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Questions tagged [independence]

Events (or random variables) are independent when information on some of them tells you nothing about the probability of occurrence (/ distribution) of the others. Please DO NOT use this tag for independent variable use [predictor] instead.

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When does $E[f(X_i)]=E[f(X_j)], i\neq j$?

Suppose we have random variables $X_1, \dots, X_N$, with joint probability distribution $F_{X_1,\dots,X_N}$. Under what conditions does the following equality holds? $$E[f(X_i)]=E[f(X_j)],\ \ i\neq ...
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2answers
39 views

How to find out if there is any real pattern in the data set?

Let's assume that we have a regression problem (in the machine learning sense). Our data set consists of pairs of features vectors and numeric targets. It might be the case that there is absolutely ...
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1answer
18 views

Interpreting the Chi Square for a 2x2 contingency table

Here is my problem. I need to implement an algorithm to find if there is a dependency between 2 categorical variables. My population can vary a lot as it is an input I don't have control over. My ...
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6 views

Spherical distribution counterexample

Let $X \in \mathbb{R}^{n \times p}$ be a random matrix. The distribution of $X$, call it $F(X)$, is said to left-spherical if for any orthogonal matrix $O$ such that $O'O = I_n$, $F(OX) = F(X)$. ...
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24 views

Conditions for independence of a random vector to a random variable

Given a random vector $Z=(X_1, \dots, X_n)$ and a random variable $Y$, and that $X_i$ and $Y$ are independent for $i=1, \dots, n$, under what conditions are $Z$ and $Y$ are independent? The answer to ...
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1answer
340 views

Sums of normal random variables

Consider a sample of n independent normal rvs. I would like to identify a systematic way of calculating the probability of having the sum of a subset of them larger than the sum of the rest of rvs. An ...
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2answers
41 views

10% rule for sample sizes

In an introductory stats book by Nicole Radziwell "Statistics the easy way with R" , an assumption used for nearly every statistical test (e.g.t-tets, anova, etc) is that the sample size should not ...
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1answer
31 views

How can observations of random variables be IID, if they are not themselves random variables?

Suppose we perform some experiment which results in an outcome $\omega \in \Omega$. A random variable $X$ maps $\omega$ to a real number, and the (discrete) distribution $P(X)$ maps $X$ to $[0, 1]$. ...
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0answers
40 views

Interpretation of low intraclass correlation coefficient (ICC) values and assumptions of independence

I have an unbalanced repeated measures data set and my main goal is to see what stats are correlated with win percentage. There is an unbalanced number of observations per player. With my data set I ...
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3answers
117 views

Is the sample correlation always positively correlated with the sample variance?

The sample correlation $r$ and the sample standard deviation of $X$ (call it $s_X$) seem to be positively correlated if I simulate bivariate normal $X$, $Y$ with a positive true correlation (and seem ...
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1answer
20 views

Conditional independence and joint distributions in graphical models

I'm reading Deep Learning by Ian Goodfellow and Yoshua Bengio and Aaron Courville. In chapter 3 about graphical models, to reduce the model complexity, we assume that certain conditional independence ...
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10 views

Why dependencies would cancel while paramterizing CPD?

Consider that in CPTs of a Bayes Net(structure is fixed) one dependence is fix(parameters are set for $P(X|Y)$) and we are parameterizing other CPDs, is it possible that the other parameters would ...
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0answers
18 views

Conditional dependence between two ordinal variables

I really hope that this question was not answered in any way before, but I couldn't find an appropriate solution to my problem - most probably due to my lack of statistical knowledge. I am trying to ...
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1answer
14 views

Setting boundaries for calculating $P(Y/X>2)$ choosing $dx/dy$ order [duplicate]

Given two independent variables $X$ and $Y$, with marginal pdfs $f_X(x)=2x, 0 \le x \le 1$ and $f_Y(y)=1, 0 \le y \le 1$, calculate $P(\frac{Y}{X} > 2)$. So this can be written as $P(Y>2X)$, ...
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42 views

Proof of Berkson's Paradox

I'd be very thankful if someone could help me with the proof of Berkson's Paradox. I found this quite helpful thread which I understand How to prove Berkson's Fallacy?. But I'm actually trying to ...
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1answer
34 views

mixed effect model?

I have 365 days of bike sharing demand data for 15 stations. I am thinking of taking each day data as a data point (n=365*15=5475) and relate daily weather variable as well as land use variable. The ...
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3answers
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Does $\mathbb{P}(XY=a)=\mathbb{P}(X=a)\mathbb{P}(Y=a)$ for $X,Y$ id?

Does $\mathbb{P}(XY=a)=\mathbb{P}(X=a)\mathbb{P}(Y=a)$ for $X,Y$ id? What confuses me is that I only find this independence result for $\mathbb{P}(X=a,Y=a)$ i.e. $\mathbb{P}(X=a \cap Y=a)$
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1answer
393 views

Relation between independence and correlation of uniform random variables

My question is fairly simple: let $X$ and $Y$ be two uncorrelated uniform random variables on $[-1,1]$. Are they independent? I was under the impression that two random, uncorrelated variables are ...
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1answer
89 views

Is the Chi-Square Test of Independence the best option for 3x2 contingency table?

I have some data which in the terms of a 3x2 contingency table look in the following way. ...
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0answers
13 views

The consequences of ignoring autocorrelation of errors for the LASSO estimator?

In ordinary linear regression, Y = X$\beta$ + $\epsilon$, if the error is autocorrelated, then the assumptions under the Gauss-Markov theorem are violated. For example, autocorrelation violates the ...
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35 views

Converting a sequence of dependent random variables to an iid sequence

I have a sequence $L = X_1, X_2, ..., X_m$ of iid random variables and another sequence $R = X_{m+1}, X_{m+2}, ..., X_{n}$ of iid random variables such that each $X_i$, for $1 \leq i \leq m$, is ...
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0answers
26 views

Covariance of two uncorrelated variables multiplied with the same random variable [duplicate]

I have a problem where I am faced with the term $\operatorname{Cov}[XY,XZ]$. However, I do not know what to do with this term. I may assume that $Y$ and $Z$ are independent and that $X$ is independent ...
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1answer
33 views

Implications of i.i.d. sample

I have the following question: I have managed to solve it but I wasn't sure if my reasoning was correct. So I can express the OLS estimator as $\sqrt{n}(\hat{\beta} - \beta) = (\frac{1}{n}\sum_{=1}^{...
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2answers
62 views

Can statistical dependence arise only at the limit?

(This has been inspired by a comment exchange with @guy). Assume we have two infinite sequences of random variables, $\{X_n\}$ and $\{Y_n\}$. Assume the RVs in $\{X_n\}$ are statistically ...
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1answer
110 views

Sampling from Gaussian mixture models, when are the sampled data independent?

Suppose I generate a Gaussian mixture model with $N$ Gaussian distributions $p(x) = \sum\limits_{i = 1}^N w_i \mathcal{N}(x;\mu_i, \Sigma_i)$ where $w_i$ are the weights. Now I sample some points $\...
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0answers
11 views

Compare categorical variables that are not(?) independent

I have some survey data in which respondents were classified in to two overlapping groups (i.e. respondents who work in a profession and people who do this same work on their own-- many people do ...
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0answers
19 views

Probability Two Independent Samples of Different Size Have Some Intersection

I've been working on generating a probability rule for taking two random samples without replacement and finding a given amount of overlap between the two. So far I've worked through simple cases ...
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0answers
17 views

LASSO Cox Model after multiple imputation

I want to develop a predictive survival model on a data set with about 8000 subjects and 38 covariates. About 4% of subjects had the event of interest. There are 21 variables with missing values, ...
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0answers
16 views

Test of independence for response frequencies with multiple responses per user

Scenario: We have users in two categories, A and B. In each category, a user can provide a response which can be X, Y or Z. A user can provide one or many responses. Therefore, every subsequent ...
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1answer
28 views

Statistical test to use to test mean equality given non-independence assumptions

In a survey conducted, 75 survey participants were asked to rate randomized keywords from groups 1-5. Every keyword belongs to a group and subjects were asked to rate each keyword. I would like to ...
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1answer
23 views

Calculate the probability of any one independent scenario occurring

I have $3$ independent scenarios and I need to calculate the probability of any one of them occurring. I’ve calculated the probability of the $i$-th event as: : $1-\big(\frac{70000-1}{70000}\big)^1 ...
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1answer
62 views

Govt. organisation favouritism in its complaint handling

I am analysing some data for a government organisation and trying to convince them that their complaint decisions are biased. I believe they favour certain organisations. I have put together the ...
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0answers
48 views

How can we analyze a dataset with a pseudoreplication problem?

I am trying to analyse a data set with a pseudoreplication problem and would be grateful if someone could give me some guidelines on the best way to deal with it. We have a set of subjects exposed to ...
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4answers
47 views

Question regarding independence of events

I am reading Wasserman's book and his class notes. In the notes, I found the following statement If A, B are disjoint, both having positive probability, then A and B cannot be independent. Now, if ...
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1answer
88 views

Using chi square on unequal groups with unequal repeated trials

My dataset includes four groups (A, B, C, D) of unequal numbers of individuals. Each individual engages in an activity that results in one of three categorical outcomes (X, Y, Z). The individuals ...
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0answers
9 views

How to calculate multiple correlation coefficients and test hypothesis?

I found the estimation of S and $\overline{x}$ of a data set to be approximately $\overline{x} = \begin{pmatrix} 185 & 151 & 183 & 149 \\ \end{pmatrix}'.$ and $S = \begin{pmatrix} 95 &...
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0answers
20 views

Trouble understanding derivation of probability for continuous time markov chain

I'm working on exercise 6.10 from "Introduction to probability models" by Sheldon M. Ross. There's an expression for the probability $P_{00}(t)$ that I don't understand. Here's the relevant ...
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1answer
25 views

What stats method to apply when measuring a trait multiple times from the same individual?

I have 2 species of wildflowers. Using scanning electron microscopy. I took images of the petal surface. On the petal surface I measured the height and width of individual cells. I have 1 to 3 ...
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1answer
44 views

Number of parameters in Bayesian Classifier

Problem Assume we have a Bayesian classifier with the three following features to determine whether a software user is a student, an ...
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0answers
26 views

How to determine events are independent?

Say we have a period of time that lasts 1000 seconds. A coin is tossed every x seconds until time runs out. Sometimes the interval between tosses is 2 seconds, sometimes it's 5, sometimes it's 20. In ...
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0answers
27 views

Making vectors independent of each other

Assume that I have three vectors $A, B, C$ containing information about a set of variables. There may be common information shared between these vectors, that is $A, B, C$ are not independent. What ...
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0answers
98 views

Putting into words conditional probabilities and the chain rule

Consider events $a, b, c, d$. $p(a,b,c,d) = p(a)p(b|a)p(c|a,b)p(d|a,b,c)$ ; by the chain rule. We derive this by repeatedly applying the definition of conditional probability. This feels a bit dry, ...
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0answers
10 views

Strategic interaction (dependencies) between observations in Cox regression?

I want to study the survival times of subjects with Cox regression, but I suspect there may be strategic interaction between the subjects. How can I account for such dependencies in a Cox model? For ...
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0answers
9 views

Why are jackknife pseudovalues treated as independent?

Treating jackknife pseudovalues as IID variables is ubiquitous among the sources I've come across. However, I never see an attempt to justify it beyond citing a "proposal" by Tukey's 1958 paper, which ...
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1answer
74 views

Finding variance of infimum of a set [duplicate]

Let $X_1, X_2, \cdots,X_n$ be independent and identically distributed random variables having an exponential distribution with mean $\frac{1}{\lambda}$. Let $S_n=X_1+X_2+\cdots+X_n$ and $N = \text{...
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1answer
22 views

Comparing proportions from overlapping samples

I'm looking to compare the read rate of an email newsletter with the read rate of another newsletter. However, many (not all) of the people who receive one newsletter also receive the other. What ...
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1answer
123 views

Covariance of products of dependent random variables

I have four random variables, A, B, C, D, with known mean and variance. As well: Cov(A,B) is known and non-zero Cov(C,D) is known and non-zero A and C are independent A and D are independent B ...
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1answer
29 views

Combining probabilities from multiple observations

Given a number of probabilities taken from a large number of recorded football games. A: Chance of WIN given 6 corners: 0.6 B: Chance of WIN given 2 goals: 0.6 C: Chance of LOSS given 2 red cards: 0....
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0answers
15 views

If I(G) = I(G') do they have the same skeleton and the same v-strucures except those in complete (sub-) graph?

I thought if two different graphs G, G' have the same skeleton and the same v-structure, then I(G)=I(G′) and I(G)≠∅. But does the converse also holds? In this case a complete graph doesn't apply ...
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2answers
119 views

Confounding variables in experimental study

We conducted a study to analyse the effect of tablet named 'xab' that help smokers to stop smoking. 5500 of smokers are selected. half of them were given different doses of tablet while the other ...