Notes on Random Processes

When we use mathematical tools, such as differential equations, probability, or systems of linear equations, to describe a real-world situation, we say that we are employing a mathematical model. Such models must be sufficiently sophisticated to capture the essential features of the situation, while remaining computationally manageable. In this chapter we are interested in one particular type of mathematical model, the random variable. Imagine that you are holding a baseball four feet off the ground. If you drop it, it will land on the ground directly below where you held it. The height of the ball at any time during the fall is described by the function h(t) satisfying the ordinary differential equation h(t) = −32 ft sec . Solving this differential equation with the initial conditions h(0) = 4 ft , h(0) = 0 ft sec , we find that h(t) = 4− 16t. Solving h(T ) = 0 for T we find the elapsed time T until impact is T = 0.5 sec.. The velocity of the ball at impact is h(T ) = −32T = −16 ft sec . Now imagine that, instead of a baseball, you are holding a feather. The feather and the baseball are both subject to the same laws of gravity, but now other aspects of the situation, which we could safely ignore in the case of the baseball, become important in the case of the feather. Like the baseball, the feather is subjected to air resistance and to whatever fluctuations in air currents may be present during its fall. Unlike the baseball, however, the effects of the air matter to the flight of the feather; in fact, they become the dominant factors. When we designed our differential-equation model for the falling baseball we performed no experiments to help us understand its behavior. We simply ignored all other aspects of the situation, and included only gravity in our