Questions tagged [probability]
A probability provides a quantitative description of the likely occurrence of a particular event.
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Expected value of min X for bernoulli success?
I take a SRS sample of size n from a population of x values ranging from 1 to N. Each selected unit also has a probability p of success or q = 1-p of failure (i.e. the probability of success/failure ...
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What is the Johnson family of distributions good for? What are their main features and applications?
I saw the Johnson family of distributions in context of reliability and demand modeling for supply chains, but I am not sure if they are bringing real benefit to pay for their relatively more complex ...
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Understanding proof of McDiarmid's inequality
I am working through Wasserman's lecture notes set 2 and I am unable to fill in the missing steps in the derivation of McDiarmid's inequality (p.5). Just like my previous question in the forum, I am ...
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Error count limit for known residual failure rate and desired false-positive rate?
We perform a known fixed number $K$ of independent experiments, each of which has known/assumed residual odds $p$ to fail. We count the number $N$ of failures. How can I choose $M$ so that $N>M$ ...
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Bound on moment generating function
This question arises from the one asked here about a bound on moment generating functions (MGFs).
Suppose $X$ is a bounded zero-mean random variable taking on values in
$[-\sigma, \sigma]$ and let $G(...
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Calculating probabilities related to order statistics
Crux of the question
Let $q \sim F$ with support $[0,1]$. Let $q_j$ be the $j$th order statistic of $N$ draws from $F$. Let $z_j \sim \text{Bernoulli}(q_j)$. See that these draws are independent, ...
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Understanding proof of a lemma used in Hoeffding inequality
I am studying Larry Wasserman's lecture notes on Statistics which uses Casella and Berger as its primary text. I am working through his lecture notes set 2 and got stuck in the derivation of lemma ...
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Calculating probability of a random sample without replacement
For this question, assume I have a standard deck of cards (52 cards, 13 of each set).
My question is: if I were to randomly draw 7 cards, how would I calculate the probability of at least one of ...
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How to interpret coefficients from a logistic regression?
I have the following probability function:
$$\text{Prob} = \frac{1}{1 + e^{-z}}$$
where
$$z = B_0 + B_1X_1 + \dots + B_nX_n.$$
My model looks like
$$\Pr(Y=1) = \frac{1}{1 + \exp\left(-[-3.92 + 0....
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Classifying new observations into two bivariate categories
I have two bivariate distributions $A$ and $B$. For some new observation $x$, I would like to calculate the probability that it belongs to $A$ and not $B$.
I can assume that $A$ and $B$ are ...
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pLSA - Probabilistic Latent Semantic Analysis, how to choose topic number?
I am learning about pLSA (Probabilistic Latent Semantic Analysis) right now, in the hopes of being able to apply it to biomolecular annotation prediction.
I have a very simple question: How do you ...
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Probability of a measurement under a continuous model
I may be measuring the length of a stick. I then want to see what is the probability of one measurement under a model.
When using a continuous model, the probability of a single number is zero:
$$
...
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The reciprocal of $t$-distributed random variable
How is $1/T$ distributed if $T$ follows a Student's $t$-distribution?
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Calculating probability of discovery
I have a planner that can evaluate N arbitrary states, and calculate their fitness. The domains it evaluates have no explicit "end state", so it has an infinite horizon.
What are good methods for ...
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Unable to wrap my head around applying HMM or other techniques to solve my problem [duplicate]
Possible Duplicate:
Trouble applying hidden Markov models
Hi and Happy New Year everyone!
I’m having a problem applying hidden Markov models to a game I’m building to learn about programming. ...
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Significance of a group answer to a question
I ask a group of 100 people a question to which each individual either agrees, disagrees or cannot answer - e.g. 70/100 agree, 10/100 disagree and 20/100 have no opinion. Is there any statistical test ...
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having trouble applying hidden markov models to my game [duplicate]
Possible Duplicate:
having trouble applying hidden markov/machine learning models
Happy New Year!
I’m having a problem applying hidden Markov models to a game I’m building to learn about ...
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Testing that a sample corresponds to an arbitrary distribution
I'm writing a set of unit test for a function that draws random numbers from an arbitrary distribution, defined as a PDF (example of such a function can be seen here). I am confused about the proper ...
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Using the Cox axioms to derive unknown probabilities from known probabilities
To strengthen my understanding of fundamental probability theory, I am working my way through Professor Aaron Hertzmann's Introduction to Bayesian Learning course notes. Section 3.8 of these course ...
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Factoring of conditional probability
I have been watching Tom Mitchell's lecture on Bayes Nets:
http://cc-web.isri.cmu.edu/CourseCast/Viewer/Default.aspx?id=bc507778-7a18-4121-b345-
83d9bab72f55
He states the following by way of the ...
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Testing against a custom probability distribution
What I want is some tool that will test an ordinal, discrete variable against a specific probability distribution, say the variable might take 4 values, 1, 2, 3, 4, and my expected probability ...
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Rolling dice problems
Consider the following little problem:
Roll an $m$-sided die $n$ times. What is the probability that all and only indices from $1$ through $k$ appear? ($k\leq n,m$)
Here's my thought (you can check ...
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Test to calculate the probability of winning
I'm running a predictive model to predict the probability of winning a certain item based on the price that I bid (other factors also). After running the model (ols) in R, I wanted to account for all ...
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Calculating probability question from a student
I have a simple question for you guys but I am studying for exams and I honestly forgot how this was done.
Here is the problem:
"A vase contains 4 white marbles, 3 red and 2 green. You draw ...
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Does this relationship have a name?
Suppose we have two finite discrete probability distributions, say $A$ and $B$. For some event $e$ in both $A$ and $B$, $f_A(e) = \frac{P_A(e)}{P_A(e) + P_B(e)}$ and $f_B(e) = \frac{P_B(e)}{P_A(e) + ...
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What is the probability that two independent random vectors with a given euclidean distance $r$ fall in the same orthant?
Consider two independent and identically distributed random vectors of dimensionality $N$, $\mathbf{x}$ and $\mathbf{y}$, where their elements are iid generated from a Gaussian with zero mean and ...
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probability of the next number of events occurring when the true probability is known [duplicate]
Possible Duplicate:
Probability of getting between
What is the probability that at least 24 of the next 50 people like to swim when the true probability of people liking to swim is 35%?
I ...
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Taking random items out of a container with replacement
I take vitamins in the morning, but one of them I only take a half tablet.
So, I have an initial container with 100 full tablets, and every morning I take out a random tablet. If it's a full ...
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Finding a minimum variance unbiased (linear) estimator
Here is a basic question that perhaps has a simple answer, but one that I was not able to find by quickly scanning the literature.
Suppose that I have a collection of $n$ unopened boxes. Each box $i$...
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Does an exact test always yield a higher P value then an approximated test?
I ran a simulation on that for mcnemar test, and the answer seemed to be yes.
I was wondering if this can always be said to be the case that the exact P value is higher (or not smaller) then the p ...
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Learning probability and statistics together
I posted a question earlier where I mentioned that I am interested in learning Machine Learning but that my background in statistics and probability is pretty weak.
Recently I previewed pages of 2 ...
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Odds that the jack of trump dealt 46% of time in pitch
In the game of Pitch what are the odds of the jack of trump being dealt in the 24 cards?
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Likelihood of errors based on a sample
I frequently do something like: load a bunch of data, and then scan some fraction of it randomly to verify that no errors occurred. The more data I verify, the greater my certainty that no errors ...
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Finding top n across multiple sets
Suppose we have a set of key value pairs, such that the keys can be ordered based on the value . We can easily extract and report the top $n$ keys. Now suppose the we havet S such sets. From each set ...
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Expected number of unseen cards when drawing $2n$ cards from a deck of size $n$
We have a deck of $n$ cards. We draw cards from it uniformly at random with replacement. After $2n$ draws what is the expected number of cards never chosen?
This question is part 2 of problem 2.12 in ...
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What is the P(A|C) if we know B depends on A and C depends on B?
Given a Bayesian network that looks like the following:
A->B->C
How do we compute P(A|C)? My initial guess would be:
...
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What is the intuition behind the score function? [duplicate]
Wikipedia tells us that the score plays an important role in the Cramér–Rao inequality. It also phrases out the definition:
$$V = \frac{\partial}{\partial \theta} \log{L(\theta; X)}$$
However, I ...
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Dice probability for Yahtzee large straight
I had a discussion with a friend of mine about what is the best course of action in the following Yahtzee scenario:
The first throw with 5 dice results in the following values:
...
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What is a normal distribution with 'common variance'?
Data Augmentation Approach in Bayesian Modelling of Presence-only Data (Divino et al.) states that the data ware sampled from a normal distribution with mean 2.0 and a common variance (section 3). ...
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Learning to create samples from an unknown distribution
I am interested in new generating samples to approximate some unknown distribution X, where each new sample is a real-valued vector.
The purpose it to be able to create a new (arbitrary large) ...
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If the n(th) moment exists does it mean all smaller moments exist too?
I would like to prove the following statement:
If the $r$th moment of a random variable $X$ exists and is finite,
then all moments $1$ to $r-1$ exist and are finite.
Edit: I mean the raw moments ...
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Tuning the birthday paradox [closed]
There are m disjoint sets Xi and a function f : [1..n] → [1..m] such that Yj = Xf(j). I am given oracles / samples from random variables xi and yj which uniformly select elements from Xi and Yj ...
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How can the F distribution be used, other than for hypothesis testing and confidence interval estimation?
I am trying to fit informed prior distributions to data using MLE, and F occasionally provides a best fit (lowest AIC value).
I am starting with only very basic knowledge of probability theory, so I ...
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"On/off problem" resources
I need to study in depth the on/off problem with different approaches: frequentist, bayesian - frequentist hybrid, profile likelihood.
The on/off problem is a counting experiment where you observe $...
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Expectation of length of a confidence interval
Let $X_1,X_2,X_3,\cdots,X_9$ be a random sample of size $9$ from a normal distribution, $\mathcal{N}(\mu,\sigma^2)$. If $\sigma$ is unknown, find the expected value of the length of a $95\%$ ...
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How to compute mean vector and covariance matrix of equal distributions?
This question is an extended version of this one.
As you can see here, two distributions are equal, I need to compute the parameters a,b,c,d and e. Could you show me a way to do that?
Assume a two-...
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What does "equal a priori class probabilities" mean?
I am trying to solve a problem about my homework. The problem says that
Assume a two-class problem with equal a priori class probabilities
Does it mean, mean vectors and covariance matrices ...
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Plain English explanation of Bernoulli mixture models?
Not exactly the most accessible explanation can be found here, but I'm looking for something more intuitive, examples of applications and so on.
Help is much appreciated.
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What does Jaynes' continuous pdf notation "$g(x)~\mathrm dx$" actually mean?
Something has been bugging me about E.T. Jaynes' treatment of continuous parameters.
In his book Probability Theory: The Logic of Science, uses notation that I am unfamiliar with when getting ...
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Probability of circles intersecting
Suppose that N circles are dropped randomly onto a square in 2-space in such a way that the center of each circle
lands somewhere in the square – even if some of the circle extends beyond the boundary ...
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