Atmospheric Flight Dynamics Summary （大气飞行动力学总结 ）.pdfVIP

Atmospheric Flight Dynamics Summary 1. Stochastic variables and processes In this chapter, we’re going to examine stochastic variables and stochastic processes. First, we look at one stochastic variable. Then, we examine multiple stochastic variables. Finally, we’re going to examine stochastic processes. What kind of properties can such processes have? 1.1 Stochastic variables and their properties 1.1.1 Probability distribution and probability density functions Let’s examine a stochastic variable x¯, also called a random variable. A stochastic variable can be seen as a normal variable x, but with uncertainty concerning its value. For example, in 1/3 of the cases its value may be 2, but in the other 2/3 of the cases, its value may be 3. Every random variable has an associated probability distribution function Fx¯ (x), also known as the cumulative distribution function. This function is defined as the probability that x¯ ≤ x. So, in formal notation, Fx¯ (x) = Pr {x¯ ≤ x} . (1.1.1) Such a function has several evident properties. We have Fx¯ (−∞) = 0 and Fx¯ (+∞) = 1. Also, the function is monotonically increasing. So, if a ≤ b then also Fx¯ (a) ≤ Fx¯ (b). There is also the probability density function fx¯ (x), abbreviated as PDF. (Note that PDF does not mean probability distribution function!) The PDF is defined as the derivative of the probability distribution function. So, dFx¯ (x) fx¯ (x) = . (1.1.2) dx It immediately follows that fx¯ (x) ≥ 0. We also have ∞ fx¯ (x) = 1, b fx¯ (x) = Fx¯ (b) and b fx¯ (x) = Fx¯ (b) − Fx¯ (a)