Gamma and negative binomial distributions Let r 2 N, t 0, pn n1 a sequence in 0 1OE with npn , and tn n1 a sequence in ZC with tn n t Show that r 0 t D lim n 1 Br pn 0 tn and interpret this result in terms of waiting times Hint First show that Br p 0 1 m D BrCm p r rC1 rCm)

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Question: Gamma and negative binomial distributions. Let r 2 N, ; t > 0, .pn/n1 a sequence in 0; 1OE with npn ! , and .tn/n1

Gamma and negative binomial distributions. Let r 2 N, ˛; t > 0, .pn/n1 a sequence in 0; 1OE with npn ! ˛, and .tn/n1 a sequence in ZC with tn=n ! t . Show that

˛;r .

0; t

/ D lim n!1 Br;pn.¹0; : : : ; tnº/ ;

and interpret this result in terms of waiting times. Hint: First show that Br;p.¹0; 1; : : : ; mº/ D BrCm;p.¹r; rC1; : : : ; rCmº) .

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