Variance concept and Calculation Formula

I. Concept and calculation formula of variance

Example 1The five quizzes for the two are as follows:

X: 50,100,100, 60, 50E(X) = 72;

 Y: 73, 70, 75, 72, 70E(Y) = 72.

The average score is the same,XUnstable. The deviation from the average value is large.

Variance describes the expected deviation of random variables from mathematics.

Single deviation is

Eliminate symbol impact

Variance is the mean of the deviation square, recordedD(X):

The formula for direct calculation is divided into discrete and continuous types, specifically:

Here is a number. Derivation of another formula

Obtain: "the mean of the variance equals the square minus the square of the mean", that is

,

Where

They are discrete and continuous calculation formulas respectively. The standard deviation or mean variance describes the degree of fluctuation.

Ii. Nature of variance

1. SetCIs a constant, thenD(C) = 0 (constant without fluctuation );

2.D(CX) =C2D(X) (Constant square extraction );

Certificate:

In particularD(-X) =D(X),D(-2X) = 4D(X) (No negative variance)

3. IfX,YIndependent of each other, then

Certificate: Note

Then

The first two items are exactlyD(X) AndD(Y).

WhenX,YWhen they are independent of each other,

,

Therefore, the third item is zero.

In particular

The sum of independent premise items can be extended to limited items.

Iii. Variance of common Distributions

1. Two-Point Distribution

2. Two-item distribution

X ~ B(N,P)

Introduce Random VariablesXI (in the I testANumber of occurrences, subject to two-point distribution)

,

3. Poisson distribution (derivation omitted)

4. Even Distribution

Another calculation process is

5. Exponential Distribution (derivation omitted)

6. Normal Distribution (derivation omitted)

~

The latter parameter of a normal distribution reflects the degree of deviation from the mean, that is, the degree of fluctuation (random fluctuation), which is consistent with the characteristics of the image.

Example 2Returns the variance of the preceding example 2.

SolutionAccording to the distribution law given in example 2 above

The number of waste products of worker B is small, the fluctuation is small, and the stability is good.