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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Independence of functions of Random variables [on hold]

I am working with two RV's X and Y X is distributed normally with mean=x and variance=x^2 Y is distributed uniform (0,1) I know this... ...
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89 views

How to prove dependence of random variables

I need to solve the following problem. Let $X$ be a normal random variable with mean $\mu$ and standard deviation $\sigma$ and let $I$, independent of $X$, be such that $\mathbb{P}(I = 2) = ...
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15 views

What to use when chi-square independence assumption is violated

I'm trying to describe the characteristics of the sexual partners of each participant. There are multiple partners per participant. All of the variables in my analysis are categorical, so originally I ...
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15 views

Mean independence and full independence

What is an example of two random variables x and u that are NOT fully independent but such that E(u|x) = 0? Many thanks!
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2answers
120 views

Characteristic function problem

Suppose $X_1$ and $X_2$ are independent random variables and suppose also that $X_1$ and $X_1-X_2$ are independent. Show that $$\mathbb{P}_{X_1}[X_1=c]=1$$ for some constant $c$. What I get so ...
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26 views

Conditional independence: conditioning on an empty set of random variables

Is $X \perp\!\!\!\perp Y$ a conditional independence, arguing that the independence is conditioned on an empty set of random variables? If so, does that mean that an unconditional independence is ...
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1answer
17 views

Why do Fisher & $\chi^2$ tests show dependence for simulated Markov chain in nonadjacent observations?

I'm trying to understand Fisher's exact test and the $\chi^2$ test of independence. Suppose I've got a Markov Chain of first order with the following TPM: $M = \begin{pmatrix} 0.3 & 0.7 \\ 0.99 ...
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1answer
58 views

Mutual Independence Definition Clarification

Let $Y_1, Y_2, ..., Y_n$ be iid random variables and $B_1, B_2, ..., B_n$ be Borel sets. It follows that $P(\bigcap_{i=1}^{n} (Y_i \in B_i)) = \Pi_{i=1}^{n} P(Y_i \in B_i)$...I think? If so, does ...
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2answers
25 views

Testing for independence between variables in poisson distribution

I'm trying to figure out how to test if variables that I think follow a Poisson distribution are independent (which is a requirement of the Poisson distribution) -- or it it matters. Let's assume I ...
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1answer
21 views

If I am comparing actual data vs. forecasted data, is the Durbin-Watson statistic useful?

We are comparing our forecast vs actual data from the same time period and was using the mann-whitney test to help provide evidence our samples were not different. Someone challenged us saying the ...
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1answer
36 views

An example of r.v.s such that their distribution has more (conditional) independencies than their directed graphical model

I was trying to form an example where I had 3 r.v.s such that the distribution describing them had more conditional independencies or independencies than the directed graphical model corresponding to ...
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1answer
78 views

What methods are used in order to prove the independence of two variables in statistics?

I have four variables (attributes) describing student. They are categorical data. I need to prove that these variables are independent. The easiest way (even not sufficient) is to see the ...
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1answer
32 views

Independent variables in Naïve Bayes

In the perfect explanation of Bayes' Theorem here as far as I know features of the class should be independent. The question is how to prove statistically that two given features are independent? I ...
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1answer
190 views

Invalid inference when observations are not independent

I learned in elementary statistics that, with a general linear model, for inferences to be valid, observations must be independent. When clustering occurs, independence may no longer hold leading to ...
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13 views

Probability that independent variables have significant regression coefficient

suppose to have a dataset $X\in \mathbb{R}^{n\times p}$ and $y\in \mathbb{R}^n$. Independent variables. In general $cor(X,y)\neq 0 $ and so if we fit a linear model we can have that some of the ...
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86 views

What does “independent observations” mean?

I'm trying to understand what the assumption of independent observations means. Some definitions are 1 "the occurrence of one event doesn't change the probability for another". 2 "sampling of one ...
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17 views

Independence of half Wishart and multivariate t?

Let $S$ be a Wishart random variable with parameter $I_p$ and $n$ degrees of freedom, and let $z$ be an independent zero mean $p$-variate Gaussian random variable. Let $V$ be the symmetric square root ...
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1answer
53 views

Is Hurlbert 1984 the best introductory overview to pseudoreplication?

I often find myself explaining (or wanting to explain but not wanting to be boorish) the basics of random sampling and the consequences of pseudoreplication, specifically the limitations and ...
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9 views

Different Correlation/dependecy measures

I'm trying to figure what kind of correlation/depency measure to use on a multivariate data set I'm working on. I've tried the (pearson's)correlation, Kendall's Tau and Spearman's Rho. However, all ...
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39 views

Are these distributions independent?

I'm trying to compare the medians of two non-normal distributions: $A$ and $B$. $A$ is the distribution of completion durations for one type of task $B$ is the distribution of completion durations ...
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572 views

Does non-zero correlation imply dependence?

We know of the fact that zero correlation does not imply independence. I am interested in whether a non-zero correlation implies dependence - i.e. if $\text{Corr}(X,Y)\ne0$ for some random variables ...
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1answer
50 views

Variance of random variable for normal distribution

How do I find the variance for $z_n=\prod_{i=1}^n(1-k_i e^{a_i x})$ where $x$ is the random variable with a normal distribution and is the same for all $i$ (which is a subscript for time dependency ...
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10 views

Determining dependancy of observations

I have 12 people, 6 in treatment group and 6 in control group. They gave ratings (1 to 7) to 8 different activities they were performing as a part of the experiment. I looked at another thread here ...
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1answer
43 views

How to justify the statistical independence among two sets of continuous multivariate observations

I have two sets of continuous multivariate observations $X=\{x_1, x_2, ..., x_d\}$ and $Y=\{y_1, y_2, ..., y_d\}$. How can I justify if they are statistically independent or not? For simplicity, I ...
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1answer
71 views

Verifying independence for two Random Variables

Could you please give me some hints for the exercise below? Suppose we toss a coin once and let $p$ be the probability of heads. Let $X$ denote the number of heads and let $Y$ denote the number of ...
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72 views

Analysis for cross-subjects correlation

Suppose that an investigator has collected some longitudinal data from each of the $n$ subjects, and that a correlation coefficient is computed on the data between any pair of subjects, leading to ...
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1answer
359 views

Are pupils in a school independent and identically distributed when compared to a national population?

I am interested in the way in which data are analysed in education. In England, pupils' reported test results are compared to national test results using z-tests. These results are then used to judge ...
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16 views

Testing for independence and strength of association

I am having difficulty at formulating properly a null hypothesis regarding the following setup: Suppose that we are interested in probing a possible association between bundled products sold at ...
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3answers
82 views

A critical proof or counterexample regarding independence

Does independence of $X^2$ and $Y^2$ imply independence of $X$ and $Y$?
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29 views

relation between stochastic independence and functional independence

We know if two variables are functionally independent then there will be stochastic independence as well. Can any one give me an example when the converse is not true i.e. X and Y are independent ...
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1answer
615 views

If X and Y are correlated, but Y and Z are independent, is X and Z always independent?

If random variables X and Y are correlated, but Y and Z are iid, is X and Z always independent? I am able to prove they are independent for specific types of structure on X and Y (for example if ...
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26 views

How do you prove that if $ X_t \sim^{iid} (0,1) $, then $ E(X_t^{2}X_{t-j}^{2}) = E(X_t^{2})E(X_{t-j}^{2})$? [duplicate]

Suppose we have a time series $X_t$ s.t. $X_t \sim^{iid} (0,1)$. How do you prove that if $ X_t \sim^{iid} (0,1) $, then $ E(X_t^{2}X_{t-j}^{2}) = E(X_t^{2})E(X_{t-j}^{2})$? Or, I guess, if ...
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44 views

Correlation and dependency of two variables

Lets say I am interested in finding the correlation of the following two metrics. They are aggregations and include the same input A in their formula. Is it possible to talk about correlation of ...
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43 views

How to Test Independence of Poisson Variables?

I have a data set of length $10,000$. Each entry, $e_i$, of my data set is associated with two values, $x_i$ and $y_i$. Let $X$ and $Y$ represent the random variables for which my data set is a ...
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2answers
98 views

Regression, Independence & Normality of x and y variables (not residuals)

I am trying to predict productivity of inbound process of a warehouse using y-value = productivity in terms of pcs/hr x-value = # pcs and ratio pcs/SKU. (where SKU = stock keeping unit) Basic ...
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1answer
38 views

Bayes rule and conditional independence

I have two conditionally independent random variables $A$, $B$ such that $$ P(A,B\mid C) = P(A\mid C)P(B\mid C) . $$ I have to find posterior formula $P(C \mid A,B)$. My result with a ...
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1answer
90 views

Hausman test for Independence of Irrelevant Alternatives (IIA), negative value

I get a negative value for the Hausman test for the independence of irrelevant alternatives (IIA) assumption. Now I find contradicting conclusions in the literature: some people say that I should take ...
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1answer
41 views

Are samples from sliding windows independent samples?

Are samples from sliding windows independent samples? E.g. if I have window size of 90 seconds counting the number of cars on a street and I output the average within the window every second for 30 ...
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20 views

Clustering on structural variables?

I'm working with land surface models. These models basically take a bunch of meteorological forcing data (downward radiation, wind, rain, humidity, etc), and run it through some ...
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290 views

Independence of a linear and a quadratic form

How can I prove the following lemma? Let $\mathbf{X}^ \prime$ = $ \left[ X_1 , X_2 , \ldots, X_n \right]$ where $ X_1, X_2, \ldots X_n $ are observations of a random sample from a distribution which ...
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55 views

Help with covariance

A deck of 52 cards is shuffled and a bridge hand of 13 cards is dealt out. Let X and Y denote, respectively, the number of aces and the number of spades in the hand. (a) Show that X and Y are ...
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1answer
39 views

How do I prove the mutual independence of a set of random variables given another set of mutually independent random variables?

Suppose $X_{1},\,\ldots,\, X_{n}$ and $T_{1},\,\ldots,\, T_{r}$ and two sets of random variables, with each $X_{i}$ being categorical, and each $T_{j}$ being continuous taking on all values in the ...
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2answers
103 views

Proving conditional independence from mutual independence

Suppose I have a Bayesian inference problem where $E$ is the result of a chemistry experiment. The chemistry experiment consists of a controlled sequential digestion of one end of the protein using a ...
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1answer
37 views

How to check for independence within a sample

I see sometimes written (see below for examples) that given a random sample from less than 10% of the population then it's reasonable to assume that the data is independent. I wonder why 10% and not ...
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12 views

Checking assumption of dependence in a generalised linear model

I'm wondering if I've violated the assumption of independence in a generalised linear model. The data I've used to make to make the model can be downloaded with this code: ...
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57 views

Are the predictors in an ANOVA assumed to be independent of one another?

I ran an ANOVA with 4 predictors (2 categorical, 2 continuous) and 1 covariate (continuous). Including the covariate did not significantly alter the main effects of the ANOVA when it was run without ...
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351 views

Functions of Independent Random Variables

Is the claim that functions of independent random variables are themselves independent, true? I have seen that result often used implicitly in some proofs, for example in the proof of independence ...
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1answer
39 views

What is wrong with this chi squared test of independence?

I'm trying to learn how to run the chi squared test of independence. I've created two fictional variables, Variable 1 and Variable 2, and I want to know if they are correlated. I'm working with the ...
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23 views

Are these events independent? [duplicate]

We flip a fair coin (independently) three times. Define the following two events: A = "the number of tails is odd" B = "the first coin comes up heads" Are the events A and B independent? I believe ...
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45 views

What exactly is a higher-order correlation? [closed]

What exactly is a higher-order correlation? Is the term just a catch-all for all dependencies that are not captured by second-order correlations? Or do people mean something more precise than ...