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I know there are some questions about notation on here, already, but it's really confusing for me. Sorry if this is a dumb question, but I don't have people around me who know or care about this stuff.

I mainly would like to know this in order to understand a bit of machine learning theory.

When I read something like this:

For $X_1, X_2,... , X_n$ iid Poisson random variables

I have two conflicting pictures in my head and it feels to me that this notation sometimes means one thing and sometimes another

1) I did n random experiments or n measurements, let's say of some time interval. I know that the values I get are poisson-distributed. So basically, if I put this in tabular form. I have a table with one columnd and n rows.

So what this actually means is this: I have one thing/phenomenon which produces or takes on values randomly. I denote this $X$. What $X_i$ means is: in my i-th expriment, the concrete realisation of $X$ was $x_i$, so it's short for $X = (X_1 = x_1, X_2 = x_2,...)$. Which, if that is true, is really confusing to me, because in my mind it should be $X = (x_1, x_2, ...)$. Why do we need the $X_1$?

2) The other way is: I actually measured n different things. For instance, for a machine learning task to classify an animal according to weight and length, I measure weight (= $X_1$) and length (= $X_2$), so I have two completely separate random variables, I may just as well have called $L$ and $G$, but because of the i.i.d assumption, to make it clearer, I use the same letter for both.

So now, in my table, I have two columns.

However, this would mean that $X_1, X_2$ means that I measured two different things once, which is kind of weird. How would I then express, that I measured 2 things n times? Another index?

EDIT And another thing:

The product of two i.i.d random variables $XY$ or $X_1$, $X_2$.

Now, in this case, they surely must refer to different thigs (e. g. height and weight?) Because otherwise, it would mean that you take the product of two realisations, right?

EDIT 2

Here's, I guess, my main point of confusion:

Let's say I roll a fair die 10 times.

This is written as $X_1, ..., X_{10}$, which I can sum up as a random vector X.

Which means, that each die roll is described by its own random variable, each with the possible outcomes ${1,2,,4,5,6}$. Because of i.i.d, they have the same distribution and so on.

The same goes for repeating a bernoulli-expriment 10 times. I have 10 randoom variables, each with the possible outcomes ${1,0}$.

Now I do something different:

I go in my garden and randomly take 10 iris flowers and measure height, weight, number of petals.

Now: Is measuring these 3 things one flower like rolling a die once? Or like rolling a die three times? If I measure 3 things on 10 different occasions, how many random experiments have I done and how do I notate this?

I hope I am getting my point of confusion across.

I the case of the die or the Bernoulli trial, in each experiment I am doing the SAME thing.

However, in the flower experiment, I am checking 3 different things. So in the flower experiment, what are my i.i.d. variables? The three things I measure or the outcomes of each measurement?

So if one measurement is written as (9.5,6.2,20), are these 3 realisations of 3 i.i.d. random variables?

user3813234
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  • While I only have a very high level understanding of ML theory, I think the first way of thinking is correct. The reason why we use $X = (X_{1} = x_{1}, X_{2} = x_{2} ...)$ is because $X_{i}$ can take multiple values from the set of all possible values for a Poisson random variable. $x_{i}$ just specifies one of those values. – Shikhar Jaiswal Mar 31 '20 at 13:43
  • @ShikharJaiswal thanks for you comment. Okay, so $X_1, X_2,...$ are basically placeholders for possible values a poisson distribution can produce? – user3813234 Mar 31 '20 at 13:55
  • $X = (X_1,\ldots,X_n)$ is a random vector, where the $X_i$ are random variables. Notation $X = (X_1 = x_1 , \ldots, X_n = x_n)$ means the random vector is considered with a fixed observation. – AlvinL Mar 31 '20 at 13:56
  • Yes, that would be correct. Keep in mind that if $X_{i}$ is (let's say) sampled from a multi-variate normal distribution, then it could itself be a tuple, and not a one-dimensional real number. – Shikhar Jaiswal Mar 31 '20 at 13:59
  • @AlvinLepik: But do these random variables all refer to the same thing (e. g. height)? Or could they refer to different things? – user3813234 Mar 31 '20 at 13:59
  • @user3813234 Such parameters are fixed by experiment and what you are looking out for. – AlvinL Mar 31 '20 at 14:03
  • @AlvinLepik So basically, if you read this in a math book without a concrete example, it could be both? – user3813234 Mar 31 '20 at 14:05
  • @ShikharJaiswal thanks for mentioning the multivariate distribution, I had not thoguht about that. – user3813234 Mar 31 '20 at 14:07
  • @user3813234 You are measuring an animal for two different things, hence you store them as separate results. So, no, your random variables $L_i$ refer to length for every $i$. Thus you are collecting random vectors $(L_i,G_i)$. You can do $(X_1 ^i, X_2^i)$ where $i = 1,\ldots, n$, that's just notation. – AlvinL Mar 31 '20 at 14:11
  • @AlvinLepik okay, so to sum up: bascially, when I read: "two random variables $X$ and $Y$", I could just as well think "$L$ and $G$" or "$X_1$ and $X_2$". – user3813234 Mar 31 '20 at 14:45
  • @user3813234 Yes, $X,Y$ are just random variables. How you denote them is of no consequence. Sometimes a certain notation may be preferable for ease of reading or something along those lines. – AlvinL Mar 31 '20 at 14:46
  • Let us continue this discussion in chat. – user3813234 Mar 31 '20 at 15:59
  • @AlvinLepik I edited my questionn one more (hopefully last) time. Could you maybe have a look at it? – user3813234 Mar 31 '20 at 16:00

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