A Expectation:

The expectation (or mean) of a r.v. X, denoted by E( X) or ? _X, is defined by

The expectation of Y = g( X) is given by

The expectation of Z = g( X, Y) is given by

Note that the expectation operation is linear, that is,

where c is a constant (Prob. 6.45).

B Moment:

The nth moment of a r.v. X is defined by

C Variance:

The variance of a r.v. X, denoted by ? ² _X or Var( X), is defined by

Thus,

The positive square root of the variance, or ? _x, is called the standard deviation of X. The variance or standard variation is a measure of the "spread" of the values of X from its mean ? _X. By using Eqs. (6.72) and (6.73), the expression in Eq. (6.75) can be simplified to

D Covariance and Correlation Coefficient:

The ( k, n)th moment of a two-dimensional r.v. ( X, Y) is defined by

The (1,1)th joint moment of ( X, Y),

is called the correlation of X and Y. If E( X Y) = 0, then we say that X and Y are orthogonal. The covariance of X and Y, denoted by Cov( X, Y) or ? _XY