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I am trying to deepen my knowledge in probability and I am having hard times to understand dependent and not identically distributed random variables.

Can someone maybe provide me a real world example for this?

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Next to the "formal" example by Xi'an, a "real-world" example might be height and weight. Already because the two are measured on different scales will they be distributed differently, but they sure are dependent, as taller people tend to be heavier.

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    $\begingroup$ Thanks @Christoph! This is exactly what I was looking for! Now I get the dependent part. So then for example for an independent and not identically distributed RVs case an example could be: throwing a dice (let's denote it as X) and flipping a coin (let's denote it as Y) would be good, right? $\endgroup$ – csongor_kocsis Sep 15 at 14:00
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    $\begingroup$ Yes, that sure is a possible example! $\endgroup$ – Christoph Hanck Sep 16 at 14:43
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Autorcorrelated processes
A variable in series that 'remembers' its previous values to some degree is not i.i.d.! Any autoregresive value depends on previous values of the variable, and the distribution changes depending on location within the series.

For example, the time series variable $y$, where $t$ indicates period of time, $y_t = \beta_0 + \beta_1 y_{t-1} + \varepsilon_t,$ and $\varepsilon \sim \mathcal{N}(0,\sigma)$ is not i.i.d. for non-zero values of $\beta_1$ (especially for $|\beta_1|\ge 1$), because the variance of $y$ is a function of $t$ (the more time passes, the more variable $y$ is). In a similar way, the expected value of $y$ at some point in the future is also a function of $t$.

Real World Examples
Ok, so is that just some statistical abstraction? Or are there actual real-world examples of autocorrelated processes? In fact, they abound! Here are some:

  • Annual marriage rates by state, province or country
  • Annual mortality rates by state, province or country
  • Daily closing value of the NASDAQ Composite, Dow Jones Industrial Average, or S&P 500 Index—all marketing indexes—in the US

What these (and other) autoregressive series have in common is that their value at one point in time 'remembers' (i.e. is a function of) their previous value or values.

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If $\varepsilon_1,\varepsilon_2$ are iid $\mathcal N(0,1)$, $$X_1=\mu_1+\sigma_1\epsilon_1\qquad X_2=\mu_2+\varrho \epsilon_1 + \sigma_2 \epsilon_2$$ is a pair of dependent RVs that are not identically distributed for most values of the parameters.

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  • $\begingroup$ Thanks for the answer, but what I was really looking for was an actual real life example (e.g. with dice or something). But thanks! $\endgroup$ – csongor_kocsis Sep 15 at 14:01
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If you randomly draw a card from a deck of playing cads, do not put it back, and draw again. Then, the probability distributions for which card will be drawn in each of the two draws are dependent and not identical.

Otherwise, if the card of first draw is put back and well shuffled before the second draw, then the distributions of the two draws are independent and identical.

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Some other "real-world"-examples:

Let $(M, F)$ be a pair of measurements on an opposite sex married couple, sampled randomly:

  • Measurement is height, will have different means.
  • Measurement is IQ, same mean, different variance.

(But maybe for this example, independence is in doubt ...) Paired data in general can be used to make many similar examples, and could save the independence assumption maybe by conditioning on some common latent variables.

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    $\begingroup$ Did I understand your $M, F$ correctly? (21st century perspective in my edit. :) $\endgroup$ – Alexis Sep 15 at 16:25
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    $\begingroup$ You did understand correctly ... the edit is totally OK, I more or less expected some reaction, but wanted to see ... $\endgroup$ – kjetil b halvorsen Sep 15 at 16:34

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