Fundamentals of Probability and Statistics for Engineers

Expectations and Moments 103However, notice that, since X is continuous, P(X ˆ x) ˆ 0 if x is a point ofcontinuity in the distribution of X. Hence, using Equation (4.47),EfYg ˆP…X < x† ˆF X …x†ˆ 12ˆ 12Zj 12 1Zj 12 11 Efe j…X x†t gdtt1 e jtx X …t†dt:t…4:62†The above defines the probability distribution function of X. Its derivativegives the inversion formulaf X …x† ˆ 1 Z 1e jtx X …t†dt;2 1…4:63†and we have Equation (4.58), as desired.The inversion formula when X is a discrete random variable isZ1 up X …x† ˆ lim e jtx X …t†dt:u!1 2u u…4:64†A proof of this relation can be constructed along the same lines as that givenabove for the continuous case.Proof of Equation (4.64): first note the standard integration formula:8Z1 u < sin aue jat ; for a 6ˆ 0;dt ˆ au2u u :1; for a ˆ 0:…4:65†Replacing a by X x and taking the limit as u !1, we have a new randomvariable Y, defined byZ 1 u 0; for X 6ˆ x;Y ˆ lim e j…X x†t dt ˆu!1 2u u1; for X ˆ x:The mean of Y is given byEfYg ˆ…1†P…X ˆ x†‡…0†P…X 6ˆ x† ˆP…X ˆ x†;…4:66†TLFeBOOK