作者ppppppppp (凹呜呜)
看板Statistics
标题[问题] 请问一题条件期望值
时间Sun Mar 26 11:06:23 2023
帮朋友问
Suppose X~exp(λ) and Y~exp(μ) are independent. Find E[X|X>Y]
我朋友的笔记写的解法如下:
Step1
P(X>t, X>Y) = e^(-λt)-(λ/(λ+μ)*e^(-(λ+μ)*t))
Step2
P(X>t|X>Y)
= P(X>t,X>Y)/P(X>Y)
=(λ+μ)/μ*e^(-λt)-λ/μ*e^(-(λ+μ)*t)
Step3
P(X<=t|X>Y) = 1-P(X>t|X>Y)
Step4
E[X|X>Y] = 1/λ +1/(λ+μ)
请问Step4是怎麽从前三个步骤推导出来的?
谢谢
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