ABSTRACT
When we sample a function f(x) we usually make some error, and the data we get is not precisely f(nΔ), but contains additive noise, that is, our data value is really f(nΔ) + noise. Noise is best viewed as random, so it becomes necessary to treat random sequences f = {fn} in which each fn is a random variable. The random variables fn and fm may or may
several topics important
The simplest answer to the question “What is a random variable?”is “A random variable is a mathematical model”. Imagine that we repeatedly drop a baseball from eye-level to the floor. Each time, the baseball behaves the same. If we were asked to describe this behavior with a mathematical model, we probably would choose to use a differential equation as our model. Ignoring everything except the force of gravity, we would write
h′′(t) = −32
as the equation describing the downward acceleration due to gravity. Integrating, we have
h′(t) = −32t+ h′(0) as the velocity of the baseball at time t ≥ 0, and integrating once more,
h(t) = −16t2 + h′(0)t+ h(0)
as the equation of position of the baseball at time t ≥ 0, up to the moment when it hits the floor. Knowing h(0), the distance from eye-level to the floor, and knowing that, since we dropped the ball, h′(0) = 0, we can determine how long it will take the baseball to hit the floor, and the speed with which it will hit. This analysis will apply every time we drop the baseball. There will, of course, be slight differences from one drop to the next, depending, perhaps, on how the ball was held, but these will be so small as to be insignificant.
