independent, identically distributed (IID) random variables [closed]

Im sure you know that iid means independent, identically distributed. I think the most prominent example is a coin toss repeated several times.

If $X_1, X_2, \dots$ designate the result of the 1st, 2nd, and so on toss (where $X_i = 1$ means that in the i-th toss you have got head and $X_i = 0$ tail), you have that $X_1,X_2,\dots $ are iid.

They are independent since every time you flip a coin, the previous result doesn't influence your current result. Edit: there is a mathematical definition of independence, but I don't think it is necessary at the moment.

They are identically distributed, since every time you flip a coin, the chances of getting head (or tail) are identical, no matter if its the 1st or the 100th toss (probability distribution is identical over time). If the coin is "fair" the chances are 0,5 for each event (getting head or tail).

Does that help?

"Independent" means for any $x_i \in X$, $P(x_0, x_1,..., x_i) = \prod_0^i P(x_i)$

For example, toss 2 dice. Let $X_1$ be event of the first being {1, 2}, $X_2$ the first being {2, 3, 4} and $X_3$ the second being {6} and $X_4$ the second being {4, 6}. It's intuitive to conclude that $P(X_1 \cap X_2, X_3) = P(X_1 \cap X_2) P(X_3) = P(X_1=2) P(X_3)= P(X_2=2) P(X_3) = \frac{1}{6} \cdot \frac{1}{6}$, so are other combinations.

However, synonyms of "identical" include "alike" and "equal". That's, the probability of every variable should be equal, or identical. In the above example $P(X_1) \neq P(X_2)$ or $\frac{2}{6} \neq \frac{3}{6}$. To make $X_1$ and $X_2$ be indentical variables, we can let $X_2$ be event of the second die being {1, 6} or the first being {4, 2} or other similar posibilities you like. Then we get $P(X_1) = P(X_2)=\frac{1}{3}$.

"Independent but not identical" as shown in the above two instances, like any combinations of evens between two dice: $P(X_1)$ and $P(X_3)$ or $P(X_2)$ and $P(X_4)$ and etc.

"Identical and independent", any event combinations between the two dice that have the same probability. For example $P(X_1=1)$ and $P(X_3=6)$ and etc.

"Identical but not independent", any two events for one die. For instance, $P(X_1=1)$ and $P(X_2=3)$. It is not independent since if the die shows you 1 you can not see a 3 at the same time, and hence the probability of seeing 1 and 3 at the same time while rolling one die is 0.

"Not identical and not independent", any two events for one die with unequal probability, for instance $P(X_1)$ and $P(X_2)$ or $P(X_3)$ and $P(X_4)$, or $P(X_2=4)$ and $P(X_1)$ and etc.