按定义算呗，由独立且指数分布可知\((X, Y)\)的联合密度是\(f(x,y;\lambda_1, \lambda_2) = \lambda_1 e^{-\lambda_1x} \lambda_2 e^{-\lambda_2y}\)。那么
$$P(X<Y)=\int_0^\infty \int_x^\infty f(x,y)dydx = \frac{\lambda_1}{\lambda_1 + \lambda_2}$$
以及
$$P(X<x, X<Y) = \int_0^x\int_s^\infty f(s,y)dyds = \int_0^x\lambda_1 e^{-(\lambda_1 + \lambda_2)s}ds = \frac{\lambda_1}{\lambda_1+\lambda_2}(1 - e^{-(\lambda_1+\lambda_2)x})$$
所以
$$P(X<x|X<Y)=P(X<x, X<Y)/P(X<Y)=1 - e^{-(\lambda_1+\lambda_2)x}$$
多说一句，这好像是生存分析指数分布模型下的基本结论，翻翻教材里面应该都有的吧？
