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Questions tagged [probability]

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

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2answers
23 views

Law of total probability with intersections

So in my intro course to stats, we ecnountered the law of total probability. The definition is $$P(A) = \sum^n_{j=1}P(A\mid H_j)P(H_j)$$ However, the definition says $$\bigcup^n_{j=1}H_j = S \quad \...
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1answer
53 views

Computing the probability density function

Suppose we have the cdf $$F_X(x) = \begin{cases} 0 \quad \quad, x<-1 \\ 0.25 \quad \quad, -1\leq x < 1 \\ 0.5 \quad \quad, 1 \leq x < 2 \\ \frac{2}{3} \quad \quad, 2 \leq x < 3 \\ 1 \quad ...
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33 views

Observational Data and Bias - A real problem

I'm hoping you all can provide some guidance. I'm working a problem with the following objectives and data set. I would like to be able to predict, for each unit, at each sampled moment, the expected ...
2
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1answer
95 views

Scikit-Learn Gaussian Mixture: How can log-probabilities be positive? [closed]

I am fitting a Gaussian Mixture model: gm = GaussianMixture(n_components=K) gm.fit(features) When I do: ...
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0answers
27 views

Distance between two distributions

I am analyzing time-series data, and I would like to detect a significant change in data distribution. I already know about Bhattacharyya distance, but that requires histograms of equal-sized bins. ...
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1answer
49 views

Non Linear transformation of Random Variable

I have a pdf say $p(x)$. Now, I apply some transformation (may be linear or non-linear) to the variable $x$ say $g(x)$. Let the new pdf be called $p(y)$. For, a small change in $x$ say $dx$, there ...
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2answers
76 views

X is Uniform $[-\theta,\theta]$ what is the distribution of $Y=\frac{1}{x^{2}}$?

X is Uniform $[-\theta,\theta], \theta>0$ what is the distribution of $Y=\frac{1}{x^{2}}$ So I've been working on some transformation questions; however, most of them have been one to one. I am a ...
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1answer
22 views

How to combine confidence and probability scores into a single metric

I have an algorithm which outputs a confidence score and a probability score that a particular user belongs to class $C_i$, for multiple values of $i$. I want to output a single class $C_o$ as the ...
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0answers
15 views

Confusion about the probability of being dealt 2 pairs

There are seemingly three ways to calculate this, and they yield different answers. Please help me understand which way is correct. 1) Having two pairs is choosing 2 ranks (namely A and B) out of 13 ...
1
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1answer
24 views

Probability distribution for draws with conditional replacement? (or how many matches does it take to beat Yugi The Destiny?)

While trying to figure out an old video game, I seem to have stumbled upon a problem that's halfway between a binomial distribution problem and a hyper-geometric distribution problem. With the hopes ...
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1answer
44 views

When drawing three number from the same distribution, what is the probability of the first to be between the two?

If I draw 3 numbers: $a$, $b$ and $c$ from the exact same distribution (unknown, but the same for each of the numbers). I want to know the probability that $a$ is between $b$ and $c$. That is: $b < ...
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0answers
24 views

Why do we use geometric distribution here in place of the multiplication rule?

For the question: "Only 4% of people have type AB blood. What is the probability we won't find a type AB donor before the 10th person?" the textbook says to use a geometric distribution. But for the ...
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2answers
56 views

Confusion about range of integration for density function

Consider the joint density function: $$f(x,y) = \begin{cases} 2 & & \text{for } 0 \leq x \leq1 \text{ and } 0 \leq y \leq 1-x, \\[6pt] 0 & & \text{otherwise}. \end{cases}$$ From ...
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0answers
91 views

$X$ and $Y$ being two independent Poisson random variables

Let $X$ and $Y$ be two independent Poisson random variables, with means $\lambda_1$ and $\lambda_2$, respectively. Then, $X + Y$ is a Poisson random variable with mean $\lambda_1+ \lambda_2$,. Arguing ...
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2answers
4k views

Did Statistics.com publish the wrong answer?

Statistics.com published a problem of the week: The rate of residential insurance fraud is 10% (one out of ten claims is fraudulent). A consultant has proposed a machine learning system to review ...
1
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1answer
81 views

Convergence in probability of $\frac{1}{n}\sum_{i=1}^n X_i^2$ when $X_i$'s are i.i.d $N(0,1)$

Question: My approach: And after this I am stuck..How do I put the modulus over here and how do I determine the appropriate value of "k" ? (here k signifies the value of convergence in probability ...
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0answers
52 views

Identifiyng mutual trends between treatments

(WARNING: Not a statistician - do not get mad) We have two different treatments $A,B$, each was tested separately ($A$ treated group vs control, $B$ treated group vs control). In each comparison, the ...
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0answers
42 views

Did I understand the usage of Gumbel-Softmax reparametrization correctly?

I am working on a deep learning model, which has a mixture of experts formulation like $\log p(y|x)=\log \sum_{z}p(y|z,x,\theta)p(z|x,\phi)$. So, each $p(y|z,x,\theta)$ is a deep learning classifier, ...
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12 views

Mathematical Probability problem on Aplia Hold 'Em

Rule: Aplia Hold 'Em is a simplification of the popular poker game Texas Hold 'Em. Aplia Hold 'Em is played with a special Aplia card deck that contains only six cards: the jack, queen and king of ...
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3answers
120 views

Does $\mathbb{P}(X < a) = \mathbb{P}(f(X) < f(a))$?

If $f(x)$ is a monotonic increasing function, then does $\mathbb{P}(X < a) = \mathbb{P}(f(X) < f(a))$? My intuition says it's true but I cannot prove the case nor find the name of the theorem.
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1answer
93 views

Optimal scaling of the Random Walk Metroplis-Hastings algorithm and the speed measure of the limiting diffusion

Let $d\in\mathbb N$ with $d>1$ $\ell>0$ $\sigma_d^2:=\frac{\ell^2}{d-1}$ $f\in C^2(\mathbb R)$ be positive with $$\int f(x)\:{\rm d}x=1$$ and $g:=\ln f$ $Q_d$ be a Markov kernel on $(\mathbb R^...
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0answers
14 views

Updating the odds / probability once user plays a bet

I am working on an assignment in which I have created a game in which user will bet on outcome of a game. Team A will win or not. if win = Invest 7 get 10 if not win = Invest 5 get 10 I have an ...
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2answers
106 views

Find $P(A^2 < B)$ where $A$ and $B$ are independent and uniformly distributed $\mathrm{Unif}(0,h)$, $h > 0$

I solved it two ways and in both the cases the answer is different and different from the actual answer. Approach 1: Since, $A$ and $B$ are independent, we can find the joint distribution of $AB$ ...
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0answers
9 views

Estimate expected power on a wind turbine based on other nearby wind turbines

I'm looking for a reliable way to estimate the power that a wind turbine should be producing, based on the power that its neighbours are producing. We use this to identify turbines that are ...
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1answer
18 views

Mass of single random variable from joint mass

I'm taking a course on probability and confused about single variable mass and joint mass. Here is a quote of the video, I'm learning from: Using the joint mass to calculate the mass of a single ...
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2answers
53 views

Transformation of Random Variables and making sense

So, I was going through Pattern Recognition and Machine learning by Bishop, page - 18, and I came across probability densities where I read about the transformation of a random variable. I couldn't ...
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2answers
54 views

clarification on Bayes theorem application

I'm experimenting with bayes rule. I have been invited to last stage interview in highly selective company. I want to use bayes rule to evaluate my chances of getting the post. Let's define the ...
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1answer
40 views

Probability that at least 3 of the 5 randomly selected questions can be solved

As my friend my studying for her final for her Philosophy class, she asked me a question. The professor assigned a list of 10 essay questions, and would be picking 5 to put on the exam. Of the 5 ...
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1answer
26 views

How is probability y = j|X calculated from an ordinal logistic regression model?

I have an ordinal logistic regression model fitted with lrm from the rms library in R, and am presenting results as prob y = j|X ...
3
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1answer
50 views

Sufficient Conditions for the Central Limit Theorem

My understanding is that the central limit theorem applies as long as the variance of the random variable is less than infinity. Is this equivalent to saying that all moments are finite? If not, what ...
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0answers
30 views

Minimum of n independent, but not identically distributed inverse Gaussians

I would like to find the probability distribution of n independent, but not identically distributed, i.e. differently parametrized inverse Gaussians. I would prefer an analytical solution. Also, would ...
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1answer
42 views

Covariance of Two Quadratic Forms

We're looking for the $\operatorname{Cov}\left[x^T A x, ~x^T B x\right]$ where $x$ is random variable and mean-centered, but not independent and $A$ and $B$ are symmetric matrices. The fundamental ...
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0answers
42 views

Lognormal Distribution Probability

I'm dealing with this question, and i didn't understand should i use the $f(x)$ formula for lognormal distribution or can i calculate it with $z(P)$? Thank you for help. And i've found probability $1....
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0answers
23 views

number of samples needed to determine quantile

I am trying to estimate 16th, 50th, and 84th percentiles of some quantity $b$, and it would be helpful to know if I have enough samples to do so reliably (say, at the 1% level). Sound like a beginner ...
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0answers
25 views

P-value on statistical tests [duplicate]

If I have a p-value of 0.01 on a statistical test, does that mean the probability of my hypothesis being wrong is 1%? If not, why not?
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0answers
16 views

Convergence radius of random power series

I have a problem with getting how I should interpret the random power series. I am given $X_n$ that are i.i.d random variables. Further the random power series, $\sum_{n=0}^{\infty} X_{n}z^{n}$ ...
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1answer
31 views

Is this sentence referring to joint or conditional probability?

The following is a quote from my textbook, in a chapter discussing the Viterbi algorithm (Durbin, Richard, et al. Biological sequence analysis: probabilistic models of proteins and nucleic acids. 1st ...
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2answers
122 views

How can I estimate the highest posterior density interval from a set of x,y values describing the PDF?

I'd like to estimate the Highest Posterior Density Interval (HPDI) of a calculated density function, rather than from empirical samples as is normally done (e.g., from an ...
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1answer
48 views

When $(X_1-X_0, X_1-X_2)\sim (X_2-X_0, X_2-X_1)\sim(X_0-X_1, X_0-X_2)$?

Consider a bivariate probability distribution $P: \mathbb{R}^2\rightarrow [0,1]$. I have the following question: Are there necessary and sufficient conditions on the CDF associated with $P$ (joint or ...
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0answers
24 views

Mean and variance of maximum of normal random variables

I'm trying to find the mean and variance of $Y = \max(X_1, ..., X_n)$ where $X_i \sim \mathcal{N}(\mu_i, \sigma^2)$. Note that the $X_i$ are independent, but not identically distributed. That is, ...
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1answer
162 views

Mutual Independence in a Multivariate Normal with Identity Covariance

Consider a random vector $X$ which follows a multivariate nomal with zero means and Identity Covariance. $X\sim \mathcal{N}_n(\mathbf 0, \mathbf I)$ We can say that the individual variables $X_1, ...
3
votes
2answers
93 views

Distribution of maximum of normally distributed random variables

I'm trying to find the closed-form CDF and PDF of $Y = \max(X_1, ..., X_n)$ where $X_i \sim \mathcal{N}(\mu_i, \sigma^2)$. My thought process so far: $$ \begin{align*} F_Y(y) &= \mathbb{P}(\max(...
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0answers
29 views

CLT and convergence of Variance

I am looking at a problem where the sum of the individual $X_i$ is $S_n=X_1+\dotsm+X_n$. The probability is given as, $P(X_i=i)=P(X_i=-i)=\frac{i^{-\alpha}}{4}$ and $P(X_i=0)=1-\frac{i^{-\alpha}}{2}$. ...
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0answers
9 views

coverage index?

Suppose I have a space of potential outcomes X with a probability distribution on it. I assume that there is a distance function between elements of X (e.g. X is a metric space). I also have a set S ...
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1answer
53 views

Basic probability theory

I was recently given the following statistics: On a particular highway, 18% of drivers are black, 63% of drivers searched by the police are black. So, a black driver is 7.7 times more likely to be ...
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3answers
4k views

Shouldn't the joint probability of 2 independent events be equal to zero?

If the joint probability is the intersection of 2 events, then shouldn't the joint probability of 2 independent events be zero since they don't intersect at all? I'm confused.
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0answers
13 views

General techniques for coupling a set of random variables with mutual dependence

Disclaimer: the usage of coupling is in the title is not of the usual definition in probability theory. Suppose I have a set of random variables $\{X_1, X_2, \dots, X_n\}$, indexed by time $t$, and ...
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0answers
26 views

Rewrite a constraint on the probability distribution using the cumulative distribution function

Consider a probability distribution $P: \mathbb{R}^3\rightarrow [0,1]$ and assume $$ (\diamond) \text{ }\int_{(x_1,x_2,x_3)\in \mathbb{R}^3 \text{ s.t. } x_3= x_1-x_2} dP=1 $$ Questions: Is there a ...
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1answer
30 views

Binary classification and target-label proportion

Suppose that we have a binary classification problem with a vector y = [1 1 0 1 0 0 1 ... 0] having the proportion: ...
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1answer
92 views

Who invented the independence notation $\perp \!\!\! \perp$?

This is more of a historical question: who invented the notation $\perp \!\!\! \perp$ for denoting (conditional) independence?