关于概率统计的题,请给详解 Distribution of minimum of random variables.Let X1 ,X2 ,· · · ,Xn beindependent random variables from an exponential distribution with the density functionf (x) =1/τ* e-x/τ(这里我打不出来,后面是e的-x/τ次方）Define the minimum of X1 ,X2 ,· · · ,Xn asX(1) = min(X1 ,X2 ,· · · ,Xn ).a.Find the distribution function of X(1) .b.What is the density function of X(1)
问题描述：
关于概率统计的题,请给详解
Distribution of minimum of random variables.
Let X1 ,X2 ,· · · ,Xn be
independent random variables from an exponential distribution with the density function
f (x) =1/τ* e-x/τ(这里我打不出来,后面是e的-x/τ次方）
Define the minimum of X1 ,X2 ,· · · ,Xn as
X(1) = min(X1 ,X2 ,· · · ,Xn ).
a.Find the distribution function of X(1) .
b.What is the density function of X(1)
答
令Z=X(1) (没什么特殊含义,就是表达起来方便)
用定义F(z)=P{min{X1,X2...Xn}z}
=1-P{X1>z}P{X2>z}...P{Xn>z}=1-[1-P{X