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Accepted

On the importance of the i.i.d. assumption in statistical learning

The i.i.d. assumption about the pairs $(\mathbf{X}_i, y_i)$, $i = 1, \ldots, N$, is often made in statistics and in machine learning. Sometimes for a good reason, sometimes out of convenience and ...
• 18k
Accepted

• 123k
Accepted

With categorical data, can there be clusters without the variables being related?

Consider the clear-cluster case with uncorrelated scale variables - such as the top-right picture in the question. And categorize its data. We subdivided the scale range of both variables X and Y ...
• 57.2k
Accepted

Independence of sample mean and sample variance in binomial distribution

$\bar x$ and $s^2$ are random variables. We can work out their joint distribution. Let's try the simplest possible nontrivial case, that of a sample of size $2$ from a Binomial$(1,p)$ distribution. ...
• 321k
Accepted

Relation between independence and correlation of uniform random variables

Independent implies uncorrelated but the implication doesn't go the other way. Uncorrelated implies independence only under certain conditions. e.g. if you have a bivariate normal, it is the case that ...
• 281k

Let A and B be two random variables, both independent from another random variable C. Is A*B also independent from C?

If all you have is pairwise independence then there is a counterexample. Suppose the following four cases each have probability $\frac14$: ...
• 38.8k
Accepted

Probability of surviving an event three times

When you write "No extra variables, each incident is isolated and does not affect the subsequent", the mathematical word for this is that they are independent. And for independent events $A$ and $B$, ...
• 23.2k
Accepted

The statement that you are asking about has two parts: If $X$ and $Y$ are independent, then $X$ and $Y$ are uncorrelated. If $X$ and $Y$ are uncorrelated, then $X$ and $Y$ are independent. Statement ...