The probability that an event A will occur, when another event B is known to occur or to have occurred. It is commonly denoted by P(A|B).

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When to use Bayes' theorem to calculate conditional probability?

Given 2 events $E, F$, I know that $P(E | F) = \frac{P(E \cap F)}{P(F)}$. However sometimes the Bayes' theorem is used instead: $P(E | F) = \frac{P(F | E) P(E)}{P(F|E)P(E)+P(F|E^{c})P(E^{c})}$. ...
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20 views

Calculating probability based on mixed variables

Assume there are $K$ people and iid. parameters $a_1,\ldots,a_K$ associated to them with $a_i \sim U(0,1)$. Person $i$ observes his own fixed $a_i$ with some noise: \begin{equation} X^{(1)}_i= a_i+ ...
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22 views

Conditional expectation with conditioning on two independent variables

Let $Y_1$ and $Y_2$ be independent r.v.s, and $X$ be another random variable. What is $E[X|Y_1, Y_2]$? Is $E[X|Y_1, Y_2]$ equivalent to $E[X|Y_1] \cdot E[X|Y_2]$? More specifically, is there a ...
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45 views

Conditional Probability equal to zero

Let $X_1$ and $X_2$ be two independent Normal variables with mean $\theta$ and variance $1$. Let $S=X_1+X_2$ and consider the following conditional probability: $$P \left[ X_1 \leq x_1; X_2 \leq x_2 ...
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17 views

Meaning of marginal prior distribution

I have a variable $\mu$ with distribution, given $\phi$ of $ \mu|\phi \sim N(\phi, \sigma^2)$, and the distribution of $\phi$ is $\phi \sim N(\nu, \tau^2)$. (Here, $\sigma^2,\nu$ and $\tau^2$ are ...
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32 views

Reconcile difference in answers for conditional probability

Disclaimer: I am a mechanical engineer, not a statistics guy, so I sometimes miss things. Background: I have a copy of "Applied Probability Models with Optimization Applications" by Sheldon M. Ross ...
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22 views

Probability of classification given two observations

If two doctors each correctly diagnose a disease 55% of the time, and both agree in a certain case there is disease, what is the probability of disease? Surely it is higher than 55%?
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48 views

Determine the probability density function

I have the following problem. Let's say g is a function of A and R. I run a Monte Carlo ...
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115 views

Given two independent normal random variables $X$ and $Y$, what is $P(X\leq x\mid X>Y)$?

As the title says, I'm looking for the distribution of $X$ given that $X>Y$.
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23 views

Assuming training data as set of target funtions

In this 10th slide of http://www.cs.cmu.edu/afs/cs.cmu.edu/project/theo-20/www/mlbook/ch6.pdf presentation The Training data set $D$ is assumed as the set of target function . Actually $D = ...
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45 views

Formulas for probabilities in Bayes theorem

In continuation to this question $p(h|D) = \frac{p(D|h)p(h)}{p(D)}$ $p(h) = $prior probability of hypothesis $h$ $p(D)$ = prior probability of training data $D$ $p(h|D)$ = probability of $h$ ...
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37 views

How do I think about conditional probability in this situation?

I am trying to understand some data relating the likelihood of a positive stock return following a certain signal. The frequency of positive returns differ across datasets (over a particular time ...
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28 views

A Conditional Probability Problem

This is a biological problem, however the problem requires a statistical solution, so therefore I'll keep it as abstract as possible in order for statisticians to be also able to contemplate it - I'm ...
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3answers
84 views

The probability of detecting a defect given defect growth when the two variables are independent

I am dealing with a real world problem and just wanted to check if I am going about this the right way. I have the probabilities that a defect will grow within a particular structure, say ${\rm ...
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1answer
35 views

Marginalising probability distribution

I am reading a book on probability theory and there is an example to denote conditional independence. A joint distribution is given as follows $$ p(a, b, c) = p(a) p(b) p(c|a,b) $$ Now, assuming ...
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149 views

Convergence in Distribution\CLT

Given that $N = n$, the conditional distr. of $Y$ is $\chi ^2(2n)$. $N$ has marginal distr. of Poisson($\theta$), $\theta$ is a positive constant. Show that, as $\theta \rightarrow \infty$, $\space ...
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12 views

Determining the state of a conditional system

I have the following situation: There are two systems, $s_A$ and $s_B$. For $s_A$ there are two states: $s_A = a,b$. For $s_B$ there are three states: $s_B = 1,2,3$. Now, what I want to do is measure ...
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12 views

Soft Margin Loss and Conditional Probabilities

The following question is taken almost verbatim from "Learning with Kernels", by Scholkopf and Smola. Consider the following soft margin loss function: $max(0,1−yf(x))$. I have the problem of needing ...
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68 views

Pattern Recognition in Practical Problem in R

I am using R for last 4 months with basic test analysis, regression models, ANOVA etc. Now I am looking into a particular problem. In words problem can be stated like, A particular machine got a ...
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1answer
43 views

How to calculate the probability of winning an election?

I'm analyzing a survey data that contains the following 3 variables (questions): If the election was today who would you vote for in the following list? (A, B, C, D, or E)? Let's imagine the ...
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48 views

ldentifying observations on the edges in heteroscedastic data

Consider bivariate data consisting of correlated variables (r ~ 0.9). It is known how both the variables are calculated. Each observation is identically distributed but non independent. Also, the ...
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1answer
48 views

What is the difference between these two probabilities?

I am working on a word sense disambiguation task and I have a dataset consisting of labelled sentences for two meanings of the word 'line'. I am trying to find 'trigger' words that are good at ...
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21 views

Dependent variables

I am faced with the following problem: In general terms: I have a number dependent measurements in two biological samples and want to test if the difference between the samples is significant. ...
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97 views

Making an inference from conditional probability values

Consider the following data about bulb failures (which require a filament replacement for repair) compared across three companies: ...
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63 views

conditional independence and joint distribution

This is something from Chris Bishop's book and I want to confirm that I have understood this correctly. So, suppose we have a joint distribution between two RVs as: $$ P(a, b) = P(a|b)P(b) = ...
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11 views

Determine if Clique tree is calibrated

A, B 0 0 50 0 1 10 1 0 20 1 1 50 2 0 15 2 1 15 The above table defines the clique potential $\phi_1(A,B)$. B C 0 0 62.5 0 1 12.5 1 0 12.5 1 1 62.5 The above table defines the clique ...
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40 views

Probability of reaching node A from node B in exactly X steps

I have a three-node matrix with two edges (A-B and A-C). I would like to determine what the probability is starting from B and ending at C in exactly 100 steps. I have only written out probabilities: ...
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1answer
24 views

Recursive definition of conditional normal distribution

I am acquiring samples that are Gaussian distributed and I need to calculate $$ p(x_n | x_{n-1}, x_{n-2}, \ldots , x_1) $$ for each sample $x_n$ as it comes in. I am trying to break this expression ...
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50 views

Is there a statistical test to show that sugar liking flies tend to belong to bins where there are flies that also likes sugar and vice-versa?

I have about 900 flies that either like sugar or are neutral (about 50% in each group). However, on closer inspection I do notice a trend that sugar liking flies are from bins 1-10 (each bin has about ...
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21 views

Is there a way to calculate the probability of drawing X+ > y% of time given X- [closed]

Say I have a bag with 10 balls (either red or blue), and I know that there are 8 balls that are red. How do I calculate the probability of drawing a a red ball $>75\%$ of the time if the first ball ...
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19 views

Learning Conditional Random Fields using EM Algorithm (from unaligned data)

I am trying to learn CRF from the unaligned data in Natural Language Understanding application. There is one paper in this field which does exactly the same, Learning conditional random fields from ...
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36 views

Conditioning on a variable

I am looking at some software code that performs conditioning on random variables. For example, one can have a set of random variables which have a multivariate normal distribution associated with ...
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1answer
35 views

Summary statistic for 'average' conditional probabilities

I have the following scenario: Let $X_i$ denote some event. Now I have the following calculated probabilities. $P(X_1 \cap X_2 \cap \cdots \cap X_9 | X_{10})$ $P(X_1 \cap X_2 \cap \cdots \cap X_{8} ...
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93 views

Probability to draw a black ball in a set of black and white balls with mixed replacement conditions

When a black ball is drawn, it is not replaced in the set, but white balls are replaced. I have thought of this, with the notations: $b$, $w$ the initial number of black and white balls $x_i = (b ...
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1answer
100 views

Conditional expectation of $(X+Y)^2 | X = x$

If the density $f(x,y) = c$ when $x>0$, $y>0$, $x+y < 1$ and $0$ otherwise, find $$E((X+Y)^2 | X = x)\text{ for } x \in (0,1).$$ How to approach this question? Can we approach it by ...
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2answers
200 views

Conditional probabilities/expectations

A coin minting machine randomly produces unbalanced coins so that the probability of getting a head in tossing a coin is a random variably $Y$. Supposed $Y$ has a pdf $f(y) = 2y$ for $0 <= y <= ...
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26 views

Expected assortativity coefficient of random graphs

I am wondering whether there exist a closed-form expression for the expected assortativity coefficient (http://arxiv.org/pdf/cond-mat/0205405.pdf) of an Erdos Ranyi random graph model $G(n,m)$, where ...
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95 views

Probability in mafia/werewolf party game

Okay, first post on this Stack Exchange board, and I'm no mathematician so I kind of lack the proper jargon. But here goes. I'm currently playing a game known as werewolves, which is based on the ...
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1answer
58 views

ARMA conditional density

Consider the ARMA(1,1) process \begin{align} y_t=a_1y_{t-1}+b_1\epsilon_{t-1}+\epsilon_t, \end{align} and assume $\epsilon_t$~$N(\mu_t,\sigma_t)$. And $\mu_t$ and $\sigma_t$ are all known. ...
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1answer
45 views

What is a conditional quantile function?

What is conditional quantile of $Y(t)$ given $Y(t-1)$ where $Y(t)$ is a univariate time series (they call it conditional value at risk in risk management). Can anybody explain this? Thanks
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13 views

Proof for bivariate conditional mean of Gaussian dist [duplicate]

I see that a lot of questions are answered here for multivariate and bivariate conditional distributions. But I did not find the proof of these equations (I need just for bivariate case). to get ...
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14answers
25k views

Amazon interview question—probability of 2nd interview

I got this question during an interview with Amazon: 50% of all people who receive a first interview receive a second interview 95% of your friends that got a second interview felt they had a good ...
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1answer
71 views

Conditional Expectations and Variances

$\textbf{Background:}$ When $\mathbb EX$ is hard to calculate, it is a common trick to use the following formula: $$\mathbb EX=\mathbb E[\mathbb E(X|Y)].$$ And similarly, $\mathbb VX$ can be ...
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3answers
177 views

Naive Bayesian Algorithm in R/SAS for categorical input variables?

Could anyone please let me know how to implement Naive Bayesian Algorithm in R or SAS?I have got a training dataset with all the categorical predictors and target variable(3 levels).I got to build a ...
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20 views

Calculate probability distribution for n of participants in common to two parties given their size

I have a set of parties $V=\left\{ {V_i, V_j,..., V_n}\right\}$. Each party has a corresponding attribute $S=\left\{ {S_i, S_j,..., S_n}\right\}$ indicating the number of people participating to the ...
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30 views
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72 views

Conditional Distribution of Poisson Variables, given $\sum X_i$

An exercise asks me to show that if $X_1,X_2,\ldots,X_n$ is a random sample from a Poisson distribution with parameter $\theta$, the conditional distribution of $X_1,X_2,\ldots, X_{n-1}$, given ...
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32 views

Derived probability formula for an event…How to deal with “unintuitive” factors?

I have been working on a problem where I would like to estimate a particular probability. Namely, the problem has to deal with estimating the probability that an element $a$ in a structure $A$ can be ...
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58 views

Probability extraction from random forest classifier

I have a random forest to perform classification. I need the real probability of the predicted class. They take a feature vector X and output a predicted class C. Additionally we can compute the ...
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23 views

Is there an efficient way to compute the ordered probabilities of a weighted sample of events?

For this example, I have 4 different events a ... d, each with a different probability of occurrence. My sample space is the following: {a a b b b b c c c d d d} p(a) = 2/12 p(b) = 4/12 p(c) = ...