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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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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30 views

What exactly is a higher-order correlation? [on hold]

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 ...
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Testing if alcohol consumption and smoking are independent

The question asks to test if smoking status and level of alcohol consumption are independent using the usual five-step procure at alpha $=0.05$: I am having trouble finding expected values. As the ...
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1answer
80 views

Chi-square test: difference between goodness-of-fit test and test of independence

Concerning the Pearson chi-square test there seems to be a subtle difference between the goodness-of-fit test and the test of independence. What is confusing is that both tests seem to be calculated ...
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55 views

Does A independent of B, and B correlated with C imply that C is independent of A?

Assume 3 random variables, $A, B, C$. If $A \perp B$, but $Cov(B,C) \neq 0$, can we say anything about $Cov(A,C)$? I think it can either be 0 or not 0, but it seems like there should be more general ...
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17 views

decomposition of independent random variables

If X and Y are independent random variables such that X = A + B and Y = C + D? Are the pairs (A, C), (B, C), (A, D), (B, D) also independent? By this I mean whether A and C are independent, B and C ...
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35 views

Overlapping sample t-test

I have 10 people working together. They work in group for 6 days. Three days in week 1, three days in week 2. In each day I don't have the full set of people, but a subset of them. In each day I ...
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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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2answers
62 views

Does randomness imply independence?

By looking at a lag plot, it seems that my dataset is random (no autocorrelation whatsoever). Should this be sufficient to infer independence of observations? As a matter of fact, I know that the ...
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1answer
15 views

Cycles in gaussian graphical models

I understand examples where two nodes can be dependent but conditionally independent given their common cause. For instance the common cause is high temperature, and the children nodes are high ...
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50 views

In a linear regression, why is the mean of all $y_i$ equal to 0?

I don't understand why the mean for all $y_i$s must equal $0$. What property/properties does the above assertion rely on? I had thought that the mean of $y_i=E(y_i)$, so: $$ E(y_1)=\alpha $$ $$ ...
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1answer
62 views

Testing that two models from two different data sets are independent

I have two different sets $(n_1=47$, $n_2=23)$ obtained under different conditions. I fit two different functions $(\text{Fu}_1$, $\text{Fu}_2)$ in MATLAB. $\text{Fu}_1$ was fit using the first data ...
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1answer
20 views

Independent vs. Dependent Compound Poisson Distributions

I have an issue with a section of some Actuarial lecture notes that I am reading. Here are the snippets from the notes: "Consider a portfolio consisting of $n$ independent policies. The aggregate ...
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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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59 views

What happens in linear regression when observations are not independent in time

Let's consider the example of whether a person's weight is correlated with their height (simple linear regression). What is the difference between running this correlation on the following two data ...
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13 views

Can mutual independence be defined in terms of conditional distributions?

Suppose there are more than two random variables $X_1, X_2, \dots, X_n$. Can mutual independence between them be defined in terms of conditional distributions? Thanks!
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78 views

$\chi^2$ test of significance vs. goodness of fit

I have 644 companies out of which 154 went bankrupt. I investigate below if the bankruptcy is related to the sector type: ...
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148 views

Checking Independence for functions in R

Given $$f_{(U_1,U_2)}(u_1,u_2)=\begin{cases} 1/2& -u_1<u_2<u_1 \text{ and } u_1 - 2 < u_2 < 2 - u_1 \text{ and } 0 < u_1 <2\\ 0& \text{otherwise}\end{cases}$$ I found that ...
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39 views

Chi-square / Fisher test variable independence for an individual

I have a data set of $\approx$100 individuals showing pulse rate before exercise, pulse rate after exercise, sex, smoker status, weight etc. for each individual. I've been asked to select an ...
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1answer
43 views

Chi-squared & Pearson correlation coefficient

I have just ended my math's assignment and my chi-squared test approved the null hypothesis that my data are independent; however, pearson's correlation coefficient is -0.23. Can they therefore be ...
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Sports winning percentage distribution

I am doing a study where I am measuring parity in sports. To do this I am looking at how well a year's winning percentages (WP) are distributed. I am currently comparing them to a binomial ...
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44 views

Correlation between independent groups of dependent data

I’m trying to correlate data between an accelerometer and a pedometer (step counts). These are worn simultaneously by 20 people for 7 days. If the two devices give me 1 datum per day at the end of the ...
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3answers
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Simplest example of uncorrelated but not independent X and Y? [duplicate]

If any hard-working student is the simplest counterexample to "all students are lazy", what is the simplest counterexample to "if random variables X and Y are uncorrelated they are independent"?
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Given independence, is the median of a product equal to the product of the medians?

Question: Assume $X$ and $Y$ are independent random variables. Is $Median(XY) = Median(X) \cdot Median(Y)$? If so, how would one prove this? If not, what conditions would be sufficient for this ...
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67 views

Does RandomForest ignore spatial independence?

I have 5 variables for each countries of the world and I need to analyze their effect and interactions on an independent variable. Random Forest would be adequate for my scope as it deals with ...
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Is “ independent and identically distributed” an assumption or a fact ?

This is in the context of two random variables. A frequent assumption (e.g. of the error term in ANOVA) is of independent and identically distributed random variables. There is a question on this site ...
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Finding the moments for the Mauchly sphericity statistic

Suppose we have $V \sim W_p(n-1,\sigma^2 I)$ In order to find the moments of the Mauchly sphericity statistic $\Lambda = \frac{|V|}{(tr(\frac{V}{p}))^p}$ i read one can split it defining $\Lambda_1 = ...
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Determining if these random variables are independent

Let $X$ and $Y$ be iid $N(0, \sigma^2)$ variables. Show that $X^2+Y^2$ and $X/ \sqrt{X^2 + Y^2}$ are independent. Hint: Compute the joint distribution of the first and the square of the ...
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How to connect nodes in full conditional model of GRN?

I have problem in intuitive understanding of full conditional model in building gene regulatory network. it's basically asking that "can correlation between two genes be explained by all other genes ...
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2answers
100 views

A problem on correlation and independence

Here is a little challenge/problem for you guys. Let $(X,Y)$ be jointly discrete RVs such that each have at most two mass points (not necessarily $1$ and $0$, i.e. not necessarily indicator ...
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How can I show $(X_{1} / r_{1}) / (X_{2} / r_{2})$ and $(X_{3} / r_{3}) / ( (X_{1} + X_{2}) / (r_{1} + r_{2}) ) )$ are independent?

In the book by Hogg and Craig 6th edition, there is a problem numbered 3.6.14. saying that If $X_{1}$, $X_{2}$, and $X_{3}$ are three independent chi-squre variables with degrees of freedom ...
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How to test whether the distribution of majors among grad students differ from that of incoming students?

A university offers degrees in the following areas: Arts/Science Engineering Business Computer Science Incoming freshmen apply for majors in Arts, Science, Engineering, Business, and Computer ...
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29 views

Calculate the likelihood of two consecutive events

I have a sequence of events for a number of users. Now I want to see if there is any correlation or a likelihood score between two consecutive events. For example, I have 10 types of events ...
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45 views

Independence between residual sum of squares and mean?

Let $Y_1, \cdots, Y_n$ be independent random variables each with the distribution $N(\mu, \sigma^2)$, then \begin{align*} S^2 &= \sum(Y_i - \overline{Y})^2 \\ &= \sum((Y_i - \mu) - ...
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How to extract covariance (error) matrix of independent parameters x,y from covariance matrix of dependent parameters x,y,z?

Suppose I have a covariance (error) matrix C_xyz for system with 3 parameters: x, y, z. Then it appears that ...
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797 views

Test of independence vs test of homogeneity

I am teaching a basic statistics course and today I will cover the chi-squared test of independence for two categories and the test for homogeneity. These two scenarios are conceptually different, but ...
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48 views

How to check linear independence

How can I check the linear independence of my variables? I have this system $Ax=b$ where A is a $N\times 4$ matrix. I want to check the linear independence between the 4 variables in matrix $A$.
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25 views

independent or dependent

I have situation in which I query different sites on the genome (biologically independent) and for each site I get a measurement. I have a number of measurements that are close together on the genome ...
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dependent samples & correction for multiple testing

I have situation in which I compare many genomic regions between two or more cell lines (CL). Each region is covered by n probes measuring the level of methylation (continuous variable). The coverage ...
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83 views

Bayesian Networks: Does the d-Separation Property originate from the basic Markov Property?

I asked the following question in order to gain some intuitive understanding about the d-Separation property in Bayesian Networks a while ago: Understanding d-separation theory in causal Bayesian ...
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How to check if two variables are dependent given a third variable?

In R, I have a dataset of many variables and based on correlation matrix I see that some of the variable are correlate with the others. For simplicity, let's assume ...
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Probability of moving house (non-independence problem)

We are analyzing a dataset where we are looking at the probability of individuals moving house, and the factors that predict moving. The problem is that a lot of people are married, and (usually) if ...
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223 views

For which distributions does uncorrelatedness implies independence?

A time-honored reminder in statistics is "uncorrelatedness does not imply independence". Usually this reminder is supplemented with the psychologically soothing (and scientifically correct) statement ...
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Dataset with many interrelated variables - remove before PCA?

I have a dataset with many variables. Some of these variables are linked to each other in various ways, but I don't know in advance those that are. For example, these are some relationships: A=B ...
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786 views

How do I test that two continuous variables are independent?

Suppose I have a sample $(X_n,Y_n), n=1..N$ from the joint distribution of $X$ and $Y$. How do I test the hypothesis that $X$ and $Y$ are independent? No assumption is made on the joint or marginal ...
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1answer
41 views

Is chi square the best approach for looking at number of cases by year

I have the number of people diagnosed with a condition in each quintile of deprivation over five different years. We are interested in whether numbers of diagnoses are going up faster in more deprived ...
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161 views

Joint distribution of two multivariate normal distributions

If we define 2 independent variables $Y_1$ and $Y_2$ as follows: \begin{align} Y_1 &= (Y_{11},Y_{12},Y_{13})^T \sim\mathcal N_3(\mu_1,\Sigma_{11}), \\ Y_2 &= (Y_{21},Y_{22})^T ...
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67 views

Prove the following are independent

I have been unable to figure out the following question $f_{Y|X}(y|x)=N(x,x^2)$ $f_X(x)=U(0,1)$ Prove that $\frac{Y}{X}$ and $X$ are independent
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Understanding tail dependence coefficients

How can I analyze the $\lambda_U$ and $\lambda_L$ results (estimated by non-parametric method)? What does higher or lower coefficients mean? Does $\lambda_U = 0.5$ mean there's some kind of linear ...