One random process-conditional mathematical expectation

During the mid-autumn festival, you have to leave something to commemorate. Let's talk about the conditional mathematical expectation. I learned this before in my undergraduate course, but at that time I had a superficial understanding. I read other books today. Science students should be well-defined and have a clear concept. The definition starts from below.

Definition:
The Conditional Expectation of X given Y = y is:
① E (X | Y = y) = Σ XF (X | Y) for discrete case
② E (X | Y = y) = repeated XF (X | Y) dx for continuous case
Note that ex is a numeric value, and E (X | Y = Y) is a function about y.

Comparison:
Ex is the weighted average of all ω, x (ω) values; and E (X | Y = Y) is limited to ω, {ω: Y (ω) = y}, x (ω) is the local weighted average. Based on different values of Y, the entire sample space Ω is divided into N incompatible events (Ω = Σ B (j )). Therefore, E (X | Y = Y) is the local weighted average of X (ω) on a {B (J), J ε n.

 

Introduction of E (X | Y)
Obviously E (X | Y = Y (1), E (X | Y = Y (2 )),.... (It is too inconvenient to enter the subscript. The "1" and "2" in parentheses are all subscripts. Let's take a look at them.) It depends on Y = Y (j ), it depends on the Global Sample Space Division. In this way, it is necessary to introduce a new random variable, which is counted as E (X | Y), from the perspective of the sample space Ω and the change to ω ε Ω ). For the random variable E (X | Y), when y = Y, the value is E (X | Y = Y), which is called the random variable E (X | Y) it is the conditional mathematical expectation of random variable X about random variable Y. Here we will use the statement in a textbook: before
We observe y, we don't know the value of E (X | Y = y) So it is a random varible which we denote E (X | Y ). random Variable E (X | Y) is a function of random variable Y. In fact, it is only a local average {e (X | Y = Y (j )), the Unified Expression of J, n.

 

It is easy to think of the mathematical relationship between ex and E (X | Y. Since E (X | Y = Y) is a local mean dependent on y, and ex is the whole mean, then E (X | Y) is averaged again, what do you get? Thus, a theorem of the name is obtained. The rule of Iterated expectations:
For random variables X and Y, assuming the expectations exist, we have that
E (X | y) = ex;
More generally, for any function R (x, y) we have
E (R (x, y) | X) = E (R (x, y ))

In fact, the background of this great theorem is extremely common, and truth often comes from life. For example, if we want to calculate the average score of a certain grade student, there are two ways:
1. The score of each student in this grade can be Σ and then divided by the total number of students. This is an extremely common method. This method corresponds to computing ex;
2. We can also calculate the average score of each class (the first average), and divide the average score of each class by the number of classes (the second average ). This is E (X | y )). In this example, each class is equivalent to Y, and the average score of each class is equivalent to a fixed y = y to calculate E (X | Y = Y), and then the class is averaged.
Obviously, the results obtained using the methods 1 and 2 are consistent. E (X | y) = ex! This is the implicit idea of the Great Theorem: local mean first, and then the overall mean. How popular it is! This is great wisdom! I think of a sentence from the junior high school class teacher: what is justice? It's what dogs know, such as the shortest straight line between two points! Of course, after thinking, we must put it into a formula, which must be expressed in the form of mathematics. Then perfect!

 
I personally think that as long as I understand the nature of partial mean expected by conditions, there is no problem with the derivation of a lot of formulas. It is nothing more than a pile of points centered on the conditional probability density function.