Geometric meaning of probability density function and distribution function

Geometric meaning of probability density function and distribution function

1. The first derivative of the distribution function f (x) is the probability density function: F (x) = DF (x) / DX
The infinite integral under the probability density curve is equal to 1, which means P {x}|

What is the significance of the peak value of the probability density function graph of X

The probability density function itself represents the probability of the "nearby" value of the corresponding continuous random variable x = X. for continuous variables, P {x = x} = 0, that is, the probability at a certain point is 0, but the probability in the small area △ X of this point is the value f (x0) * △ X of the density function at this point. Therefore, the meaning of density itself is the ratio, which means that in this area, The higher the probability density in the unit interval, the greater the probability in the cell. Especially, when the approximation is applied, when the cell is selected as 1, that is, △ x = 1. For example, when calculating the image histogram, the statistical value on the histogram can be approximately regarded as the probability density function on the gray level of the image, In this case, when △ x = 1, the probability density function value is equal to the corresponding probability value of this interval

The quotient of a divided by B is 15 and the remainder is 8. Given that the sum of a, B, quotient and remainder is 225, what is the number of a? It can't be solved by algebra

225-15-8=202
202-8=194
194/16=12.125
12.125*15+8=189.875
A = 189.875

As shown in the figure, the radius of the cone bottom is known to be 10 cm, and the length of the generatrix is 40 cm. (1) find out the central angle and surface area of its side expansion. (2) if a line starts from point a and goes along the side of the cone to the midpoint B of generatrix SA, please think about the shortest route it takes? Why?

(1) N π × 40180 = 2 π × 10, the solution is n = 90. The surface area of the cone is π × 102 + π × 10 × 40 = 500 π cm2. (2) as shown in the right figure, the shortest route that the beetle takes from point a along the side of the cone to the midpoint B of bus SA is the length of line ab

Given the function f (x) = log2 (AX + b), if f (2) = 1, f (3) = 2, find f (5)

From F (2) = 1, f (3) = 2, log2 (2a + b) = 1, log2 (3a + b) = 2, that is, 2A + B = 23a + B = 4, ∧ a = 2B = − 2, ∧ f (x) = log2 (2x-2), ∧ f (5) = log28 = 3

High school absolute value inequality (urgent)
Solve the inequality about X: a (x-1) (absolute value in brackets) > A + 2, a belongs to real number

a(X-1)>a+2
(x-1）>1+2/a
-2

The radius of the big circle is equal to the diameter of the small circle. It is known that the area of the big circle is 9.42 square decimeters more than that of the small circle, and the area of the small circle is______ ．

Let the radius of the small circle be r, then the radius of the big circle is 2R, the area of the big circle is: π (2R) 2 = 4 π R2, and the area of the small circle is: π R2, so the area of the big circle is 4 times of that of the small circle, then the area of the big circle is 4-1 = 3 times larger than that of the small circle, 9.42 △ 3 = 3.14 (M2). A: the area of the small circle is 3.14dm2

It is known that the four numbers a, B, C and D are proportional, and a and D are external terms. Verification: the point (a, b), (C, d) and the coordinate origin o are on the same line

It is proved that: let the straight line passing through points o and (a, b) be y = KX, then B = AK, then k = Ba, let the analytic formula of the straight line passing through points o and (C, d) be y = MX, then d = cm, and the solution is: M = DC, ∵ a, B, C, D are proportional, ∵ AB = CD, ∵ Ba = DC, ∵ k = m, then the straight line y = KX and the straight line y = MX are the same straight line, that is, the point (a, b), (C, d) and the coordinate origin o are on the same straight line

There are two monomials, which satisfy the following conditions at the same time: 1. All polynomials contain only the unknowns x 2. The absolute value of each coefficient in each polynomial is 1
3. The sum of the two polynomials is a quartic binomial, and the difference of the two polynomials is a quadratic binomial
What are the two polynomials?
In a hurry~·

x^4+x^2；x^4-x^2
1. The variables are only x; 2. The absolute values of the coefficients are 1,1,1, - 1; 3. The sum of the two is x ^ 4, and the difference is x ^ 2

If the radius of the sector is R and the area is s, then what is the arc length?

S=rl/2
Arc length L = 2S / R