Question $1(20\%)$ You have purchased a light bulb. The box of the bulb states that

the life time $T$ of the product in years defined as $P(Tt)=e^{-\frac{t}{4}}$ for all $t0.$

$($a$)$ This is an example of a continuous probability model. Write down the

sample space $S($write it as an interval$).$

$($b$)$ Make sure this is a valid probabilistic model by finding $P(T0)$ and

$\underset{t}{lim}P(Tt).$

$($c$)$ Prove that if then $P(T{t}_1)P(T{t}_2).$ Why $do$ you think this

has $tobe$ true?

$(d)$ Find the probability that the product breaks down within four years $of$ the

purchase time.

$(e)$ Find the probability that the product breaks down $in$ the second year, $i.e.,$

find