Dtyjlui

I am having trouble understanding IID random variables. I've tried reading http://scipp.ucsc.edu/~haber/ph116C/iid.pdf, http://www.math.ntu.edu.tw/~hchen/teaching/StatInference/notes/lecture32.pdf, and http://www-inst.eecs.berkeley.edu/%7Ecs70/sp13/notes/n17.sp13.pdf but I don't get it.

Would someone explain in simple terms what IID random variables are and give me an example?

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.

.So basically you will consider events where the outcome in one case will not depend on the outcome of the other cases .It is called identical because in every case u consider the possible outcomes will be same as the previous event .Some one has suggested yes tossing of coin is a good example .I will try to be a statistician here .You should go through few statistical distributions like normal ,gamma etc and then see additive property .There while proving the additive property u will consider independent events initially and then prove that the addition of all the independent variates also follows the respective distribution using the MGF of that particular distribution and then you will extend your property to see what if the variates are made similar .Hope you got your answer