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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61 views

Poor phrasing in probability problem, what is the problem even asking for?

If $m$ things are distributed among $a$ men and $b$ women, show that the chance that the number of things received by man is$${1\over2} {{(b + a)^m - (b - a)^m}\over{(b+a)^m}}.$$ I'm not asking for a ...
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52 views

We have a density function: $p_{X,Y}(x,y)=ce^{-y}, 0\leq x<y<x+1$

$$p_{XY}(x, y)= ce^{-y},\quad 0\leq x \lt y\lt (x+1)$$ I need to find $c$. My try: $$\int_0^{\infty}\int_x^{x+1}ce^{-y}dydx = \int_0^{\infty}\int_0^{y}ce^{-y}dxdy = \ldots =1$$ Is the idea correct, ...
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1answer
44 views

Uniformly distributed conditional probability

We have $X \in (0,2)$ and we have $Y \in(0,2X)$, both uniformly distributed. I need to find conditional probabilites. My try: We know that uniform distribution has density $\frac{1}{b-a}:$ $$p_X = \...
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1answer
29 views

ordered vs unordered sample space

In the game of bridge, the entire deck of 52 cards is dealt out to 4 players. What is the probability that each player receives 1 ace? I tried to solve this problem by trying to solve what I thought ...
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31 views

how to calculate the probability of two of a kind in a throw of three dice [closed]

There are 3 dice. What is the probability, in a throw, of a specific two of a kind(three of a kind is included)? What is the probability, in a throw, of all two of a kind(three of a kind is included)? ...
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40 views

bounding probability with Hoeffding's inequality

I was asked the following question in an assignment: An airline has collected an i.i.d. sample of 10000 flight reservations and figured out that in this sample 5 percent of passengers who made a ...
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2answers
63 views

Calculating: $\sum_{s=0}^\infty\sum_{t=0}^s e^{-3}st\frac{1}{t!}\frac{2^{s-t}}{(s-t)!}$

After a considerable time of trying to calculate the following: $$\sum_{s=0}^\infty\sum_{t=0}^s e^{-3}st\frac{1}{t!}\frac{2^{s-t}}{(s-t)!}$$ Assuming 0≤t≤s.I succeeded to reach an answer by using the ...
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53 views

A formula to equalize win chances

I writing a game that has characters with few parameters: agility, strength and luck. Agility allows a person to escape a hit, luck allows to do a critical hit (double the hit damage) and strength is ...
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1answer
42 views

$X$ is a random variable, $\alpha \in \Bbb R$ is constant, $Y=a-X$. Prove that: $R(X,Y)=-1$

$X$ is a random variable and $\alpha \in \Bbb R$ is constant. $Y=a-X$. $R(X,Y)=\,? \,\,\,$ (The correlation) My guess was that the answer is $-1$. Why? Because $X$ and $Y$ are correlated, according ...
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51 views

Evaluating $\mathbb{E}[X^2 Y^2]$, where $(X,Y)$ is a gaussian vector [closed]

I have the vector $(X,Y)$, I know that the expectation of both $X$ and $Y$ are zero, and $\text{Var}(X)=\frac{1}{2}$, $\text{Var}(Y)=\frac{9}{2}$, $\text{Cov}(X, Y)=\frac{3}{2}$. How can I evaluate $\...
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66 views

$X_1,X_2,...,X_n$ independent random variables are uniformly distributed on $[0,1]$. $P(X_1<X_2<...<X_n)=?$

$X_1,X_2,...,X_n$ independent random variables are uniformly distributed on $[0,1]$. So, we say that $P(X_1<X_2<...<X_n)=\frac{1}{n!}$ But why is that the correct answer? How do we calculate ...
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52 views

Estimating $n^2$ after observing $X\sim Bin(n,p)$ and knowing $p$

Let $n\in\mathbb N$ be unknown and let $p\in (0,1]$ be known. Suppose that we observe $X\sim Bin(n,p)$. This allows us to estimate $\widehat n= X/p$ and we can use the Chernoff inequality to bound the ...
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1answer
88 views

if $P(A \cap B \cap C)=P(A)P(B)P(C)$, does it necessarily means $A,B,C$ are independants?

Lets say we have the events $A,B,C$. We know that if they are independants, then the following occurs: $$P(A \cap B \cap C)=P(A)P(B)P(C)$$ But does it work the opposite way? If some $A,B,C$ events ...
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1answer
46 views

Expected value of sin of sum of n random angles

Consider the following problem. Let $θ_1,θ_2,...θ_n ∈[0, \frac{π}{2}]$ be independent and uniformly distributed variables. Find $E[sin(θ_1 + ... + θ_n)].$ I was able to solve for $n=1$ (of course), ...
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1answer
61 views

Can Uniform random variables be viewed as "more random" than Normal random variables?

Since a Normal random variable has a mean which has a higher likelihood of observation compared to its tails, would it be correct to say that a Uniform random variable (where mean and tails share the ...
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1answer
62 views

Distributing $r$ balls into $n$ cells. What is the probability that exactly $m$ cells contain exactly $k$ balls?

$r$ balls are randomly distributed into $n$ cells (the balls are indistinguishable). What is the probability that there is exactly $m$ cells that contains exactly $k$ balls (each one)? That is, the ...
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27 views

What does it mean when variance is 1?

I was studying skewness. We know that the sign of the third moment of a random variable $\mu_3^*=E(X-E(X))^3$ is the sign of skew. But we can not measure the amount of skewness with that because it ...
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1answer
64 views

Find the probability of U.

If $U$ and $V$ are two independent events with $P(U)<P(V)$, the probability that both $U$ and $V$ occur is $6/25$, and $P(U|V)+P(V|U)=1$. I want to find $P(U)$. $U$ and $V$ are independent so; $P(U|...
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72 views

How to solve the problem of giving money to N people?

There are $N$ people and the amount of money hold by the $i_{th}$ person is a prior given parameter denoted by $x_{1,i}\in\mathbb{R}$, where $i=1,\cdots,N$. Here we assume that the wealth can be ...
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1answer
41 views

What is the probability of hitting 2 pair on the flop with upaired cards in Holdem poker?

I have mainly seen that the probability is 2.02%, but it always somewhere said it does not include if boards pairs. So I was thinking is the probability of getting 2 pairs if board pairs is equal to: ...
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1answer
29 views

Calculate number of faces and dices from toss samples

I have a sample of data. That data comes from tossing n dices with k faces and adding the values of each dice. How can I calculate k and n? My approach: I know that the expected value of 1 dice of k ...
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37 views

How to find the PDF for this problem?

In the problem I am trying to solve, I am asked to find the PDF of the second highest variable out of $U_1,U_2,U_2$ that are independent and follow the same uniform distribution in $[0,1]$. I ...
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39 views

Why $(t, \omega)\mapsto X_t(\omega)$ is not measurable in following example?

I want to find an example that r.v. $X_t:\Omega\to R$ is measurable but $(t, \omega)\mapsto X_t(\omega)$ is not measurable. If take a non-measurable set $B\subset [0,T]$, let $\Omega=\{-1, +1\}$. Then ...
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1answer
58 views

If $x$ and all $Ax$ are identically distributed, when are $x_1, \ldots, x_n$ identically distributed?

Let $\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$, let $x \in \mathbb{F}^n$ be a multivariate random variable, and let $\mathcal{A} \subset \mathbb{F}^{n \times n}$ be a family of matrices. For every $A \...
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1answer
47 views

What is the PDF of $Y = |X|$ given $X\sim \mathcal{N} (0,\sigma ^2)$ [duplicate]

I've faced this question and found a solution for it, so I decided to share it with others; maybe it'll help them. Also, the question solves somehow the generality of it. Question: Given $X\sim \...
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46 views

A random walk on a circle with $n$ points. What is the probability that the walk will end at point $i$ ? ($0 \le i \le n-1$) [duplicate]

$n>1$ points numbered $0,1,2,...,n-1$ are placed on a circle. A random walker starts his journey at point $0$ and at each step, he steps randomly on the circle to one of the two closest points. For ...
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1answer
44 views

Using a martingale property to deduce orthogonality of some increments

I know that if $(M_t)_{t\ge0}$ is a continuous Martingale with $EM_t^2<\infty$ for every $t\ge0$, then its increments are orthogonal. Is there a way to deduce from this result directly that $g(B_{...
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1answer
31 views

Is my proof that a sequence of binomial variables of fixed mean don't converge in probability to the Poisson distribution correct?

$\newcommand{\pmf}{\operatorname{pmf}}\newcommand{\d}{\,\mathrm{d}}\newcommand{\pr}{\operatorname{Pr}}$It is not hard to see that the probability of the sequence of binomially distributed variables ...
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1answer
58 views

$5$ dots on a $8$x$8$ board

We randomly distribute $5$ dots on a $8$x$8$ board, such that no square contains more than $1$ dot (excuse me for the spelling, i'm not sure if 'square' is the right word. Hope you know what i meant - ...
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1answer
32 views

Discontinuous pdf of 2 branch Random Variable Transformation

Given the pdf of a random variable $X$ $$f_X(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$$ I want to find the pdf of random variable $Y$ defined: $$ Y = g(X) = \left\{ \begin{array}{ll} 4X & X \geq ...
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1answer
52 views

A probability inequality of events

$A_i$ is events, $P(A_i)$ is not all zeors, $i=1,\cdots,n$. Proof: $$ P\left( \bigcup_{i=1}^n A_i \right) \ge \frac{\left( \sum\limits_{i=1}^n P(A_i) \right)^2 }{ \sum\limits_{i=1}^n P(A_i) + 2\sum\...
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35 views

Is this question a Bayes' Theorem question? and What is the answer? [closed]

You want to know if it will rain. You check 3 different weather websites to see if it will rain. Based on history, each weather website has 2/3 probability of being correct and 1/3 probability of ...
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1answer
57 views

(Conditional) coin flips mean and variance

Roll a fair die to obtain a random number $1 ≤ n ≤ 6$, then flip a fair coin $n$ times. Let $X$ be the random variable that expresses the number of heads in the coin flips. Find the mean and variance ...
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1answer
39 views

Find joint bivariate distribution using double integral (Wackerly 5.5a)

This is problem 5.5a from Mathematical Statistics with Applications 7th Edition - Wackerly et al. The background to the problem is that we have a tank which is filled with gas once a week. $Y_1$ is ...
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1answer
37 views

What is the probability that $X$ is located closer to $Y$ than to $0$?

X and Y are uniform random variables. X uniformly distributed on $[0, \frac{L}{2}]$ and Y is uniformly distibuted on $[\frac{L}{2}, L]$. What is the probability that $X$ is located closer to $Y$ than ...
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1answer
60 views

Calculation of probability based on population distribution [closed]

In one city there are ninety percent who like peas, and ten percent who do not like peas, in that city there is a road driven by one hundred people who live in the city at any one hour, can I claim ...
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1answer
45 views

Probability that the player wins a pass line bet with a 4 on the first roll

Precalculus textbook problem (self-study): In the game of craps, there are two ways a player can win a pass line bet. The player wins immediately if two dice are rolled and their sum is 7 or 11 . If ...
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1answer
33 views

Probability of picking balls without replacement

Jar contains 14 balls - 5 reds,4 blue,3 green and 2 yellow. I grab 4 balls at once. What is the probability I get 2 reds,1 blue and 1 green balls (order does not matter). I can compute the probability ...
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1answer
14 views

estimation of the distribution of the difference of ordered samples

I came across an interesting problem listed below: Consider an interval from 0 to 1 with ten uniformly distributed points in it. What is the distribution of the difference between the 6th and the 5th ...
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3answers
73 views

Realization vs Random Variables in Formulae

Currently reading through: https://web.math.princeton.edu/~rvan/APC550.pdf. Page 8 Section 1.2.1 says: If $X_1$, $X_2, ...$ are i.i.d random variables, then $$ \frac{1}{n}\sum^n_{k=1} X_k - \mathbf{E}[...
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0answers
49 views

stop rolling dice until the cumulated sum is perfect square

You have a six-sided dice, and you will receive money that equals to sum of all the numbers you roll. After each roll, if the sum is a perfect square, the game ends and you lose all the money. If not, ...
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1answer
58 views

What is the probability that EA > EC?

I found this question in one of my books: Square $ABCD$ has a length of $1$. Point $E$ is selected inside the square. Segment $EA$ represents the distance between point $E$ and one of the vertices of ...
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Should I determine how long it takes to fix a problem or should I just fix it? [closed]

I was looking at the good-turing equation and was thinking about problem solving, and if: I should learn how long it takes to fix something or I should just fix it It has come up today in a ...
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34 views

Integral bounds for $x\geq yz$

I am having trouble understanding the integral bounds. From what side should my understanding go (first or second?): first: as $z$ is between zero and one, $y$ is also between zero and one, thus $x$ ...
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1answer
99 views

What is $\binom{n}{2} \binom{n}{1} + \binom{n}{3} \binom{n}{2} + \ldots + \binom{n}{n-1} \binom{n}{n-2} + \binom{n}{n} \binom{n}{n-1}$?

A bag contains $n$ white and $n$ black balls, all of equal size. Balls are drawn at random. Find the probability that there are both white and black balls in the draw and that the number of white ...
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1answer
40 views

A soft question about the differene between guassian distribution and uniform distribution over a circle

Choosing $n$ independently and uniformly distributed points from a unit circle is relatively easy and there are many ways to do it. One way for instance is the rejection method; choosing points in the ...
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1answer
42 views

Triple integral probability problem

We have density functions: $p_X = e^{-x}$ $p_Y = 2e^{-2y}$ $p_Z = 3e^{-3z}$ We need to find the density function of $p_{x+y+z} $. All three variables are independent. Because they are independent I ...
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2answers
126 views

Showing that $X(t)=\int_0^tB(s)ds $ is a Gaussian process.

The time integration of Brownian motion is given by $X(t)=\int_0^tB(s)ds$. In order to show that $X(t)$ is a gaussian process I work with the definition that for any $t_0<t_1 < t_2<\ldots &...
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0answers
18 views

How do we name the X when we have an independent and identically distributed random variables X_1, X_2, ..., X_n? Also how to name data?

I need to be sure about some jargon/naming. In this page we have "For i.i.d. random variables $X_1, X_2, ..., X_n$, the sample mean, denoted by $\bar{X}$, is defined as $\bar{X}=\frac{X_1 + X_2 + ...
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0answers
32 views

probability of a sample being in the the top N of all samples [closed]

Given a distribution $f$ of which i have already drawn $n$ random samples with real values, what is the probability of the next sample being in the top k of all samples? Edit: With the suggestion of ...

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