A simple explanation of the probability density function
Can you briefly explain what the probability density function is?
The probability density function (p.d.f., probability density function) describes the probability distribution of random variables and is the derivative function of the cumulative distribution function. [Edit] Definition For a one-dimensional real random variable X, any function that satisfies the following conditions can be defined as its probability density function:
A beam of particles is dispersed by an obstacle (located at x = 0). Its wave function is as follows:
Ψ(x, t) = Ae-iEt/h[when x
Ψ(x, t) = e-iEt/h( Beikx Ce-ikx) [when x> 0 ]
Where E = h2k2/( 2m ) and k > 0, A, B and C are complex coefficients.
﹝The "h" is "h-bar", which is the horizontal line above h﹞
(a) Calculate its probability density p(x, t) when x
(b) Calculate its probability flow density j(x, t) when x
(c) Calculate its probability density p(x, t) when x > 0.
(d) Calculate its probability flow density j(x, t) when x > 0.
(e) The wave function above contains three different parts, three coefficients A, B and C. Tell whether each one moves right or left. The three of them represent incidence, reflection and emission respectively. Which one is that?
Note: The answers to p(x, t) and j(x, t) must be real numbers.
How to distinguish between probability function and probability density
Mathematical definition of probability density
For random variable X, if there is a non-negative integrable function p(x) (﹣∞
Continuous random variables are often intuitively described by their probability density functions. The probability density function f(x) of continuous random variables has the following properties:
This refers to one-dimensional continuous random variables, and multi-dimensional continuous variables are similar.
Probability density function of random data: represents the probability that the instantaneous amplitude falls within a specified range, so it is a function of amplitude. It varies with the magnitude 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 The problem-solving process is as follows: Extended information Probability density method: Suppose the random variable Among them, α=min(g(-∞), g(∞)), β=max(g(-∞), g(∞)), h(y) is the inverse function of g(x).
Suppose the probability density function of random variable X N0 1 Y |x| Y
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