使用经验被积函数计算积分

Question

我有一个经验概率函数p(z)。在第一列z中，第二列包含p(z)值。数据如下：

data.cat

+0.01234 +0.002816
+0.03693 +0.003265
+0.06152 +0.003551
+0.08611 +0.006612
+0.1107 +0.008898
+0.1353 +0.01041
+0.1599 +0.01269
+0.1845 +0.01404
+0.2091 +0.01616
+0.2336 +0.01657
+0.2582 +0.01865
+0.2828 +0.01951
+0.3074 +0.02024
+0.332 +0.02131
+0.3566 +0.0222
+0.3812 +0.02306
+0.4058 +0.02241
+0.4304 +0.02347
+0.4549 +0.02461
+0.4795 +0.02306
+0.5041 +0.02298
+0.5287 +0.02212
+0.5533 +0.02392
+0.5779 +0.02118
+0.6025 +0.02359
+0.6271 +0.024
+0.6517 +0.02196
+0.6762 +0.02155
+0.7008 +0.02314
+0.7254 +0.02037
+0.75 +0.02065
+0.7746 +0.0211
+0.7992 +0.02037
+0.8238 +0.0198
+0.8484 +0.01984
+0.873 +0.01959
+0.8976 +0.01869
+0.9221 +0.01873
+0.9467 +0.01861
+0.9713 +0.01714
+0.9959 +0.01739
+1.02 +0.01678
+1.045 +0.01633
+1.07 +0.01624
+1.094 +0.01543
+1.119 +0.01494
+1.143 +0.01547
+1.168 +0.01445
+1.193 +0.01384
+1.217 +0.01339
+1.242 +0.01384
+1.266 +0.01298
+1.291 +0.0109
+1.316 +0.0122
+1.34 +0.0111
+1.365 +0.0109
+1.389 +0.009592
+1.414 +0.01114
+1.439 +0.0111
+1.463 +0.009061
+1.488 +4.082e-05

我必须通过某种插值使用经验概率密度来计算以下积分：

或者它可以重写为

其中定义为

和 a 以

的形式给出

我想知道如何计算中间存在经验概率密度函数的复杂积分？

我需要进行某种插值吗？