在應用雙變量機率分布時，通常會因為資料具有某些特殊限制，使得現有的雙變量機率分布的值域與資料不符合，因而造成機率分布無法合理地解釋資料。例如，一份探討四週內幼童因生病而未上學的天數與因病而在床上休息的天數之調查中，大多數幼童幾乎未生病或生病仍上學，故上述兩個天數大多為零，即所謂的零膨脹(zero-inflated)。此外，因生病而未上學的天數一定會大於或等於在床上休息的天數，此即資料所具有的特殊限制，所以Cheung 和 Lam (2006)將雙變量卜瓦松-卜瓦松分布(bivariate Poisson -Poisson distribution)，以變數變換(change of variables)的方式推廣出在此限制下的機率分布。然而以此機率分布配適這組資料不如預期的好，所以本論文使用截斷(truncated)機率和變數變換的想法來推廣雙變量卜瓦松-卜瓦松分布和雙變量零膨脹卜瓦松分布(bivariate zero-inflated Poission distribution)，使這些機率分布的值域具有跟資料相同的特殊限制。最後，以對數概似值(log-likelihood value)及卡方配適度檢定(goodness of fit test)來比較這些機率分布配適資料的結果。

Due to the constraints of the data, the range of the present bivariate probability distribution don’t correspond to the data when applying the bivariate probability distribution, and thus the probability distribution can not adequately explain the data. For example, a survey exploring the number of days that children couldn’t go to school because of illness and the number of days that children stayed in bed because of illness found that most children still went to school regardless of illness or no illness in the four weeks. Based on the above results, the number of days that children couldn’t go to school and the number of days that children stayed in bed were both zero mostly, which can be seen as zero-inflated. Besides, the number of days that children couldn’t go to school was absolutely greater or equal to that children stayed in bed, which is the constraint of the data. Therefore, Cheung and Lam (2006) developed a probability distribution within the constraints of the data by the change of variables from the bivariate Poisson-Poisson distribution, which didn’t result in the data allocating as well as expected, so finally the bivariate Poisson-Poisson distribution and the bivariate zero-inflated Poisson distribution were developed using truncated probability and the change of variables in this research in order to make the range of the probability distribution match the constraints of the data .Finally, this research adopted log-likelihood values and goodness of fit tests to compare the results of the data allocation of the probability distributions.

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