Fundamentals of Probability and Statistics for Engineers

84 Fundamentals of Probability and Statistics for Engineerswhich the value at Y ˆ y i isEfXjY ˆy i g. Hence, EfXjYg is itself a randomvariable, and one of its very useful properties is thatEfXg ˆEfEfXjYgg…4:13†If Y is a discrete random variable taking on values y 1 ,y 2 ,..., the above statesthatEfXg ˆXEfXjY ˆ y i gP…Y ˆ y i †;…4:14†andiEfXg ˆZ 1EfXjygf Y …y†dy;1…4:15†if Y is continuous.To establish the relation given by Equation (4.13), let us show that Equation(4.14) is true when both X and Y are discrete. Starting from the right-hand sideof Equation (4.14), we haveXXEfXjY ˆ y i gP…Y ˆ y i †ˆXx j P…X ˆ x j jY ˆ y i †P…Y ˆ y i †:iSince, from Equation (2.24),iP…X ˆ x j jY ˆ y †ˆP…X ˆ x j \ Y ˆ y i †i ;P…Y ˆ y i †jwe haveXXEfXjY ˆ y i gP…Y ˆ y i †ˆXx j p XY …x j ; y i †iijˆ X Xx j p XY …x j ; y i †jˆ Xx j p X …x j †jˆ EfXg;iand the desired result is obtained.The usefulness of Equation (4.13) is analogous to what we found in using thetheorem of total probability discussed in Section 2.4 (see Theorem 2.1, page 23).TLFeBOOK