$4.$B$.8$ The random variable $x$ measures the lifetime of a $($random$)$ light bulb and is Weibull distributed with cumulative distribution function $F_x(x)=1-e^{-x^2}($for $x0).$

a$.$ This lamp is subject to wear and tear. Express this mathematically.

b$.$ Prove that the lamp is subject to wear and tear.

Clue: $(t+s{)}^2{t}^2+s^2$ for $t,s0.$

c$.$ Show that $E[x]=\frac{\sqrt[\phantom{2}]{}}{2}$ and Var$[x]=1-\frac{}{4}.$

d$.$ Approximate the probability that there is more than $95$ hours of light if $100$ such lamps burn in succession and independently of each other $($each time until they break$).$