Theorems for expectations For linear operations the following theorems are used For a constant <c>=c Linear operator <cH(x)>=C<H(x)> Summation <g+h>= <g>+<h> Covariance: the relationship betwe variables fx,(x,y) is joint probability distribution ∫(x-)y-)fn(x,y)h Correlate Estimation on moments Expectation and variance are the first and second moments of a probability distribution ∑xn1N=J∫x(d (x-H,)N=∑x-A)2/ As n goes to infinity these expressions approach their expectations. ( Note the N-1 in form which uses mean03/12/03 12.540 Lec 10 17 Theorems for expectations – For a constant <c> = c – Linear operator <cH(x)> = c<H(x)> – Summation <g+h> = <g>+<h> xy s xy =< (x - mx )(y - my ) >= (x - mx )(y - my ) f xy Ú (x, y)dxdy rxy = s xy /s x s y • For linear operations, the following theorems are used: • Covariance: The relationship between random variables f (x,y) is joint probability distribution: Correlation : 9 03/12/03 12.540 Lec 10 18 • moments of a probability distribution • As N goes to infinity these expressions approach their expectations. (Note the N-1 in form which uses mean) mˆ x ª xn n=1 N Â /N ª 1 T Ú x(t)dt sˆ x 2 ª (x - mx n=1 N Â ) 2 /N ª (x - mˆ x n=1 N Â ) 2 /(N -1) Estimation on moments Expectation and variance are the first and second