By Karl Bosch
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Extra resources for Elementare Einfuehrung in die Warscheinlichkeitsrechnung
XjÀ1 ; xjþ1 ; . . ; xn Þd xj À1 ¼ pðx1 ; . . ; xk jxkþ1 ; . . ; xjÀ1 ; xjþ1 ; . . ; xn Þ ð2:117Þ In particular, the following formula (playing a prominent role in the theory of Markov processes) can be obtained ð1 pðx1 jx2 ; x3 Þpðx2 jx3 Þd x2 ð2:118Þ pðx1 jx3 Þ ¼ À1 All the considered deﬁnitions and rules remain valid for the case of a discrete random variable, with integrals being reduced to sums. Random variables, 1 ; 2 ; . . ; n are called mutually independent if events f1 < x1 g; f2 < x2 g; .
193) that pA; ðA; Þ ¼ pA ðAÞp ðÞ and thus the phase and the magnitude are independent. 191), m 1 1 cos A þ m À 2Am A m 4À ðA; Þ ¼ exp 2 2 2 2 2 pA; 2 2 3 m1 2 þ sin m 5 ! A A2 þ m2 À 2 A m cosð À 0 Þ ¼ exp À 2 2 2 2 ð2:194Þ Here tan 0 ¼ m1 2 m1 1 ð2:195Þ Further integration over the phase variable produces the Rice distribution for the magnitude ! ð ! A A2 þ m 2 A m cosð À 0 Þ A A2 þ m 2 Am exp À exp exp À d ¼ I 0 2 2 2 2 2 2 2 2 2 2 À ð ð ð2:196Þ 42 RANDOM VARIABLES AND THEIR DESCRIPTION Similarly, integration of pA; ðA; Þ over A produces a PDF of the phase with the following form !
Indeed, it is shown in  that cumulant coefﬁcients n of a random variable n ¼ n n=2 2 ¼ n n ð2:234Þ must satisfy certain (non-linear) inequalities. For example, skewness 3 and curtosis 4 must satisfy the condition  4 À 32 þ 2 ! e. 3 2 ðÀ1; 1Þ; 4 must exceed À2. Restrictions on higher order cumulants are still an area of active research. 8 CUMULANT EQUATIONS It was shown earlier that the characteristic function Âð j uÞ can be deﬁned if an inﬁnite set of cumulants k ; k ¼ 1; 2; . , is given.