这道题我再概率论与数理统计2018年4月考试真题上看到。颇考计算。看了答案是0.5。但我的答案和它不一样,但步骤肯定是对的。
解:
因为相互独立,X~N(1,2),Y~N(3,4),所以可知,X+Y~N(4,6)。
所以我们根据下面的公式,可以得到对应的X+Y的概率分布函数:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/a50f4bfbfbedab64d0076ba4f936afc379311e58?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
我们设Z=X+Y,所以:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/2f738bd4b31c870151b49282297f9e2f0708ff50?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
以此我们可以知道对应的分布函数:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/d4628535e5dde711f4aa6192a9efce1b9d16615d?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
的积分为
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/71cf3bc79f3df8dc4d38a2f5c311728b47102848?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
最后,P{X+Y≤4} = F(4) 。
我们将z=4代入后,得出
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/728da9773912b31b53e0e04d8818367adbb4e1c2?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
但正确答案为0.5。我这个和它是有出入的。但我坚信我的步骤肯定是对的。
如有疏忽错误的地方,希望可以告知。我也很困惑