Questions tagged [probability]

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

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19 views

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 πœ€=Β±1 with probability 1/2 each, independently of 𝑋. Let π‘Œ=πœ€π‘‹...
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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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1answer
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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 ...
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2answers
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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Compute your payoff for the game and decide if it's worth playing

Suppose you are offered to play the following game: with probability 0.20 you win and receive $\text{\$}$40 and with probability 0.80 you lose $\text{\$}$5. The fee of playing the game is $1. What is ...
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Standard deviation and covariance of time series in slightly different time

Is it possible to stdev of multiple time series which have different length and different times? For example we have a data set, let's say a stock price or a family income ...
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Every second customer converts better. Can this be by chance?

At a bank customers are applying for a loan and then the customer gets a loan-offer which they can choose to accept or not. This bank has been running for 11 weeks and by coincidence a developer saw ...
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Model that takes two percentiles as input - what is the percentile of the output value

For some analysis I have two input variables with some (unknown) probabilities distributions. Of both the input variables I know the (assumed) 10th, 50th and 90th percentile. I have some simple model ...
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Can a Bayesian estimator perform better than an MVUE?

According to wikipedia: In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any ...
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Does a conjugate prior always exists? [duplicate]

Are there distributions where no conjugate prior exists? Is there a necessary and/or sufficient condition which guarantees the existence of a conjugate prior? Edit: Why has this question been closed? ...
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Deterministic function and conditional independence

Given a finite state probability $P(X,Y)$ and a deterministig function $f: Y \rightarrow Y'$, does conditional independence $p(x,y' \mid y) = p(x \mid y) \cdot p(y' \mid y)$ hold for any $f$?
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Multivariate normal's convergence to chi-square distribution [closed]

$Y \sim N(0,\Sigma)$, $\text{and} \, \sum p_i=1,p_i\geq 0.$ Prove that $Y^\top DY$ convergence in distribution to chi-square with $df=(k-1)$ as $n \to \infty$. $D=\text{diag}((p_1)^{-1},......,(p_k)^{...
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1answer
48 views

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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1answer
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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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1answer
26 views

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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1answer
31 views

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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1answer
26 views

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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38 views

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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0answers
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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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1answer
56 views

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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5 views

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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5 views

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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1answer
34 views

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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1answer
32 views

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 ...
2
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1answer
221 views

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.
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Why choose a Dirichlet distribution in Latent Dirichlet Allocation (LDA)?

I am studying Latent Dirichlet Allocation (LDA), but I don't have much knowledge of stochastic processes. From Wikipedia: LDA assumes the following generative process for a corpus $D$ consisting of $...
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2answers
61 views

Boundary of $E\left[\frac{\prod_{i=1}^n x_i}{\prod_{i=1}^n x_i+\prod_{i=n+1}^m x_i}\right]$

Suppose $X_i$ are i.i.d. In addition, $X_i>0$ and $E[X_i]>1$. Suppose $E[X_i]$ is known, could we find upper bound or lower bound for the following expectation: $$ E\left[\frac{\prod_{i=1}^n x_i}...
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1answer
61 views

Random variable without pdf but with a cdf?

In this video, the professor says that some random variables have no pdf but do have a cdf. Also, in my course material, I studied that converging in mean was stronger than converging in cdf which ...
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1answer
26 views

The student is sure of 4 questions. How likely is it to write the test and answer 6 out of 10 correctly? [closed]

The test contains 10 questions, each question is offered 4 different answers (a, b, c, d).
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Connection between factor analysis in theory and in practice

By Andrew Ng's lecture notes (which you can find at http://cs229.stanford.edu/notes2020spring/cs229-notes9.pdf), the factor analysis model has the following structure: Written out mathematically like ...
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9 views

How to produce a confusion matrix probability distribution in python? [closed]

Using different machine learning algorithms(xgboost & lightgbm) I generate binary predictions. The predictions (0 or 1) come with probabilities. With these predictions and the actual values I ...
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Probability variant [closed]

A coin is tossed three times. What is the probability that exactly two heads occur, given that (a) the first outcome was a head? (b) the first outcome was a tail? (c) the first two outcomes were heads?...
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1answer
35 views

What is the proper (measure theoretic) definition of a random variable?

(1) If we believe the following link: http://www.columbia.edu/~md3405/DT_Risk_2_15.pdf Then a random variable is a map from a probability/measurable space to a metric space, i.e. (2) If we believe ...
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Comparing win loss records or test results for statistical difference

I was reviewing some game statistics of wins and losses and was wondering what is the best approach to differentiate between how likely a person or team who had come in first is statistically that ...
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26 views

Ensemble Forecast Distributions

Say multiple forecast methods each generate full forecast distributions. When using point forecasts one may average the point forecasts to get a ensemble forecast. What is the safest way to ensemble ...
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2answers
43 views

Probability that 100 die rolls result in A 1s and B 2s

I am trying to calculate the probability above, and I think that my logic is a bit off. I thought that $$P(1s=a, 2s=b)=\frac{(\frac{1}{6})^{a+b}\times(\frac{4}{6})^{100-a-b}*100!}{a!b!(100-a-b)!}$$ ...
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10 views

Statistics - is these nbinom distribution

Are these question the same(except the parameters of course) : First - What's the number of trials needed until having at least 1 defective product, with a probability >= 85%, with pDef = 0.03. ...
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19 views

Question on working out a pdf from a posterior distribution

Need help with part (a) of this exercise. The exact step I'm concerned about is calculating the pdf from the relationship given in the exercise. I will appreciate any explanations on how should I ...
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1answer
34 views

Statistics - which distributions fit the description

I need to determine (then sample from) two distribution $X$ and $Y$, which have the following requirements: $X$ has 15k samples, $Y$ has 20k $X$ has a mean of 6k, $Y$ has a mean of 4k $X$ and $Y$ ...
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1answer
41 views

Bayesian decision boundary for bivariate uniform distributions

I would like an analytical derivation of the Bayesian decision boundary between 2 bivariate uniform distributions. Let me explain with an example. Suppose the two distributions are $$U_1 = 0.25 \text{ ...
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0answers
12 views

Information gain for unequally sized time series

I have some time series data, $\mathbf{x}_{1:T} = \{ x_1, \dots, x_T \}$ where the observation at time $t$, $X_t$, is a continuous random variable. Let $Y_t$ denote a discrete random variable at time $...
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26 views

95 % Quantile for anomaly detection

I'm building a model to detect anomalies in for multivariate data. At first stage I run VAR model every 10 min that predicts the next values, and when I get the values I calculate the Euclidean ...

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