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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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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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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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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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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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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
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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
24 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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24 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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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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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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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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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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Finding variance of infimum of a set

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
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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
36 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
25 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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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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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 ...
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Do all P-Maps for a distribution share the same skeleton or/and v-structures?

I learned that a distribution may have multiple P-Maps because even if we are sure about the independences in a graph we cannot gurrantee the causal direction, then I just wonder if those P-Maps share ...
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1answer
22 views

Independence imply d-separation?

In the context of Bayesian networks if two random variables (i.e. nodes) are d-separated, they are independent. However, is there any example of random variables being independent but not d-separated?
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Will Chi Square Test work for high number of categories?

I have a dataset of 1000+ records. In this dataset, I have two categorical features, Ticket-Label and Survived. Ticket-Label has 54 unique categories and Survived has 2 categories. The chi square test ...
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4answers
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How do I intuitively understand that independence is always symmetric?

Independence between two events, $A$ and $B$, is a symmetric relation, that is, if $P(A \mid B) = P(A)$, then $P(B \mid A) = P(B)$. The proof is very simple and can be found at the ProofWiki. ...
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Visualizing separability / independence

I’d like to visually ‘see’ the independence of random variables. I tried plotting f(x), f(y), and f(x, y) for independent and dependent pairs of variables. However, the difference is still not ...
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In linear regression, are the noise terms independent of the coefficient estimators?

In the Wikipedia article on the bias-variance tradeoff, the independence of the estimator $\hat f(x)$ and the noise term $\epsilon$ is used in a crucial way in the proof of the decomposition of the ...
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2answers
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Suppose $\mathbf{X, Y}$ are independent random vectors. Are their components independent? [duplicate]

Let $\mathbf{X} = (X_1, \dots, X_p)^\top$ and $\mathbf{Y} = (Y_1, \dots, Y_p)^\top$ be independent. Does it then follow that $X_i$ is independent with $Y_j$ i.e. cov$(X_i, Y_j) = 0$?
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Independence of Beta ratios of Gamma variates

If $X= x_1/(x_1+x_2+x_3)$ and $Y= x_2/(x_1+x_2+x_3)$ where $x_1, x_2, x_3$ are independent $\chi^2$-distributed random variables with d.f. $-n_1,n_2, n_3$ respectively. Are $X$ and $Y$ independent? I ...
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109 views

Independence of ratios of independent variates

If $X= x_1/(x_1+x_2)$ and $Y= (x_1+x_2)/(x_1+x_2+x_3)$ where $x_1,x_2,x_3$ independent chi-square variates with d.f $n_1,n_2,n_3$ respectively, are $X$ & $Y$ independent? I know the condition ...
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Independence and identical distribution of the error observations [duplicate]

if I know that the observation (xi,yi) are i.i.d. this should not imply that the r.v. are independent, indeed there could be some correlation between the same vector of observation i.. so could I ...
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2answers
291 views

Quantifying dependence of Cauchy random variables

Given two Cauchy random variables $\theta_1 \sim \mathrm{Cauchy}(x_0^{(1)}, \gamma^{(1)})$ and $\theta_2 \sim \mathrm{Cauchy}(x_0^{(2)}, \gamma^{(2)})$. That are not independent. The dependence ...
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Importance of the right-continuity of filtration in definition of strong Markov Property

Taking the definition from wikipedia, With $X = (X_t : t \geq 0) $ as a stochastic process on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with natural filtration $\{ \mathcal{F}(t) \}_{t \...
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Independence Among Observations in Machine Learning Dataset

The Machine Learning algorithm I would like to implement assumes that observations are obtained independently. What test could I perform in order to validate this assumption? Would that be a ...
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2answers
37 views

Is the error term a sum of r.v.?

`If in a econometric model I have: $y = \beta x + u$ where u is the error term, we have: $u = y - \beta x$ Supposing that $\beta=1$, $y\sim N(0,1)$, $x \sim N(0,1)$ and $x$, $y$ are independent. ...
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1answer
48 views

Independence and conditional distribution

In a problem that I'm solving I find that: "Let data (yi,xi) be sampled randomly from a two-dimensional distribution such that y|x is N(ɑ,x^2σ^2)". Are y and x i.i.d? maybe just identically ...
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2answers
108 views

Independence of events in real-life data

Most of statistical methods (if not all) rely on independence of events. How do we know that this assumption is valid in real-life problems like clinical trials or web crawling? What might be the ...
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Independence relation between observation and error [duplicate]

Having a simple model like: Y = a + bX + u where u represent the error term, if I know that the observation of X and Y are i.i.d, also the error term is i.i.d?
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3answers
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Is it possible for two independent variables to be correlated, by chance?

I'm reading Introduction to Probability, 2nd Edition by Tsitsiklis and Bertsekas. On page 218, they write "Thus, if X and Y are independent, they are also uncorrelated." Is it possible that there ...
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Does mutual information depend on the number of data points?

I am playing with mutual information in scikit-learn. ...
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2answers
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How can I estimate the confidence interval of correlations possibly dependend with time?

I have a multivariate problem (with solar data from different meteorological stations) that I am working on my engineering master thesis. I would like to estimate the correlations of different ...
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23 views

Testing statistical independence before using ICA

First a little bit of background. I'm interested in exploring the performance of Independent Component Analysis (ICA) in the context of disentangling intracranial EEG signals. These signals are ...
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19 views

Degree of dependence between individuals of a variable

I'd like to know if there is a measure that computes the dependence between individuals of an only variable. For example, let $X$ be such that $X = \{x_1, \ldots, x_n\}$, I'd like $dependence(x_1, \...
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1answer
23 views

Gaussian distribution: moments, independence and rotation

I have a few questions with respect to the gaussian distribution, its moments and independence. So a gaussian distribution is fully specified by its first two moments, the mean and variance (or ...
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1answer
219 views

Mutual Independence in a Multivariate Normal with Identity Covariance

Consider a random vector $X$ which follows a multivariate nomal with zero means and Identity Covariance. $X\sim \mathcal{N}_n(\mathbf 0, \mathbf I)$ We can say that the individual variables $X_1, ...
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Proving that $ (\hat{\beta} - \beta)' (X' X) (\hat{\beta} - \beta)$ is independent with SSE

Exercise: Prove that $ \mathbf{(\hat{\beta} - \beta)' (X' X) (\hat{\beta} - \beta)}$ and SSE are independent for a Least Squares Regression Model. Attempt: Note that by $'$ I denote the transpose ...
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How to derive $\operatorname{var}[(X_i−\mu)^2]=2\sigma^4$ where $X$ is distributed normally

I have $X_1,...,X_n$, i.i.d. $N(\mu,\sigma^2)$ and I would like to calculate $\text{var}[(X_i−\mu)^2]$. I know that the solution is $2\sigma^4$. However, I can't derive it. Any suggestions?
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Prove that the sum and the absolute difference of 2 Bernoulli(0.5) random variables are not independent

Let $X$ and $Y$ be independent $Bernoulli(0.5)$ random variables. Let $W = X + Y$ and $T = |X - Y|$. Show that $W$ and $T$ are not independent. I know that I have to show that $P(W, T)$ is not equal ...
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1answer
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Conditional independence of two events

1) If events A and B are independent on given condition C, then does it implies that those two events A and B are independent without the condition C? 2) If events A and B are independent events, ...
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Statistical test to compare dependent samples

I am comparing by race the number of limbs (4 total - including both arms and both legs) affected by condition X. Some patients contribute 4, some 1, and some somewhere in between. I want to compare ...
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To find the covariance given the joint probability density function.

Question: I was solving some question papers and got stuck in this problem. My problem: I know how to find "marginal probabilities" from a joint probability density function and also know how to ...
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2answers
84 views

Does independence between random variables imply independence between related events?

Say I have two random variables X1 and X2 and that they are independent. Am I guaranteed that the events "X1 is less than x1" and "X2 is less than x2" are independent? If not, under which conditions ...
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Association rule lift ratio converging to 0.5, is it not independent?

As far as I know, lift is a measure used in association rules to see whether there is a positive association between two items(instances?) I've done a simple (possibly wrong) simulation with lift ...