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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Place balls in bins with specific distributions algorithm [on hold]

I've searched for an answer for my problem on the forum but is still not answered. Let's say I have N balls and want to place them in M buckets without replacement, BUT.. These buckets have all ...
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1answer
35 views

Paired samples or independent sample hypothesis test for two time periods

I want to know if employees in an organization are surveyed in 2013 and again in 2014 if the samples from the two time periods are considered related and dependent or independent if they are asked the ...
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1answer
20 views

Reg: MGF and independence

If three random variables X, Y, and Z are pairwise independent, whether U = aZ+bY, V = aY+bX are independent? I tried using MGF technique and arrived (i don't know its correct or not) as U and V are ...
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1answer
23 views

Independence within groups - which test?

I have a table of data with three columns: ...
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2answers
52 views

Can someone explain what it means for an observation, from a group of data, to be independently and identically distributed?

I don't quite understand what the difference is between an observation being independent and it also being identically distributed. Can someone explain please?
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2answers
26 views

Determining unusual binomial results from a sub-population

If I have a population of 1948 students with a success rate of .154 (301 successes) and I break the 1948 students into their respective majors, how to I determine if the sub-population success rate is ...
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1answer
28 views

Are identical samples independent?

Are results of two different tests performed on exact same sample independent or dependent? For example, if the same group of rats were injected some drug A, results collected, then long enough time ...
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0answers
18 views

How to determine if these values are different?

I don't have much stats background (I am a programmer) but want to learn more. I have five long columns of numbers that represent conversion rates (ex. how many successes vs. how many attempts) for 5 ...
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1answer
110 views

Conditional independence iff joint factorizes

I have proven that: $X⊥Y|Z\ {\rm iff}\ p(x,y|z)=p(x|z)p(y|z)$ for all $x,y,z$ such that $p(z)>0$. The next question is to prove an alternative definition: $X⊥Y|Z$ iff there exist functions $g$ ...
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2answers
50 views

Transform two correlated random variable to independent variables without knowing correlation

I am thinking about this interesting question which arises in the following realistic setting. For example, in one medical experiment one drug and one placebo are applied to two randomized groups of ...
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1answer
61 views

Assume (x,y) are drawn from independent & identical distribution when y=f(x)

Sometimes we say the following: $X$ is some training data given by $X:=\{(x_1,y_1),...,(x_l,y_l)\}\subset R^d \text{x}R$. Assume that the training data had been drawn from independent and identical ...
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1answer
51 views

Indepent variables and these functions [closed]

Random variables $x_1, x_2,...,x_n$ are independent. Then I want to show whether these functions $$y_1=f_1(x) \\ y_2=f_2(x) \\ ... \\ y_n=f_n(x)$$ are independent or not . How to prove this?
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48 views

joint distribution, probability, calculating probabilities under false independent assumption, when the random variables are actually dependent

Suppose I have random variables $X,Y,Z$ and I would like to compute the probability that random variable $X$ is smaller than $Y$ and $Z$: $$ \pi_X \overset{def}{=} Pr(X < Y, X < Z) = \int Pr(x ...
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1answer
40 views

Is there a difference between degrees of freedom and independent variables? [duplicate]

They seem to represent the same idea as far as I can tell - can they be used interchangeably?
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2answers
53 views

Derive the LLN for a certain sequence

I have a sequence of dependent random variables $X_1, X_2...X_n$. Each RV is correlated with two other RVs and uncorrelated with the others.The ones that are correlated satisfy the condition ...
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0answers
14 views

Is there a test of pair-wise independence for a sequence of non-identical Bernoulli random variables?

Given a sequence of $n$ instances of Bernoulli random variables $x_1,\ldots,x_n$, I am interested in testing whether $x_i$ is independent of $x_{i-1}$ for $i=2,\ldots,n$ (i.e. ...
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0answers
14 views

Probability that difference between demand for two skills in a population sample is due to chance

I am trying to measure the demand for two skills in group of companies that are hiring software developers. One skill might be e.g. Java and the other e.g. C++. If out of 100 companies, 30 report they ...
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1answer
41 views

Example for uncorrelated, not independent , but with same distribution functions random variables

I am looking for an example of two random variables X and Y on $\Omega=\{-2,-1,0,1,2\}$ with the following properties: a) X and Y have the same distribution b) X and Y are uncorrelated c) X and Y ...
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46 views

Will something terrible happen if my data points aren't independent?

Suppose I run a bunch of experiments, and get a bunch of datapoints, but several of the datapoints were derived from the same experiment. (In the extreme case, I just repeat the same datapoint 10 ...
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2answers
97 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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0answers
35 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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23 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
138 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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43 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
64 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
33 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
28 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
41 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
87 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
40 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
199 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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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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202 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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86 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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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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41 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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685 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
51 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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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
58 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
73 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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74 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 ...
2
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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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84 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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36 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
650 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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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 ...