Statistics and Probability

Several definitions and concepts from statistics, probability theory, and linear algebra are collected in nearly alphabetical order [1-5], all of which might not have been used in this book; however, they will be very useful in general for aerospace science and engineering applications.

B1 AUTOCORRELATION FUNCTION

If x(t) is a random signal, then it is given as R^j) = E{x(t)x(t +t)}; here, t is the ‘‘time-lag,’’ and E is the expectation operator. For a stationary process, Rxx is not dependent on ‘‘t,’’ and has a maximum value for t = 0. This is then the variance of the signal x. With increasing time if Rxx shrinks, then it means that the nearby values of the signal x are not correlated and hence are not dependent on each other. The autocorrelation of the white noise process is an impulse function. The autocorrelation of discrete-time residuals is given as

1 N—t

Rrr(t) =———- r(k)r(k + t); t = 0, … , tmax (are discrete time lag)

iV 1 k=1

B2 BIAS IN AN ESTIMATE

Bias is given as (8) = 8 — E(8), the difference between the true value of the parameter 8 and the expected value of its estimate. The estimates would be biased if the noise were not zero mean. The idea is that for a large amount of data used for estimation of a parameter, an estimate is expected to center closely on the true value, and the estimate is called unbiased if E{8 — 8} = 0. The bias should be very small.