CHI-SQUARE DISTRIBUTION AND TEST

The random variable x2 given by x2 = x + x2 + ••• + хП, where x; are the normally distributed variables with zero mean and unit variance, has the PDF (probability density function) with n degrees of freedom: p(x2) = 2~n/2r(n/2)~1(x2)2~1 exp(-X2/2). Г(п/2) is Euler’s gamma function, and E(x2) = n; s2(x2) = 2n. In the limit the x2 distribution approaches the Gaussian distribution with mean n and variance 2n. Once the probability density function is numerically computed from the random data, the x2 test is used to determine if the computed probability density function is Gaussian.

For normally distributed and mutually uncorrelated variables x;, with mean mt and with variance s;, the normalized sum of squares is formed as s = ^ni=1 (xi ~m;) . Then, s obeys the x2 distribution with n DOF. The x2 test is used for hypothesis testing.

B6 CONFIDENCE INTERVAL AND LEVELS

A confidence interval for a parameter is an interval constructed from empirical data, and the probability that the interval contains the true value of the parameter can be specified. The confidence level of the interval is the chance that this interval (that will result once data are collected) will contain the parameter. In estimation result, requirement of high confidence in the estimated parameters or states is imperative. This information is available from the estimation-process results. A statistical approach is used to define the confidence interval within which the true parameters/ states are assumed to lie with 95% of confidence. This signifies a high value of probability with which the truth lies within the upper and lower confidence intervals. If P{l < b < u} = a, then a is the probability that b lies in the interval (l, u). The
probability that the true value, b, is between I (the lower bound) and u (the upper bound) is “a.” As the interval becomes smaller and smaller, the estimated value b is regarded more confidently as the true parameter.