Fundamentals of Probability and Statistics for Engineers

Expectations and Moments 89P roof of P ropert y 4. to 2: show Property 4.2, let t and u be any real quantitiesand form…t; u† ˆEf‰t…X m x †‡u…Y m Y †Š 2 gˆ 20 t 2 ‡ 2 11 tu ‡ 02 u 2 :Since the expectation of a nonnegative function of X and Y must be nonnegative,(t,u) is a nonnegative quadratic form in t and u, and we musthave 20 02 2 11 0; …4:25†which gives the desired result.The normalization of the covariance through Equation (4.24) renders auseful substitute for 11. Furthermore, the correlation coefficient is dimensionlessand independent of the origin, that is, for any constants a 1 ,a 2 ,b 1 ,and b 2with a 1 >0and a 2 > 0, we can easily verify that…a 1 X ‡ b 1 ; a 2 Y ‡ b 2 †ˆ…X; Y†:…4:26†P ropert y 4. 3. If X and Y are independent, then 11 ˆ 0 and ˆ 0: …4:27†P roof of P ropert y 4. let 3: X and Y be continuous; their joint momentfound from11 is 11 ˆ EfXYg ˆZ 11Z 1xyf XY …x; y†dxdy:1If X and Y are independent, we see from Equation (3.45) thatandf XY …x; y† ˆf X …x†f Y …y†; 11 ˆZ 11ˆ m X m Y :Z 1 Z 1 Z 1xyf X …x†f Y …y†dxdy ˆ xf X …x†dx yf Y …y†dy111Equations (4.23) and (4.24) then show that 11 ˆ 0 and ˆ 0. A similar resultcan be obtained for two independent discrete random variables.TLFeBOOK