Let f (x) = x ^ 2-4x + 3 / X (x ^ 2-1) be the first kind of discontinuous point X=___ Please give details of the process, thank you Let f (x) = (x ^ 2-4x + 3) / X (x ^ 2-1) be the first kind of discontinuous point X=___ I forgot to enter the brackets But can you write down the three steps

Let f (x) = x ^ 2-4x + 3 / X (x ^ 2-1) be the first kind of discontinuous point X=___ Please give details of the process, thank you Let f (x) = (x ^ 2-4x + 3) / X (x ^ 2-1) be the first kind of discontinuous point X=___ I forgot to enter the brackets But can you write down the three steps

Because f (x) = (x ^ 2-4x + 3) / X (x ^ 2-1) = [(x-1) (x-3)] / [x (x + 1) (x-1)], although the function f (x) is not defined at x = 1, it can be reduced after factorization, so x = 1 has a limit (when x tends to 1, f (x) tends to - 1), which is the first kind of discontinuity

How to judge the break point between the first type and the second type
For example, y = [cos (1 / 2)] ^ 2, x = 0, which type of discontinuity point is it

When it is a discontinuous point, the existence of both left and right limits is the first kind, and at least one nonexistence is the second kind

Integrable functions can have a finite number of discontinuities. Are these discontinuities of the first or the second kind

If there are finite discontinuities in integrable function, these discontinuities can be of the first or the second kind
On the other hand, it may be more clear that a function on a closed interval [a, b] has only a finite number of discontinuities and is continuous elsewhere
1. If these discontinuities are of the first kind or removable, then the function is integrable
2. If these discontinuities are of the second kind, the function may or may not be integrable
There is a second kind
Integrable, such as: F (x) = sin (1 / x) in [- pi, PI], X not = 0, f (0) = 0
For example: F (x) = sin (1 / x) * 1 / x ^ 2 in [- pi, PI], X not = 0, f (0) = 0

Is the oscillation discontinuity of higher numbers the first or the second?

The second kind, because the first kind has limit and the second kind is infinite or oscillatory, has no limit

What is the covariance?

Cov（X,Y）=E(XY)-E(X)E(Y)=E[X-E(X)][Y-E(Y)].

What is the covariance

For two-dimensional random vectors (x, y), the expected e (x), e (y) only reflect the respective mean values of X and y, and the variance D (x), D (y) only reflect the degree of deviation from their respective mean values. They do not provide any information about the relationship between X and Y. we know that when x and y are independent of each other, there is e ((x-e (x)) (y-e (y)) = 0

How to calculate covariance?
cov(x+y,x-y)=cov(x,x)-cov(x,y)+cov(y,x)-cov(y,y)
What formula or theorem is used in this step?

CoV (x + y, z) = cov (x, z) + cov (y, z) and cov (ax, by) = AB * cov (x, y) [where x, y, Z are variables and a, B are constants]
Combining the two, your formula can be divided into two parts
cov(x+y,x-y)
=cov(x,x-y)+cov(y,x-y)
=cov(x,x)-cov(x,y)+cov(y,x)-cov(y,y)

Calculate the sample covariance

Divide by N, first of all, treat these two groups of data as two-dimensional random variables (x, y). The covariance cov (x, y) is required to have the formula cov (x, y) = e {[x-e (x)] * [y-e (y)]} x0d = e (x * y) - E (x) * e (y) \ \ x0d, because the expected expression is e (x) = ∑ Xi * PI \ \ x0d, because there are fewer elements in the sample, the probability of each element can be regarded as equal, which is 1 / N \\, E (x) = (∑ XI) / n-x0d similarly, e (y) = (∑ Yi) / n-x0de (x * y) = (∑ Xi * Yi) / n-x0d

What's the variance?

A set of data x1, X2 Xn, average first
Variance = 1 / n [(x1 mean) ^ 2 + (x2 mean) ^ 2 +... " +(xn average) ^ 2]

Calculate variance
A 12 13 14 15 10 16 13 11 15 11
B 11 16 17 14 13 19 6 8 10 16
How to use the scientific calculator which imitates Casio's half, not full screen dual line display to calculate! After answering, I have a big reward. I've seen a lot about how to use the scientific calculator, or tell me to use the calculator that comes with windows to calculate oxygen!

The first type: Casio model. This model is characterized by the word "CAISO" on the top of the calculator; double line display; the detailed model data of the test model is "CAISO Fx-82MS student calculator s-v.p.a.m." 1. Press [mode], [2] to enter the statistical mode after startup; 2. Press [1], [M +], [2], [M +] ,[4...