u> 0$ is the initial capital of the insurance company and $c> 0$ is a premium rate. We know that the number of ...
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3answers
178 views

a simple argument?

This argument appeared in a proof in a paper (I am paraphrasing), and I am not sure why it is true. It should be rather simple. Let $(\Omega,P)$ be some sample space. Let $X$ be a random variable ...
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2answers
113 views

probability of heads

What is the probability of getting $3$ heads in a row? Would it be $\frac 18$? assuming the coin is a fair one.
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1answer
2k views

density function of U when Y has a beta distribution

Assume that Y has a beta distribution with parameters a and b. Find the density function of U = 1 - Y. I know how to do then when they give the density function of Y, but i'm confused here. Thanks!
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1answer
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Probability on spreading of rumors

A little help here. Exercise 21, Ch. 2 from Feller's book reads In a town a $n+1$ inhabitants, a person tells a rumor to a second person, who in turn repeats it to a third person, etc. At each ...
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3answers
363 views

Gaussian Distribution + Hash Tables

The original question I posted on StackOverflow. I think it's more mathematically inclined so I posted it here again. In terms of math. There's a class of students. Each students has a score between ...
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3answers
5k views

How do I Generate Doubly-Stochastic Matrices Uniform Randomly?

A doubly-stochastic matrix is an $n\times n$ matrix $P$ such that $\displaystyle\sum_{i=1}^n{p_{ij}}=1$ and $\displaystyle\sum_{j=1}^n{p_{ij}}=1$ where $p_{ij}\ge 0$. Can someone suggest an ...
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1answer
93 views

What probability methodology is electronic payment using?

Just curious. You pay electronically, and you get a string of numbers. My question, what is the likelihood for a person to type in the same serial number to say, top up their mobile credit? What is ...
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1answer
94 views

Continuity of $E[\max(x+Z,0)]$ as a function of $x$

Let $Z$ be a random variable with finite mean and consider the function $F: R \rightarrow R$ defined by $$ F(x) = E[\max(x+Z,0)]$$ How to prove that $F(x)$ is a continuous function of $x$? This is ...
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6answers
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Probability of dice sum just greater than 100

Can someone please guide me to a way by which I can solve the following problem. There is a die and 2 players. Rolling stops as soon as some exceeds 100(not including 100 itself). Hence you have the ...
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2answers
238 views

Particle moving at constant speed with Poisson setbacks

Consider a particle starting at the the origin and moving along the positive real line at a constant speed of 1. Suppose there is a counter which clicks at random time intervals following the ...
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3answers
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probability of getting 50 heads from tossing a coin 100 times

folks, i am new to this forum and not a math expert. so please bear with me if am asking silly questions. The question is "probability of getting 50 heads from tossing a coin 100 times". So the ...
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4answers
998 views

Probability of the maximum (Levy Stable) random variable in a list being greater than the sum of the rest?

Original post on Mathoverflow here. Given a list of identical and independently distributed Levy Stable random variables, $(X_0, X_1, \dots, X_{n-1})$, what is the is the probability that the maximum ...
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2answers
1k views

Given a function $f(x)$ where $x$ is uniformly distributed between $a$ and $b$, how do I find the probability density function of $f$?

For example, if $f(x) = \sin x$ and $x$ is uniformly distributed on $[0, \pi]$, how is the equation found that satisfies the probability distribution function of $f(x)$? I imagine the distribution ...
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2answers
466 views

expectation of $ \left(\sum_{i=1}^n {x_i} \right)^2 $

If $x_i$ is exponentially distributed $(i=1,...,n)$ with parameter $\lambda$ and $x_i$'s are mutually independent, what is the expectation of $\left(\sum_{i=1}^n {x_i} \right)^2$ in terms of $n$ and ...
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2answers
2k views

Does equality in distribution imply equality of expected value?

In other words, if X = Y in distribution, is it true that EX = EY? I think this must be true, but I've tried to prove it a few times and I always get stuck. Thanks in advance for any hints or ...
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1answer
271 views

Maximum value of probability

This is my first post here. Apologies if there are any misformats or mis-tagging. Also if the post doesn't belong here, sorry, and I will delete it. I derived this formula from a probability ...