Mika伤害

2023-09-12 15:05 作者:Baobhan_Sith 0人读过 | 我要投稿

关于夏莱DB计算器中, 所计算的mika平均伤害有没有每段分别计算衰减, 可以直接下一个定论: 夏莱DB计算了每段的衰减, 所以此数据可以直接使用.

关于如何计算mika均伤

如果使用蒙特卡洛实验来计算均伤是对计算资源的浪费, 因为这里可以直接利用全期望公式解析地算出均伤.

假如使用11项的数列a来表示mika每段伤害

$a_n%2C%20n%3D1%2C2%2C3%2C...%2C11%0A%0A$

众所周知, mika有11段, 其中前十段基础伤害浮动最低为a, 最高为b, 而第11段浮动最低为10a, 最高为10b, 那么可以列出伤害的递推式

$%5Cfrac%7Ba_1%7D%7B1%2B%5Cfrac%7B%5Cmathrm%7BcurrentHP%7D%7D%7B%5Cmathrm%7BtotalHP%7D%7D%7D%5Csim%20U%5Cleft(%20a%2Cb%20%5Cright)%20%0A$

$%5Cfrac%7Ba_2%7D%7B1%2B%5Cfrac%7B%5Cmathrm%7BcurrentHP%7D-a_1%7D%7B%5Cmathrm%7BtotalHP%7D%7D%7D%5Csim%20U%5Cleft(%20a%2Cb%20%5Cright)%20%0A$

$...$

$%5Cfrac%7Ba_n%7D%7B1%2B%5Cfrac%7B%5Cmathrm%7BcurrentHP%7D-%5Cleft(%20a_1%2B...%2Ba_%7Bn-1%7D%20%5Cright)%7D%7B%5Cmathrm%7BtotalHP%7D%7D%7D%5Csim%20U%5Cleft(%20a%2Cb%20%5Cright)%20%0A$

$...$

$%5Cfrac%7Ba_%7B10%7D%7D%7B1%2B%5Cfrac%7B%5Cmathrm%7BcurrentHP%7D-%5Cleft(%20a_1%2B...%2Ba_9%20%5Cright)%7D%7B%5Cmathrm%7BtotalHP%7D%7D%7D%5Csim%20U%5Cleft(%20a%2Cb%20%5Cright)%20%0A$

$%5Cfrac%7Ba_%7B11%7D%7D%7B1%2B%5Cfrac%7B%5Cmathrm%7BcurrentHP%7D-%5Cleft(%20a_1%2B...%2Ba_%7B10%7D%20%5Cright)%7D%7B%5Cmathrm%7BtotalHP%7D%7D%7D%5Csim%20U%5Cleft(%2010a%2C10b%20%5Cright)%20%0A$

我们想求的是11段总伤的期望, 也就是

$%5Cmathrm%7BE%7D%5Cleft(%20a_1%2Ba_2%2B...%2Ba_%7B10%7D%2Ba_%7B11%7D%20%5Cright)%20%0A$

使用全期望公式, 并记 $%5Cmu%20_0%3D%5Cfrac%7Ba%2Bb%7D%7B2%7D%0A$ , 可以得到

$%3D%5Cmathrm%7BE%7D%5Cleft(%20%5Cmathrm%7BE%7D%5Cleft(%20a_1%2Ba_2%2B...%2Ba_%7B10%7D%2Ba_%7B11%7D%7Ca_1%2Ba_2%2B...%2Ba_%7B10%7D%20%5Cright)%20%5Cright)%20%0A$

$%3D%5Cmathrm%7BE%7D%5Cleft(%20a_1%2Ba_2%2B...%2Ba_%7B10%7D%2B%5Cleft(%201%2B%5Cfrac%7B%5Cmathrm%7BcurrentHP%7D-%5Cleft(%20a_1%2B...%2Ba_%7B10%7D%20%5Cright)%7D%7B%5Cmathrm%7BtotalHP%7D%7D%20%5Cright)%20%5Cleft(%205a%2B5b%20%5Cright)%20%5Cright)%20%0A$

这启发我们可以通过找递推关系来计算:

$%5Cmathrm%7BE%7D_n%3D%5Cmathrm%7BE%7D%5Cleft(%20a_1%2Ba_2%2B...%2Ba_n%20%5Cright)%20%0A$

$%3D%5Cmathrm%7BE%7D%5Cleft(%20%5Cmathrm%7BE%7D%5Cleft(%20a_1%2Ba_2%2B...%2Ba_n%7Ca_1%2Ba_2%2B...%2Ba_%7Bn-1%7D%20%5Cright)%20%5Cright)%20%0A$

$%3D%5Cmathrm%7BE%7D%5Cleft(%20a_1%2Ba_2%2B...%2Ba_%7Bn-1%7D%2B%5Cleft(%201%2B%5Cfrac%7B%5Cmathrm%7BcurrentHP%7D-%5Cleft(%20a_1%2B...%2Ba_%7Bn-1%7D%20%5Cright)%7D%7B%5Cmathrm%7BtotalHP%7D%7D%20%5Cright)%20%5Cmu%20_0%20%5Cright)%20%0A$

$%3D%5Cleft(%201%2B%5Cfrac%7B%5Cmathrm%7BcurrentHP%7D%7D%7B%5Cmathrm%7BtotalHP%7D%7D%20%5Cright)%20%5Cmu%20_0%2B%5Cleft(%201-%5Cfrac%7B%5Cmu%20_0%7D%7B%5Cmathrm%7BtotalHP%7D%7D%20%5Cright)%20%5Cmathrm%7BE%7D%5Cleft(%20a_1%2Ba_2%2B...%2Ba_%7Bn-1%7D%20%5Cright)%20%0A$

$%5Cmathrm%7BE%7D_1%3D%5Cmathrm%7BE%7D%5Cleft(%20a_1%20%5Cright)%20%3D%5Cleft(%201%2B%5Cfrac%7B%5Cmathrm%7BcurrentHP%7D%7D%7B%5Cmathrm%7BtotalHP%7D%7D%20%5Cright)%20%5Cmu%20_0%0A$

容易算出数列通项为

$%5Cmathrm%7BE%7D_n%3D%5Cleft(%20%5Cmathrm%7BtotalHP%7D%2B%5Cmathrm%7BcurrentHP%7D%20%5Cright)%20%5Cleft(%201-%5Cleft(%201-%5Cfrac%7B%5Cmu%20_0%7D%7B%5Cmathrm%7BtotalHP%7D%7D%20%5Cright)%20%5En%20%5Cright)%20%0A$

代回第十一项的递推关系

$%5Cmathrm%7BE%7D%5Cleft(%20a_1%2Ba_2%2B...%2Ba_%7B10%7D%2Ba_%7B11%7D%20%5Cright)%20%3D%5Cleft(%201%2B%5Cfrac%7B%5Cmathrm%7BcurrentHP%7D%7D%7B%5Cmathrm%7BtotalHP%7D%7D%20%5Cright)%2010%5Cmu%20_0%2B%5Cleft(%201-%5Cfrac%7B10%5Cmu%20_0%7D%7B%5Cmathrm%7BtotalHP%7D%7D%20%5Cright)%20%5Cmathrm%7BE%7D_%7B10%7D%0A$

$%3D%5Cleft(%201%2B%5Cfrac%7B%5Cmathrm%7BcurrentHP%7D%7D%7B%5Cmathrm%7BtotalHP%7D%7D%20%5Cright)%2010%5Cmu%20_0%2B%5Cleft(%201-%5Cfrac%7B10%5Cmu%20_0%7D%7B%5Cmathrm%7BtotalHP%7D%7D%20%5Cright)%20%5Cleft(%20%5Cmathrm%7BtotalHP%7D%2B%5Cmathrm%7BcurrentHP%7D%20%5Cright)%20%5Cleft(%201-%5Cleft(%201-%5Cfrac%7B%5Cmu%20_0%7D%7B%5Cmathrm%7BtotalHP%7D%7D%20%5Cright)%20%5E%7B10%7D%20%5Cright)%20%0A$