P(X>μ)=P(X<μ)的充分条件X的概率密度为f(x)=(1
X的概率密度为f(x)= (1/2)e^[-|x-μ|] x∈R 是P(X>μ)=P(X<μ)的充分条件吗？
X的概率密度为f(x)=(1/2)e^[-|x-μ|] x∈R 是P(X>μ)=P(X<μ)的充分条件吗？ 是的，因为X的概率密度函数关于直线x=μ对称。 P(X<μ)=∫<-∞,μ>(1/2)e^[-|x-μ|]dx =(1/2)∫<-∞,μ>e^(x-μ)dx[令t=x-μ] =(1/2)∫<-∞,0>e^tdt =(1/2)e^t|<-∞,0>=(1/2)(1-0)=1/2 P(X>μ)=∫<μ,+∞>(1/2)e^[-|x-μ|]dx =(1/2)∫<μ,+∞>e^[-(x-μ)]dx[令t=x-μ] =(1/2)∫<μ,+∞>e^(-t)dt =-(1/2)e^t|<μ,+∞>=-(1/2)(0-1)=1/2 P(X>μ)=P(X<μ)