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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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Conditional Independence

I have a joint probability, which factors as follows: $P(A,B,C,D) = P(A,B) \cdot P(C|A) \cdot P(D|B)$ So I know that $C$ and $D$ are independent given $P(A, B)$ right? I want to infer $P(A,B|C,D)$....
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A little help with probability calculation [on hold]

i have a question, if we know P(A) and P(B),how can we calculate the $P(A) \bigcap P(B)$? for example let say P(A)=2/3 and P(B)=1/6
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When do these two definitions of KL-divergence match?

Suppose $P$ and $Q$ are two distributions on a space ${\cal H}$ (could be a subset of an infinite dimensional function space) with p.d.fs denoted by the same letter then one can define the $KL$ ...
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Why use Convolution of probability function instead of cross-correlation?

First, let me quickly remind you of the two operations: convolution and cross-correlation between 2 function $f$ and $g$, assuming continuous domain. Cross-Correlation $f \star g$ : $\int f(\tau)g(\...
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How to compute the loss normal function (not standard normal distribution)

I am struggling with computing the expression of the following term: $E[x-Q]^+$ where $x$ is a normal r.v with mean $\mu$ and variance $V^2$. I want to express it as a function of $\mu$, $V^2$ ...
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192 views

When to use empirical cdf?

I saw in some papers that people sometimes use empirical cdf or complementary cdf to quantify the distribution of data. What is the advantage of using them and what interesting can they tell about ...
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Can someone come up with a concrete example of ML estimation with uniformly distributed noise?

I am currently reading Boyd's book on convex optimization of maximum likelihood estimation problem http://web.stanford.edu/class/ee364a/lectures/stat.pdf I have zero background in statistics so I am ...
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Probability that rolling N dice that the two larger dice sum to X [on hold]

New here. I'm trying to figure out how to come up with a roll system for a game, where i want the probability of rolling N dice or rolling one dice N times and then taking the two largest outcomes, ...
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Is probability density itself a random variable?

I'm trying to understand random variables a bit better. As I spent time thinking about, it occurred to me that the probability density of a random variable could itself be considered a random variable....
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How to find maximum probability [on hold]

Members of a mafia group gathered at a secret hideaway. The city police come to know about the meeting and plan to arrest the leader of the group. The police know that the members of the mafia will ...
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I've heard that ratios or inverses of random variables often are problematic, in not having expectations. Why is that?

The title is the question. I am told that ratios and inverses of random variables often are problematic. What is meant is that expectation often do not exist. Is there a simple, general explication of ...
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Lottery entry for convention panels - grouping vs going solo

The upcoming Star Wars convention in Chicago is testing out a new system that allows people to enter a lottery to try and get a seat at the big panels. This has lead to a lot of math and pseudo math ...
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Twist to 3 prisoners problem applying Bayes rule

T, J and B work for a company but the chairman has decided to fire one person randomly chosen through 3 cards. The chairman decides to fire with unequal probabilities -- T with probability of 15%, B ...
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Probability question using Bayes rule

I have a probability question here which I believe I need to apply Bayes rule to solve it. Here is the question: A specific enzyme, QQ, which is designed to quickly help cows gain weight, is ...
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Posterior pdf is proper if and only if the conditionals are

Note: This is a homework problem so PLEASE DO NOT PROVIDE A COMPLETE SOLUTION. Some gentle hints would be good. Let pdf $f(x\mid\theta,\, \phi) = g(\theta;\, x)$ be given and unidentifiable in $\phi$....
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Should the <s> and <e> be included in the vocabulary while calculating probability of a sentence in a Bigram model with Laplace smoothing?

I am working through an example of Add-1 smoothing in the context of NLP Say that there is the following corpus (start and end tokens included) ...
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Support of joint pdf being the Cartesian product of supports of the one-dimensional conditionals

Note: This is a homework problem so please DO NOT PROVIDE A COMPLETE SOLUTION. Some gentle hints would be good. Suppose $X_1,\, \ldots ,\, X_p$ have a $p$-variate joint pdf whose support is the ...
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What is the difference between “likelihood” and “probability”?

The wikipedia page claims that likelihood and probability are distinct concepts. In non-technical parlance, "likelihood" is usually a synonym for "probability," but in statistical usage there is a ...
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How to find CDF of a function of continuous joint distribution from PDF of joint distribution?

Here's what I think I should proceed: Make a level curve for the function keeping the constraints given in the PDF of joint distribution in mind. Find the area of interest keeping the constraints ...
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1answer
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Which curve to select for finding the CDF of a function of a continuous joint distribution?

I came across a question which required to find the CDF of a function of a continuous joint distribution: $W=XY$. The following is the joint PDF: $$f_{X,Y}(x,y)=\begin{cases}\frac{xy}{4000}&,\, ...
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Why does “explaining away” make intuitive sense?

I recently learned about a principle of probabilistic reasoning called "explaining away," and I am trying to grasp an intuition for it. Let me set up a scenario. Let $A$ be the event that an ...
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Favorite sequence in 10 flip tosses

I have the following question. Somebody likes to flip coins. In particular, this person is delighted to get a sequence HTTH. Assuming the person flipped coins 10 times, what is the probability that ...
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Intuition behind product distribution pdf

Say we have two distributions $X$ and $Y$. I know that the pdf of the distribution $Z = X + Y$ is given by: $f_Z(z) = \int_{-\infty}^{\infty}f_X(x)f_Y(z-x)dx$ The intuition is that you sum up the ...
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Football Pool Chances

I am trying to confirm the probability of being a weekly winner in an NFL football pool at least once during a given NFL season. The pool is set up as the following: Each player picks five teams ...
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cannot prove simple conditional probability

Is the next true? If so, why? P(A∩B|C) = P(A|B∩C) x P(B|C) I saw as part of solution of an exercise, but I can't prove that is true, how is it possible to arrive to that?
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Level curves and functions of pair of Random Variables?

I came across the following question: I tried solving it, the following is my 1st attempt:(2nd method at the end) 𝑃[𝑊≤𝑤]=𝑃[𝑋𝑌≤𝑤]=𝑃[𝑌≤𝑤/𝑋] And then I simply double integrated keeping the ...
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Minimum CDF of random variables

I know that the joint cumulative function of two random variables X and Y is defined as: $F_{X,Y}(x,y)=P(X≤x,Y≤y)$. How can I find the CDF for $F_{X,Y}=\{x,x\}$. In other words is what will be $Pr\{...
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1answer
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Three-prisoner problem and Bayes rule

Here is the wiki of Three-prisoner problem, in which only one prisoner is pardoned, and the Bayes solution is given in the wiki. My problem is pretty much the same, except that only one prisoner is ...
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1answer
44 views

Probability of normal random variable yielding highest value among other normal variables

Let's say I have a competition with $N$ participants. Each participant yields a score that is normality distributed with unique means and unique variations. Each participant gets to post one score (we ...
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2answers
82 views

What can we say about $N_{i}$ where $N=N_{1}+\cdots+N_{m}$, $N\thicksim Geom(\frac{1-p}{p})$ and conditional distribution of $N_{j}$ is binomial

Suppose that the number of events $N$ is a Geometric random variable with mean $\frac{1-p}{p}$. Further suppose that each event can be classified into one of $m$ types with probabilities $p_{1},p_{2},\...
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Sufficient conditions for variance convergence in CLT

More generally, if $\{X_n\}_{n\in\mathbb{N}}, X$ are real random variables with finite variance such that $X_n\xrightarrow{d}X$, what are some sufficient conditions to assure that $\operatorname{Var}(...
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Why do we take a function of x as the limit of integration over y while calculating the marginal pdf of x?

I searched through similar questions but couldn't find one answering my question. I know the following is the way of finding marginal pdf from joint pdf. $$ f_x(x)= \int_{-\infty}^{\infty} f_{x,y}(x,y)...
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Expectation Maximization for a 2D Normal Model

I'm working through an example in Richard Duda's Pattern Classification on Expecation Maximization Algorithm. Specifically I'm trying to understand the expectation part, and how the parameters get ...
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How do I reconcile the output of a decision tree visual vs the output of the DecisionTreeClassifier class?

I have a decision tree model stored in dt. I also plotted and can visually see the decision tree itself. Given a new, unseen customer, I can use the visual tree to ...
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Calculate probability based on part of features

I have a knowledge base, where each row represent a class and columns represent features of classes: ...
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I have 3 gene datasets, 2 genes are sig in all 3, what is the probability of this being by chance?

I have three large RNA-seq datasets, two genes were found to be significant in all three datasets, what is the probability of this happening by chance? not sure what info i would need so here is ...
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Why is the relationship between max(X,Y) and X and Y the way it is?

My textbook says that the above follows from the observation: $\{W\leq w\}=\{X\leq w\},\{Y\leq w\}$, How do we prove the above observation?
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Interpretation of Naive Bayes Probabilities

I'm dealing with a Naive Bayes approach to a Multiclass Classification problem with 9 different classes in the target variable. Let's assume the following: I've fitted a model to my training data and ...
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Expected value of a variable given a probability distribution

I have fitted some data with a normal and gamma distribution. But I need to find the expected value of my variable. For example is x is my data (mine is not normal) and fit them into a normal and ...
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How is $s^k$ in p.g.f. likened to Dirac measure $\delta_k$?

How is $s^k$ (sometimes $z^k$, $\theta^k$) in probability generating function likened to Dirac measure $\delta_k$? I don't see how these are similar. Yet I see them equated. Clearly the Dirac ...
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An urn contains n unique marbles. I draw k of them with replacement, and get r duplicates (r>0). What is the MLE for n?

What is the MLE for n which is the total number of marbles in the urn? What is the expected value of n? And what if r = 0 (no duplicates were drawn)? Can anything be said about n? A duplicate ...
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1answer
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number of option to choose 5 couples of men and women [on hold]

I need help with the following question: There is 10 mans and 12 womens, how many options there is to make 5 couple (women,men)? I tought maybe (10 choose 5) - mens and (12 choose 5) women and 5! ...
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26 views

Probability of having a match in an existing database

Imagine you are an airport authority in CountryLand and you have two passenger types: human passengers and animals. There is no way for you to distinguish between human passengers and animals except ...
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46 views

Using binomial approximation to calculate probability

Question is as follows: In a shipment of $20$ engines, history shows that the probability of any one engine proving unsatisfactory is $0.1$. a) Use the Binomial approximation to calculate the ...
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Emergence of Lognormal distribution for the concentration of chemical compounds

I'm currently reviewing the literature about lognormal distributions describing/approximating the variability of a given chemical compound across different cells/ samples etc etc. The main argument ...
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2answers
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How to simulate, if a customer will buy

If customer 1 will buy a product with a probability of 0.6 and every following customer 0.7, how can I simulate a system with n customers? The result should show which customers bought and which ...
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Alternative to Hoeffding Bound in Streaming for Normally Distributed Variable?

I am reading a paper (Hellinger Distance Trees for Imbalanced Streams by R. J. Lyon, J. M. Brooke, J. D. Knowles, B. W. Stappers) on using the Hellinger Distance for Very Fast Decision Tree (VFDT)...
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How do I calculate the probability that an event occurred by 1 of 2 independent processes when they occur together?

Suppose that I know the probability of an event occurring by either process 1 or process 2: process 1, p(event)=1-exp(-ax) process 2, p(event)=1-exp(-by) With this information, is it possible to ...
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Joint probability distribution of some Gumbel differences

I would like your help to double check my derivations below involving the joint probability distribution of some Gumbel differences. Consider $K$ i.i.d. random variables $\epsilon_1,...,\epsilon_K$, ...
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1answer
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proof of independence of X-Y and X+Y when X,Y come from bivariate normal

I have a bivariate normal distribution: $$\begin{pmatrix} X\\Y\end{pmatrix} \sim \mathcal{N}(\mu, \Sigma)$$ where: $$\mu = \begin{pmatrix}\mu_X \\ \mu_Y\end{pmatrix}$$ $$\Sigma = \begin{bmatrix} \...