Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [probability]

For basic questions about probability and for questions about calculating a probability, expected value, variance, standard deviation, or similar quantity. 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.

0
votes
0answers
9 views

Let $T$ be a normal random variable that describes the temperature…

Let $T$ be a normal random variable that describes the temperature in Rome on the 2nd of June. It is known that on this date the average temperature is equal to $µ_T = 20$ centigrade degrees and that $...
2
votes
2answers
36 views

Why is the sum of two independent geometrically distributed RVs not geometrically distributed again?

Let X = Geo(1/4) and Y = Geo(1/4) (both independent) and let Z = X + Y. Why is X + Y not Geo(0.5) for example and how could I, as an example, calculate P(Z = 3)?
0
votes
1answer
23 views

Probability of set of coins with different bias

Suppose I have a set of $n$ biased coins with the same probability $p$. I throw them at once. The probability to have $k$ successes is the pmf of the binomial distribution $f(k,n,p)=\binom{n}{k}p^k (...
0
votes
0answers
12 views

States of the world/Game theory and Beliefs

This post consists on 3 parts: the question itself, hint and a table. The question will make sense to you only after you have read the tables and the hint attached. The problem is about beliefs of a ...
0
votes
2answers
15 views

probability of dependent events of box of balls

I have a box full of $N$ unique balls. The first time, I pick $k$ balls out of this box (I inspect them and put them back). On the second time, I also pick $k$ balls out of the same box (that has $N$ ...
1
vote
1answer
25 views

Are there similar theorems to the Infinite Monkey Typewriter Theorem?

I was thinking about the Infinite Monkey Theorem and wondered if there’s any similar theorems but with a finite set? I know the infinite size is a critical assumption in the Monkey theorem, but are ...
0
votes
0answers
11 views

Independence of Records of Permutation

For any permutation $\sigma = (\sigma_1,\dots,\sigma_n)$, call $\sigma_k$ a record if $\sigma_k>\sigma_i$ for $1\le i\le k-1$. Let $\alpha_k$ be the indicator for the event that $\sigma_k$ is a ...
0
votes
0answers
14 views

In a sports competition, ten athletes of different nationalities challenge…

In a sports competition, ten athletes of different nationalities challenge each other in the javelin throwing. Let $X$ be the distance covered by the javelin of the Italian athlete and suppose it is a ...
0
votes
0answers
8 views

Expectation of Gaussian r.v. conditioned on positive r.v.s with positive covariances is positive

Suppose that $\Delta_1,\dotsc,\Delta_K \sim \mathcal{N}(0, \Sigma)$, with $\mathrm{cov}(\Delta_i, \Delta_j) > 0$ for all $i,j$. Does $$ \mathbb{E}[\Delta_K \mid \Delta_1> 0, \dotsc, \Delta_{K-1}...
1
vote
1answer
24 views

Figuring out the variance of the birthday paradox

Given n people, if I want to estimate how many of them are likely to have an overlapping birthday with any other person, how do I calculate the variance? So far I have $E[X]=n\cdot p$ with $n$ as the ...
-1
votes
0answers
14 views

Joint PDF of two dependent exponential random variables

I need help in following Let's assume we have two identically distributed exponential random variables $X$ and $Y$. What will be the joint pdf of those random variables, i need to consider two cases, ...
11
votes
2answers
268 views

What is the expected convex depth of a set of $m$ randomly chosen points in the unit square?

Definition. Let $X$ be a set of points in $\mathbb{R}^2$. Then the vertex sequence of $X$ is defined by $X_{0}=X$, and $X_{i+1}=\left\{x\in \operatorname{Conv}(X_{i}): x \notin \...
0
votes
1answer
938 views

Is my entropy calculation correct? Clustering entropy example

I would like to calculate entropy of this example scheme http://nlp.stanford.edu/IR-book/html/htmledition/evaluation-of-clustering-1.html Equation of entropy Then the entropy is (the first line) ...
0
votes
1answer
14 views

Given the distribution function of a continous random variable $X$…

Given the distribution function of a continous random variable $X$: $F(x) = \begin{cases} 0, & x<0 \\ cx^2+x/2, &0<=x<1 \\ 1, &x>=1\ \end{cases}$ Is $c=9/4$? How can I find ...
0
votes
2answers
25 views

Probability of winning the lottery twice

I'm struggling to get the answer to this question. I believe I have almost solved it but am having trouble getting the final answer. What is the probability of someone winning the lottery twice? It ...
1
vote
0answers
15 views

Obtaining a probability distribution after a change of variables.

I have two random variables $x_1,x_2$ whose distributions are unknown. I define $y_1=g(x_1,x_2)$ and $y_2=f(x_1,x_2)$ where $f(\cdot)$ and $g(\cdot)$ are known and the probability distributions $p(y_1)...
1
vote
1answer
997 views

Proof of if two random variables have the same distribution then they have the same moment generating function.

I am trying to prove that if $X$ and $Y$ have the same distribution, then they have the same moment generating function: $M_X(t) = M_Y(t)$ for all $t \in \mathbb{R}$. I came up with a proof, but am ...
1
vote
2answers
26 views

The probability of meeting in a tournament

$2^n, n\in\mathbf N$ tennis players compete in a tournament. In the first round, they partition into a set of $2^{n-1}$ disjoint pairs. The two players in each pair compete against each other. The $2^{...
0
votes
0answers
15 views

Maximum Likelihood Method for continuous distribution

I am currently working on an assignment for my data and models lecture. We have just begun learning to use the maximum likelihood method to create a model for a probability experiment. Our script ...
2
votes
0answers
13 views

Hitting and excursion times for biased random walk on the hypercube

I consider the following update rule for a random walk $\{X_{t}\}_{t \geq 0}$ on the hypercube $\{0,1\}^{n}$: At each time step, I sample $I \in \{1,2,\ldots,n\}$ uniformly at random and $U \sim ...
0
votes
0answers
12 views

Probability when selecting documents randomly

Setup Background context, see latest comment on answer dated 'Apr 30 at 20:46'. You have documents $d_0$ to $d_{n-1}$ each randomly assigned an integer $k$ from the range [0, $2^{64}$), with a new ...
2
votes
2answers
951 views

Pig Wheel question

A friend of mine was playing the bar game Pig Wheel recently and posed some interesting questions to me. He was playing with others as a group of four and, acting collectively, they came out about ...
6
votes
3answers
63 views

What's the probability that the teacher teaches her class?

Hi, thanks for reading! I really need help with this question. I'll post all my progress below - I tried really hard being as thorough as possible, but if I don't meet the guidelines for how a ...
0
votes
0answers
15 views

Unconditional probability from conditional probability

Could someone explain to me how I can get unconditional probability? If $Y_1, Y_2, ..., Y_m$ are i.i.d random variable U(1,N)and $P(X_i|Y_i=t)\sim U(t-1,t)$, what is $P(X_i)$?
0
votes
0answers
16 views

Example of a pair of random variables

I have this example of random variables let $ \ \ \ f_{XY}( x,y) = \begin{cases} cx \ \ \ \ 0 \leq x \leq 1 \ , \ 0 \leq y \leq 1 \\0 \ \ \ \ \ other \end{cases} $ a) determine the $c$ ...
0
votes
1answer
20 views

Classify the 1000000 elements on the basis of 1000 tests

There are 1000000 elements with parameters (a,b,c,d,e,f). I have to classify them into 2 classes (whether f(a,b,c,d,e,f) less than N or not). The calculations for ...
0
votes
1answer
29 views

probabilility point closer to distance circle

the probability that the point is closer to the distance to the center of the circle than to the circumference is $\frac{1}{4}$ find probability: (A) When several points are selected sequentially ...
0
votes
0answers
11 views

combine several dependent variables

I have 3 boolean variables X,Y,Z (indicating if the same event has happened). Let's assume the error of each of them is 0.3. This means, for example, that if X is true then the probability of an ...
0
votes
2answers
934 views

Probability question using PIE

Five people check identical suitcases before boarding an airplane. At the baggage claim, each person takes one of the five suitcases at random. What is the probability that every person ends up with ...
0
votes
1answer
31 views

Expectation E(XY) of two dependent variables

If X and Y are 2 dependent variables, how does their combined expectation look? For example, if flipping a fair coin n times, with X representing the number of heads and Y representing the number of ...
0
votes
0answers
19 views

calculate the probability of error in a array of bits

I need to calculate the probability in a certain problem. So there are 555 random bits [1 0 1 0 ... 1 0 0 1 1]. These 555 bits are divided in 37 parts of 15 bits each. Of the 555 bits, X bits flip at ...
1
vote
0answers
18 views

the independence property of conditional expectation

I read a proof of following property: suppose $X$ is a random variable on the probability space $(\Omega,F,P)$, $A$ and $B$ are sub $\sigma$-field of $F$, $B$ is independent of $\sigma(X,A)$, then $E(...
1
vote
1answer
15 views

find probability distirbution of defective item

when 2 defective items are extracted one by one in a partition consisting of 6 products and checked, Let x be the number of test before finding the last defective items fing probability distribution ...
0
votes
0answers
26 views

simple one variable probability question

A Lego shop is open between 10-18.00. question 'a': At the 15.th of December last year, 16 Star Wars lego was sold. What's the probability that between 10.00 and 12.00 more than five legos were sold(...
3
votes
1answer
67 views

Expectation of dependent Bernoulli sum

I want to estimate the expected value of the following sum of random variables, $$ Y = \sum_{i=1}^N X_i $$ where each $X_i$ is a Bernoulli random variable. In particular, $$ X_1 = \begin{cases} 1, &...
0
votes
1answer
48 views

“expectation of sum is sum of expectation”, is this claim true? if yes, how to justify this claim?

this post is saying linearity of expectation gives following equation $$\mathbb{E} [\sum_{j\neq i} Y_i Y_j] = \sum_{j\neq i} \mathbb{E} [Y_i Y_j]$$ per wiki, Linearity of Expected_value is ...
2
votes
1answer
29 views

conditional probability about gambler winning x amount of coins

A gambler plays seven games one after the other and the chance to win each of them is $\frac{1}{3}$, independently of the others. For $k = 1, ..., 7$, if the gambler wins game number $k$, then the ...
0
votes
0answers
15 views

Inverse transform sampling from Poisson to Exponential

I have a Poisson disributed r.v. $$X\sim Poiss(\lambda)$$ and want to transform it to an exponentially dist. r.v. $$Y\sim Exponential(\lambda),$$ by using the inverse transform sampling method. I ...
0
votes
2answers
22 views

probability of drawing two card simultaneously

For e.g what is the probability that both card drawn from a deck of card will be king. Is drawing two card simultaneously and drawing one after the other(not replaced back), the same thing. Because ...
0
votes
0answers
19 views

What mean : $R^x$ is the probability that $Y_t$ start at $x$?

Here the context : I'm a bit confuse by the notation, since he defines $Y_t=Y_t^x$, and at the bottom, he says that $R^x$ is the probability that $Y_t$ start at $x$, and after he writes $$R^x(Y_{t_1}\...
0
votes
2answers
1k views

Sample Space & Combinations/Permutations

This is the question: "List the sample space for three individuals chosen at random to vote either Democrat or Republican. List the distinct combinations and permutations." These are what I got for ...
1
vote
0answers
19 views

Probability of a name to be in the “Firstname Lastname” order

I am trying to figure if a name is in the "Firstname Lastname" order, knowing the probability of each name to be either a firstname (class $A$) or a lastname (class $B$). In other words, given two ...
0
votes
0answers
8 views

probability limit of log chi sqaured

I have the following: $Prob\left(T \cdot ln(1+\chi_{1})>ln(T)\right) \rightarrow 0$ where $\chi$ is a chi-squared distribution with 1 degree of freedom and $T$ is an arbitrary real number and the ...
0
votes
1answer
19 views

If $\tau=\inf\{t>0\mid B_t\notin B(0,R)\}$ does $|B_{\tau}|\leq R$ or $|B_\tau|\leq R$ a.s.?

Let $(B_t)$ a Brownian motion starting at $0$ and let $B(0,R)$ the open ball of radius $R$. Let $\tau=\inf\{t>0\mid B_t\notin B(0,R)\}$. Does $|B_\tau(\omega )|\leq R$ for all $\omega $ or we only ...
0
votes
0answers
12 views

Cindy spins this spinner 300 times. work out an estimate for the number of times that Cindy will get L?

I spin a spinner 80 times. The table shows information about the results Outcome/Frequency J / 39 K / 25 L / 16 Cindy spins this spinner 300 times. ...
0
votes
0answers
18 views

Conditional summations in probability

Consider the following problem, from Tijms's Understanding Probability: Nobel airlines has a direct flight from Amsterdam to Palermo. This particular flight uses an aircraft with $N =150$ seats. ...
0
votes
1answer
23 views

Generate various PDFs (probability theory, statistics)

How can one most easily generate PDFs (probability density functions) with various shapes (say with 1,2,3,... etc. local maximums)? I mean... let's say I want to generate various continuous ...
2
votes
0answers
38 views

If $S_n$ is Binomial $(n,p)$ then $\mathbb P(S_n=k)\approx \frac{(np)^k}{k!}e^{-np}$.

I was reading this post, and I have to admit that I was quite confused. The question was : If $S_n$ is a Binomial r.v. with parameter $(n,p)$ s.t. $n$ large, $p$ very small and $np$ not to big (for ...
0
votes
0answers
14 views

Simplifying the summation of fractions of Gamma functions

I am trying to simplify this summation, but I have no idea how. I have tried using the fact that $$\sum_{i = 1}^n \frac{\Gamma\left(a + i\right)}{\Gamma\left(i\right)} = \frac{n\Gamma\left(a + n + 1\...
1
vote
2answers
1k views

How do we prove the union bound inequality (Boole's inequality) for infinite sets?

I am aware that Boole's inequality can be proved by induction. How do we extend these results to an infinite set? (since induction cannot be applied in this case) P($\bigcup\limits_{i=1}^{\infty} A_i$...