Linked Questions

0 votes
1 answer

Covariance matrices and independency [duplicate]

If we have a diagonal covariance matrix does that guarantee independency?
wageeh's user avatar
  • 241
53 votes
7 answers

Why zero correlation does not necessarily imply independence

If two variables have 0 correlation, why are they not necessarily independent? Are zero correlated variables independent under special circumstances ? If possible, I am looking for an intuitive ...
Victor's user avatar
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50 votes
8 answers

Simple examples of uncorrelated but not independent $X$ and $Y$

Any hard-working student is a counterexample to "all students are lazy". What are some simple counterexamples to "if random variables $X$ and $Y$ are uncorrelated then they are independent"?
16 votes
4 answers

Can somebody illustrate how there can be dependence and zero covariance?

Can somebody illustrate, as Greg does, but in more detail, how random variables can be dependent, but have zero covariance? Greg, a poster here, gives an example using a circle here. Can somebody ...
user11883's user avatar
  • 161
4 votes
5 answers

Why are $X \sim U(-1,1)$ and $Y=X^2$ dependent?

Suppose we have two continuous random variables $X \sim U(-1,1)$ and $Y=X^2$. I don't understand why they are dependent. $$E[X] = 0$$ $$E[Y] = \int_{-1}^{1} x^2 dx = 2/3$$ $$E[XY] = \int_{-1}^{1} x^3 ...
Ray Siplao's user avatar
8 votes
2 answers

Sampling with or without replacement?

I don't know a lot about sampling methods. I have a large population of size 2,000,000. I used one of those sample size calculators. It says that I need sample size of approximately 10,000. I am ...
Martin Velez's user avatar
5 votes
2 answers

Independence and orthogonality

I know what it means to say two variables are independent, but can't understand what does it mean to say two variables are orthogonal.Can anyone help me?
arnab's user avatar
  • 581
3 votes
2 answers

Is there a difference between a causal relationship and a DIRECT causal relationship?

The following site ( defines a causal relationship as one where one variable 'directly' affects the other, but without the other ...
Mathematician's user avatar
4 votes
1 answer

Computing covariance matrix from the given variances?

How can I obtain the covariance matrix when all the variances of the variables are known? This is from the paper $$V_1 \sim \mathcal{N}(0,...
user26767's user avatar
  • 143
1 vote
1 answer

Difference between $\mathrm{Poisson}(x_1)$, $\mathrm{Poisson}(x_2)$ and $\mathrm{BPoisson}(x_1, x_2)$

I am trying to find out the difference between treating two random variables as poisson distributions, $\mathrm{Poisson}(x_1)$, $\mathrm{Poisson}(x_2)$, and using a bivariate poisson, $\mathrm{...
JMzance's user avatar
  • 274
2 votes
1 answer

Covariance Zero Equals Independent?

We know that $cov(X,Y)=0$ does not warranty $X$ and $Y$ are independent. But if they are independent, their covariance must be $0$. My question is: what kind of distribution must $X$ and $Y$ be for ...
Alicia Hampton Coleman Toledan's user avatar
1 vote
0 answers

Interpretation of (diagonalized) inverse covariance matrix

There are several threads here about covariance matrix and inverse covariance matrix interpretation (here, here or here). However, I was wondering how to interpret the inverse covariance matrix (or ...
max's user avatar
  • 11
1 vote
1 answer

What is wrong with this proof/derivation?

Regarding simple linear regression $y = a + bx + \epsilon$ where $\epsilon$ is uncorrelated, E$[\epsilon]=0$, and Var$[\epsilon]=\sigma^2$, the definition of the residual sum of squares is $...
cwackers's user avatar
  • 215
0 votes
1 answer

Independence of random variables and its relation to the expectation

For sochastic processes $X_n$ and $Y_n$ it holds that if they are indenpendant for all $n$ then $$E[X_nY_n] = E[X_n]E[Y_n]$$ But can you go the other way. I mean if $E[X_nY_n] = E[X_n]E[Y_n]$, does ...
KinkyLaura's user avatar