How to distinguish probability function from probability density?
Mathematical definition of probability density
For a random variable X, if there exists a non-negative productable function p(x)(-∞ < x < ﹢∞) such that for any real numbers a, b(a < b), there is (the formula is shown to the right) , then p(x) is said to be the probability density of X.
Continuous random variables are often visualized by their probability density function, the probability density function f(x) of a continuous random variable has the following properties:
This refers to a one-dimensional continuous random variable, and multidimensional continuous variables are similar.
Probability density function of random data:It represents the probability that the instantaneous amplitude falls within a specified range, and is therefore a function of the amplitude. It varies with the amplitude of the range taken.
The density function f(x) has the following properties:
(1) f(x)≧0;
(2) ∫f(x)d(x)=1;
(3) P(a<X≦b)=∫f(x)dx
How to determine whether a function is a density function
The sufficient necessary conditions for f(x) to be the density of the distribution of some one-dimensional random variable are:
1. f(x) is non-negative and integrable;
2. the value of the definite integral of f(x) over the entire real number axis (i.e., negative infinity to positive infinity) is equal to 1.
Expansion
As the the value of the random variable X depends only on the integral of the probability density function, the value of the probability density function at individual points does not affect how the random variable behaves.
More precisely, a function can also be a probability density function of X if there are only a finite, countably infinite number of points at which the function and the probability density function of X take on different values or measure 0 (are a zero-measure set) with respect to the entire real number axis.
The probability that a continuous-type random variable takes values at any point is 0. As a corollary, the probability that a continuous-type random variable takes values on an interval is independent of whether the interval is open or closed. It is important to note that the probability P{x=a} = 0, but {X=a} is not an impossible event.
Properties of probability densities
Nonnegativity f(x) ≥ 0, x ∈ (+∞,-∞), normality. These two basic properties can be used to determine whether a function is a probability density function of some continuous random variable. Probability refers to the chance of an event occurring randomly, and for a uniformly distributed function, the probability density is equal to the probability of a segment of an interval (the range of values of the event) divided by the length of that segment of the interval, which has a non-negative value and can be very large or very small.
Probability Density Physical Concepts
The state of motion of an electron is described by a wave function Ψ,|Ψ|² which represents the probability of an electron occurring in a unit volume somewhere in the space outside the nucleus, i.e., the probability density. Electrons in different states of motion have different |Ψ|, and |Ψ|² is of course different. A higher density results in a greater distribution of events occurring, and vice versa. If the sparseness of the black dots is used to represent the size of the probability density of individual electrons, then the denser black dots where |Ψ|² is large have a large probability density, and vice versa. The small black dots distributed outside the nucleus of an atom seem like a negatively charged cloud that surrounds the nucleus, and people call it an electron cloud.
In 1926, the Austrian physicist Schrödinger used partial differential equations to establish a fluctuation equation describing the motion of microscopic particles, known as the Schrödinger equation. From the Schrödinger equation can be seen, for a mass of m, in the potential energy of V potential field for the movement of particles, there is a wave function ψ associated with the movement of this particle, this wave function is a reasonable solution to the Schrödinger equation, each of which corresponds to the corresponding constant E, which is the energy of the particles in this state of motion (or energy level). |Ψ|² denotes the probability density of the occurrence of an electron at a point P(x,y,z) in the space outside the nucleus of an atom, i.e., the probability of the occurrence of an electron per unit volume at that point. Used to represent the probability density of the geometry is commonly known as the electron cloud, the electron cloud is not a large number of electrons dispersed in the extra-nuclear space, but the image of the probability density of electrons appearing everywhere in the extra-nuclear space.
Properties of Probability Density Functions
Nonnegativity: f(x) ≥ 0, x ∈ (-∞, +∞). Normality: ∫f(x)dx=1. These two basic properties can be used to determine whether a function is a probability density function of some continuous random variable.
In mathematics, the probability density function of a continuous random variable (which can be abbreviated to density function when not confusing) is a function that describes the likelihood that the output value of this random variable will be in the vicinity of some definite point of value.
And the probability that the value of the random variable falls within a certain region is the integral of the probability density function over that region. When a probability density function exists, the cumulative distribution function is the integral of the probability density function. The probability density function is usually labeled in lower case.
There is no practical significance in speaking of the probability density alone; it must be predicated on a definite bounded interval. You can think of the probability density as a vertical coordinate, the interval as a horizontal coordinate, the integral of the probability density to the interval is the area, and this area is the probability of the event occurring in this interval, the sum of all the areas is 1. So analyzing the probability density of a point alone does not have any significance, and it must have the interval as a reference and comparison.