To solve this we need to know that…\(E[X_i] = 1/\lambda_i\) and \(E[minX_i]=1/\lambda\)
\(1/\lambda_i = 1000\)
\(\lambda_i = 1/1000\)
\(\lambda = \lambda_1+\lambda_2...+\lambda_{100}\)
\(\lambda=1/10\)
\(E[minX_i]=10\)
Basic PDF \(\lambda e^{-\lambda x}\)
To complete this problem using what we’ve learned in chapter 7; we can use the method of convolution.
\(fZ(z) =\int_{-\infty}^{\infty}fX_1(z+x_2)fX_2(x_2)dx_2\)
Insert exp func
\(fZ(z) =\int_0^{z}\lambda e^{-\lambda (z+x_2)}\lambda e^{-\lambda x_2}dx_2\)
…
\(fZ(z) =\int_0^{z}\lambda^2e^{-\lambda z}dx_2\)
…
\(fZ(z) =\lambda/2e^{-\lambda |z|}\)
This is all we’ll need.
Lets turn it into a function.
CE = function(e){
ub = (100/3)/e^2
if(ub>=1){print(1)}
else{print(ub)}
}
CE(2)## [1] 1
CE(5)## [1] 1
CE(9)## [1] 0.4115226
CE(20)## [1] 0.08333333