在本文中，我們將考慮一種隨機化的Pólya甕模型。假設在一個甕中起初有a個白色的球和b個黑色的球，給定一種特殊的二階加球隨機變數矩陣序列{An,n≥1}其中，{(X(1,n),X(2,n) ),n≥1}、{(Y(1,n),Y(2,n) ),n≥1}為非負整數隨機變數向量序列，且滿足X(1,n)+X(2,n)=Y(1,n)+Y(2,n)=M，M,c為固定的正整數，n=1,2,…我們第n次實驗中，從甕中隨機拿出一個球，記錄其顏色，並將球放回甕中。若此次抽到的為白色球，則向甕中加入cX(1,n)個白色球和cX(2,n)個黑色球，若此次抽到的為黑色球，則向甕中加入cY(1,n)個白色球和cY(2,n)個黑色球，不斷重複此過程，我們將會研究甕中白球的分佈情況，並將此模型嵌入一個多類型連續時間馬爾可夫分支過程中進行分析。最後，我們會將此模型應用於藥物臨床實驗之中。

In this paper, we will consider a randomized Pólya urn model. Suppose there is a urn with a white balls and b black balls in it initially. Given a special randomized add-ball matrix sequence {An,n≥1}
{(X(1,n),X(2,n)),n≥1} and {(Y(1,n),Y(2,n)),n≥1} are randomized vector sequences which satisfied X(1,n)+X(2,n)=Y(1,n)+Y(2,n)=M, for some fixed positive integers M,c, and n=1,2,… We randomly take a ball out from the urn, recording its color, and put it back into the urn at n-th experiment. If the white ball is drawn, then cX(1,n) white balls and cX(2,n) black balls were added into the urn. If the black ball is drawn, then cY(1,n) white balls and cY(2,n) black balls were added into the urn. Repeating this process, and we will study the distribution of white balls in the urn. We will also embed this model into a multitype continuous-time Markov branch process to study. Finally we will apply this model for medical trials.

[1] KOTZ.S,MAHMOUD.H AND ROBERT.P,(2000).On generalized Pólya urn mod-els.Statistics & Probability Letters 49,163-173.
[2] FRIEDMAN.B,(1949).A simple urn model. Vol.2, Issue.1,59-70.
[3] ATHREYA.K,(1967). Limit theorems for multitype continuous time Markov branch-ing processes, I. The case of an eigenvector linear functional. To appear in Z. Wahr-scheinlichkeitstheorie und Verw. Gebeite.
[4] ATHREYA.K,(1967). Limit theorems for multitype continuous time Markov branch-ing processes, II. The case of an arbitrary linear functional. To appear in Z. Wahrschein-lichkeitstheorie und Verw. Gebeite.
[5]ATHREYA.K,(1968).Some results on multitype continuous time Markov branching processes. The annals of Mathematical Statistics.Vol.39.No.2,347-357.
[6] ATHREYA.K AND KARLIN.S,(1968).Embedding of urn schemes into continuous time Markov branching processes and related limit theorems.The annals of Mathematical Statistics.Vol.39.No.6,1801-1817.
[7] WEI.L.J,(1979).The generalized Pólya’s urn design for sequential medical trails. The annals of Statistics. Vol.7.No.2,291-296.
[8] ZHANG.L.X,HU.F,CHEUNG.S.H AND CHAN.W.S,(2011).Immigrated urn models-Theoretical properties and applications. The annals of Statistics. Vol.39.No.1,643-671.
[9] CHEN.M.R AND WEI.C.Z,(2005).A new urn model. J.Appl.Prob.42,964-976.
[10] CHEN.M.R AND KUBA.M,(2013).On Generalized Pólya urn mod-els.J.Appl.Prob.50,1169-1186.
[11] CHEN.M.R AND HSIAU.S.R,(2014).A new two-urn model.J.Appl.Prob.51,590-597.
[12] AOUDIA.D AND PERRON.F,(2012).A new randomized Pólya urn model.Applied Mathematics,2012,3,2118-2122.
[13] CONSUL.P,(1974).A simple urn model dependent upon predetermined strategy. Sankhyā:The Indian Journal of Statistics.Vol.36,Series B.Pt.4,391-399.