there isnoconvenientformulafor the c.d.f.ofabinomialrandomvariable.However,itispossibletoﬁnd a simple formulafor the c.d.f.ofageometric randomvariable X.Moreover, the methodfor ﬁnding thissimple formulaissimple tooifweuse alittle trick: ﬁrst ﬁnd an expression forthe probability P (X>x)and thenobtain an expression forthe c.d.f. F (x)=P(X≤x)=1−P(X>x). Now, P (X>x)isthe probabilitythatmore than x trials areneeded to obtain asuccess, or equivalently,thatall theﬁrst x trials resultinfailure. This probabilityisequal to (1 − p) x .Itfollowsthat F (x)=1−P(X>x)=1−(1 − p) x . This is the formulaweare seeking forthe c.d.f. of ageometric distribution, stated in thebox below. Thec.d.f.ofageometric distribution If therandomvariable X hasageometric distribution with parameter p,thenthe c.d.f. of X is given by F (x)=P(X≤x)=1−(1 − p) x , (5) for x =1,2,3,....
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