# Questions tagged [probability]

A probability provides a quantitative description of the likely occurrence of a particular event.

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### Does $P(x_1, y_1,x_2, y_2) = P(x_1, y_1)P(x_2,y_2)$ imply $P(y_2 \mid x_1, y_1, x_2) = P(y_2 \mid x _2)$?

Say $(x_1, y_1)$ and $(x_2, y_2)$ are independent. So that $p(x_1, y_1, x_2, y_2) = p(x_1, y_1)p(x_2, y_2)$. Does this imply that $p(y_2 \mid x_2) = p(y_2 \mid x_1, y_1, x_2)$?
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### Confusion about two Gaussian distributions

From here, it says that, linear combination of two Gaussian distribution, are always Gaussians. However, Let 𝑋 be standard normal and 𝜀=±1 with probability 1/2 each, independently of 𝑋. Let 𝑌=𝜀𝑋...
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### Histogram showing twice the probabilities

I have this list of numbers: ...
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### Combinatorics - Arranging boys/girls

Six girls are to enter a dance with 10 boys to form a ring so that every girl is between two boys: (a) What is the probability that some specified boy remains between 2 boys? (b) A spectator notices ...
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### Infinitesimal generator

I have been studying continuous time markov chains through Dobrow's book. Everything went fine until the author introduced the concept of infinitesimal generator, which he refers to as $\textbf{Q}$. ...
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### Calculating the number of excursions in a Gaussian

I've written an algorithm designed to solve the following problem: An m x n array, where m and ...
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### Probability of a sample with known characteristics have been sourced from a specific population

Suppose there is a room with $d$ drawers. A drawer $i$ has $n_i$ balls from which $b_i$ are black and $n_i - b_i$ are red. Imagine a random drawer is selected and $n_w$ balls are randomly picked from ...
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### Drawing numbers from a set, quantile

We have a set containing $20$ numbers from $1$ to $20$. Each time we draw only one number, and repeat it $15$ times (without replacement). Let's denote $X-$ the largest drawn number. Find the smallest ...
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### Probability Problem :: [closed]

Six girls are to enter a dance with 10 boys to form a ring so that every girl is between two boys:(a) What is the probability that some specified boy remains between 2 boys? (b) A spectator notices ...
84 views

### Baysian probability of false positive COVID-19 test

I am wondering what the probability of a false positive COVID-19 test would be in my city. I'm attempting to use Bayes Theorem to calculate this, however I'm getting very different results based on ...
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### Integral of difference of density functions of two Continuous Random Variables goes to 0

The problem says : Let $(X_n)_{n=1}^\infty$ be a sequence of continuous random variables with probability density functions $(f_n)_{n=1}^\infty$ , and let $X$ be another continuous random variable ...
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### Can the posterior mean always be expressed as a weighted sum of the maximum likelihood estimate and the prior mean?

See this question. Is this always true? Can the posterior mean always be expressed as a weighted sum of the maximum likelihood estimate and the prior mean (after choosing some appropriate prior)?
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### Why do I get $E[Y^2] > 0$ with one approach but $E[Y^2] = 0$ with another approach for $Y|X \sim N(0,\sigma_X^2)$?

Suppose $Y|X \sim N(0,\sigma_X^2)$ where $\sigma_X^2$ is the variance of $X$. Does this mean $Y$ itself has a normal distribution? Because it seems like this would cause a problem. If $Y$ has a normal ...
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### How likely is a difference of final conversion between two randomly split groups who have passed the same sales funnel?

I'm having a discussion with a friend regarding an interview question (neither of us got the job :): An ecommerce gets 40,000 visitors on 1 day. They are randomly split into 2 groups (A/B) of 20 000 ...
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### Distribution of Functions of One or Two Random Variables

I just wanted to confirm my understanding related to the distributions of functions of random variables. Can someone please tell me if all of my points are correct and make sense? I also have an ...
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### Sum of generalized gamma variables

This question is in a way duplicating this question, but I'm not happy with the final answer and feel like I need a deeper dive. Assume that $X_j \sim \Gamma(\alpha_j,\beta_j)$, where $\alpha, \beta$ ...
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### Choosing numbers from an interval

We draw $10$ numbers from an interval $[0,5]$. Compute probability, that at least two of these numbers will land in $[1,3]$. So if $A-$ at least two numbers land in an interval $[1,3]$, then: $A'-$ ...
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### Why do elliptical copula densities contain $x_1$ and $x_2$, but Archimedean copula densities contain $u_1$ and $u_2$?

$$c\left(u_{1}, u_{2}\right)=\frac{1}{\sqrt{1-\rho_{12}^{2}}} \exp \left\{-\frac{\rho_{12}^{2}\left(x_{1}^{2}+x_{2}^{2}\right)-2 \rho_{12} x_{1} x_{2}}{2\left(1-\rho_{12}^{2}\right)}\right\}$$ is the ...
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### J-S Divergence as a Percentage

Is there a way to interpret the Jensen-Shannon divergence which is normalized to be between 0 and 1 between two probability distributions as a "percent difference", i.e., there is a x% ...
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### Why doesn't the entropy decrease with an increasing number of observations?

I'm trying to think more about entropy. I have the following toy example: Consider a coin flip. Case 1: I think p_h = 0.5 The entropy of this is 0.5 ln(0.5) x 2 = ln(0.5) Case 2: I don't know what p_h ...
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### Strict Sense Cyclostationary and Shifting the X with $\theta$

Fellow stackexchangers, I did my best to put a topic that describes the question that I am going to ask. I am reading Probability, Random Variables, and Stochastic Processes by Papoulis and I am ...
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### Distance between two clusters vis-a-vis distribution of data

This was a question asked to me in an interview for graduation. I am interested to pursue in side of probability, statistics and machine learning. Professor asked me how would you calculate distance ...
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### The expectation of the inverse of a negative binomial random variable?

Suppose $X \tilde{} NB(n,p)$ and $\mathbb{P}(X=x) = \binom{x-1}{n-1}p^n(1-p)^{x-n}$. Then what is $\mathbb{E}(\frac{1}{X}) = \sum_{x=n}^\infty \frac{1}{x}\binom{x-1}{n-1}p^n(1-p)^{x-n}$? Many thanks
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### Front door formula - calculation in practice

I am reading about the front door criterion and I want to make sure that I understand what is calculated in the second part of the formula, where the backdoor path between $Z$ and $Y$ is blocked by ...
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### Show that maximum of two random variables is a random variable

If we have that $X$ and $Y$ are random variables, how do we prove that $Z=max(X,Y)$ is also a random variable ? I want to do this by showing that $Z$ is measurable, but I don't know how to do this.