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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.
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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