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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.

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Example for Lévy's continuity theorem

I am searching a sequence of RV $(X_n)$ for which we prove a convergence in distribution to a RV X, using the fact that the characteristic functions $(\varphi_n)_n$ converges pointwise to some ...
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Probability to win at least 1 time, playing 3 times.

I have this problem: In a game, the probability of win is $1/3$ and of lose is $2/3$, ¿What is the probabilty of win at least 1 prize playing 3 times? The probability of win at least 1 prize, ...
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Let $X_1$ and $X_2$ be uniform on $n$-spheres. What is the distribution of $\| X_1+X_2\|$?

Suppose we have two independent random variables $X_1$ and $X_2$ distribution on $n-1$-sphere of radius $r_1 $ and radius $r_2$, respectivly. Assume $r_1>r_2$. Recall, that the $n-1$-sphere of ...
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Find the conditional distribution of $X$ given that $Y=y$

Let $X$ and $Y$ be two random variable with density $f_{X,Y}(x,y)=\begin{cases} \frac{1}{y}, & \text{for } 0<x<y<1, \\[8pt] 0, & \text{otherwise}.\end{cases}$. To find the ...
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Cramér-Lundberg model for on-demand insurance

I am looking for inspiration and perhaps guidance on the following as I’ve been stuck for a while now: Context: I am working on a practically oriented project to adjust the Cramér-Lundberg model ...
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21 views

Puzzling birthday match probability

OK I've been searching the Web for an answer to a burning question. And finally this page has.. Or almost had an answer! In a group of 19 people. What are the odds that.... 2 specific pairs of ...
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1answer
30 views

probability of rigged coin for exact heads?

I have 6 coins with probability for heads in toss as: 0.51 0.52 0.53 0.57 0.48 0.49 What is the probability of getting exactly 3 heads in 6 tosses? I can do it for a normal coin but a rigged coin ...
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Binomial Random Variable Intuition

I have the following question from a stats course about binomial distribution: A multiple choice test has 10 questions, each with 5 possible answers, only one of which is correct. A student who did ...
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Gambling and Kelly criterion

Consider gambling between a player $A$ and an infinitely wealthy opponent. Suppose the expectation of the game is negative. Then by Law of Large Numbers, I know that $A$ is almost surely losing all of ...
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Expected time for two new customers in a $(\lambda_A=1,\lambda_S=2)-M/M/2$ in steady state.

Two customers enter a $(\lambda_A=1,\lambda_S=2)-M/M/2$ in steady state. Find the expected time that it will take for both to make through queue and have their service done. My attempt: if this is a ...
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How could I calculate the probability of $\mathbb{P}\left(A^{c} | B\right)$ and $ \mathbb{P}\left(B^{c} | A^{c}\right)$

Good morning! The task is to calculate the probability of $\mathbb{P}\left(A^{c} | B\right)$ and $ \mathbb{P}\left(B^{c} | A^{c}\right)$. Given are the probabilities of $\mathbb{P}(A)=2 / 3, \mathbb{...
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What is the base measure in measure theory?

I see the term "base measure" used frequently about measures. I do not completely get what that exactly means: Some examples are: Let $\cal F$ be the space of all probability density functions ...
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Can a stochastic process be neither adapted to filtration nor previsible?

The idea behind the question arises from my intuition about the concepts of 'adapted to filtration' and 'previsbility'. If a process is adapted, it essentially means that the evolution of the ...
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1answer
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Using Markov chain instead of total probability

This question was given to me as a review for an upcoming exam: If a baseball team wins a game, they have a 40% chance of winning the next game due to getting overconfident. If they lose the previous ...
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Sample two individuals and find the probability of the following events

Blood can be classified according to ABO-type: $A$, $B$, $AB$ and $O$, but also according to Rh-type, $P$ (positive) and $N$ (negative). Suppose that every individual has one Rh-type and one ABO-type ...
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How can I apply a Markov model to estimate transition probabilities bewteen areas in robot foraging simulation?

I wish to apply a Markov model to a robot in a 2d simulated foraging exercise to estimate the probabilities of transitioning between "landmarks" represented by coloured patches. The idea is that the ...
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From the infinitesimal generator of a stochastic process, can we find the finite dimensional distribution of it?

Let $X=(X_t)_t$ and $Y=(Y_t)_t$ a stochastic process. I know that if $X$ and $Y$ has the same infinitesimal generator, then the finite dimensional distribution (fdd) of $X$ and $Y$ coincide, i.e. $$\...
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24 views

Probability of a random subset of Z

I'm stuck in this question, could someone give me a hand? I'll post what I've done so far. Question 9: Let $A=(1,2,3,4)$ and $Z=(1,2,3,4,5,6,7,8,9,10)$, if a subset B of Z is selected by chance ...
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Probability on a disc

I was solving some questions of probability and I came across the following one: Question: Given an arbitrary disc with radius $r> 0$. A point is chosen randomly on the disk. Determine the ...
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Odds of two runners ending up with the same average rank across multiple races?

Imagine a race with $n$ runners, all equally skilled so that the outcome is just based on luck. If the race is run $m$ times, what are the chances that two runners end up with the same exact average ...
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probability apples and oranges

I was solving some questions of probability and I came across this one. I have already made the following calculation for now. Question: In a basket there are fifteen apples and ten oranges. Knowing ...
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1answer
13 views

find the density function of $(U,V)$

Let $X=(X_1,X_2)^T$ be a random vector with 2-dimensional normal distribution, $E(X_1)=E(X_2)=0 , Var(X_1)=Var(X_2)=1$ and $Cov(X_1, X_2)= \nu$ with $|\nu| <1$. And let $Z \sim Bin(1,\alpha)$ be ...
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Intuition behind combinations

Example problem: Suppose you have to select 5 cards from a deck: what is the probability of getting 4 diamonds and 1 spade. The solution should be : $$\ \frac{\binom{13}{4} \binom{13}{1}}{{52}\...
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If $X_n$ and $Y_n$ are independent does $(X_n,Y_n)\overset{d}{\rightarrow}(X,Y)$?

More formally: If $X_n\overset{d}{\rightarrow}X$ and $Y_n\overset{d}{\rightarrow}Y$ and also $X_i$ and $Y_j$ are independent for all i,j; does $(X_n,Y_n)\overset{d}{\rightarrow}(X,Y)$? I am aware of ...
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Simple probability equality proof

For given continuous random variable A,B,C and arbitrary continuous function f, and probability density function p, can you help prove/disprove following equality? p(A, B, f(A,B)) = p(A, B) p(A, B, ...
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Probability that maximal elements has the same position in samples from correlated random variables

Let $x$ and $y$ be two correlated random variable (say, standard normal) with correlation coefficient $\rho>0$. Let $X= \{x_1, x_2, ..., x_L\}$ and $Y= \{y_1, y_2, .. y_L\}$ be samples of size $L$ ...
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362 views

Analytic proof of area probability

$X,Y$ are i.i.d. $unif(-1,1)$ random variables. Prove that $$P(X^2+Y^2\leq 1)=\frac{π}{4}$$ Geometrically, I understand how that happens. $(X,Y)$ is a random point in square having centre at origin ...
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1answer
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How can I prove that a complex variable does not follow a normal distribution from $R$ and $\Phi$ distributions

I am trying to prove that a complex variable $Z = R.\exp(i.\Phi) = R.\cos(\Phi) + i.R.\sin(\Phi) = X + i.Y$ does not follow a normal distribution when $R\sim \mathcal{N}\left(\mu_R, \sigma_R^{2}\...
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1answer
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Why $\mathbb P(B_t\geq L)=\mathbb P(B_t\geq \ell, \tau\leq t)$?

Let $(B_t)$ a Brownian motion. I want to prove that for all $L\geq 0$, $$\mathbb P(\sup_{0\leq s\leq t}B_s\geq L)=2\mathbb P(B_t\geq L).$$ The proof start by : let $\tau=\inf\{t\geq 0\mid B_t= L\}$. ...
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Probability, Permutation and combination

One team consisting of 11 employees. Each team should contain one TL and one DM. In all, there are 18 team. What will be the probability that the TL and DM would be repeated.
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1answer
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1D Random walk with reflective barriers — average time to return to origin

I have very limited probability background, but I came across a problem in an engineering application: Is there a formula that computes the average number of steps taken for a particle beginning at ...
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Sum of dependent Binomial distributions

In one of my classes, we stated that if $X_i$ are independent Bernoulli random variables with p proportion of success, then the distribution of the sum $\sum X_i$ is Binomial(n,p). I already proved ...
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Determining whether counting events is a renewal process

$\begin{array}{l}{\text { Consider a renewal process with mean interarrival time } \mu . \text { Suppose that each }} \\ {\text { event of this process is independently "counted" with probability } p ....
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1answer
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Dart board probability using line method with Poisson application

You randomly throw darts at a dartboard, one dart every second. Suppose that every dart independently hits the dartboard at distance X from the center, where X is a Unif[0,30] random variable. Your ...
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Calculate number of failures until success where the probability changes after a failure.

How do you calculate the expected value of the number of failures until the first success is reached where the probability will change after a failure. Let $p$ be the probability of success. Let $X$ ...
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How to use Wald's equation to determine expectation in gambling model?

$\begin{array}{l}{\text { In each game played one is equally likely to either win or lose 1. Let } S \text { be your }} \\ {\text { cumulative winnings if you use the strategy that quits playing if ...
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Brownian motion Wiener process expected first passage [duplicate]

1) For the standard Brownian motion, {W(t),t≥0}, what is the expected first passage time, E(τa), for a>0, where τa=inf{t:W(t)≥a}?
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27 views

Likelihood of getting flush, straight, etc

On Planet X, cards can take on a numerical value from $1$ to $7$ (inclusive) and their suit can be either red or blue. In a game of poker, each player gets three cards. 1) What is the probability of ...
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1answer
24 views

Distribution of arrival times of Poisson point processes

Let $(M_{t})_{t\geq 0}$ and $(N_t)_{t\geq0}$ be two independent Poisson point processes with rate $\lambda$ and $\mu$ respectively. Let $\tau$ be the first arrival time for the process $N_{t}$. Find: ...
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2answers
26 views

Expected value of the sum of three different die

Suppose you have a 4-sided die, a 6-sided die, and a 12-sided die. You roll the three dice and add up the numbers that show up. What is the expected value of the sum of the rolls? My attempt solution ...
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1answer
53 views

Sum of two random variables uniformly distributed on circles

Suppose we have two independent random variables $U_1$ and $U_2$ unfiorm on \begin{align} S_i = \left\{ (s_1,s_2) \in \mathbb{R}: \sqrt{s_1^2+s_2^2} =r_i \right\} \end{align} respectily. Assume $...
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Policy gradient base line function

On the bottom of page ten of the following paper on probabilistic reinforcement learning, there are 3 equations where is author manipulates the policy gradient $\nabla_\theta J(\theta)$. Can someone ...
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3answers
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The paradox of random occurrence the family problem

we have 100 families: 10 families have no children, 40 families have 1 child for each one, 30 families have 2 children for each one, 10 families have 3 children for each one and 10 families have 4 ...
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Coin Puzzle with infinite paradox

Here’s the puzzle/problem: Let’s presume we are best friends. I own the house next to you. I make a gaming proposition along the following lines: You throw a coin over and over again. So long as it ...
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Probability a students is present [on hold]

A student is presented in the exam knowing 85 questions of 100 possible questions. In the exam, randomly, three questions are withdrawn. Find the probability of the event A: "the student will know the ...
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1answer
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Probability and Linear Combination

If you want to write all possible linear combinations of the following two mixed strategies (probabilities): $\Big(\frac 23,0,\frac 13\Big)$ and $\Big(\frac 12,\frac 12,0\Big)$ Would it be correct ...
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Sow that one probability distribution can't be preferred to others.

Let $R= ${ $r_1, r_2, r_3, ...$} be a countable set of rewards, and let $U$ be a utility function on R. Let $P_1, P_2, P_3, ...$ be a sequence of probability distributions on R. For each distribution ...
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Probability of at least $n/2$ coin flips

We define p-coin as having a $p$ probability to land on Tails and $1-p$ to land on Heads. $X$ is a random variables that given $n$-flips results, gives the number of tails ($T$) that we got: $$X(...
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28 views

What is the probability that the original sign was plus?

A slip of paper is given to A, who marks it with either a plus or a minus sign; the probability of his writing a plus is $\frac{1}{3}$. He then passes the slip to B, who may either leave it or change ...