Expectation

[Oriel]

A system made up of 7 components with independent, identically distributed lifetimes will operate until any of 1 of the system's components fails. If the lifetime X of each component has density function $f(x) = \cases{\frac{3}{x^4} & , x > 1 \cr 0 & , otherwise} $
What is the expected lifetime until failure of the system?

A) 1.02
B) 1.03
C) 1.04
D) 1.05
E) 1.06

[t.B.]

Hint:
lifetime until failure = $min(X_1,...,X_7)$
$P(min(X_1,...,X_7)<x)=1-P(min(X_1,...,X_7)>x)=1-P(X_1>x)P(X_2>x)...P(X_7>x)$
$E[min(X_1,...,X_7)]=...$