The continuous random variable TTT is used to model the lifespan in hours, ttt, of a specific type of chemical reactant after it is activated.
The probability that the reactant lasts for more than ttt hours is given by
P(T>t)=100(t+10)2,t≥0 P(T > t) = \frac{100}{(t+10)^2}, \quad t \ge 0 P(T>t)=(t+10)2100,t≥0Show that the cumulative distribution function (CDF) of TTT is given by
F(t)={1−100(t+10)2t≥00otherwise F(t) = \begin{cases} 1 - \frac{100}{(t+10)^2} & t \ge 0 \\ 0 & \text{otherwise} \end{cases} F(t)={1−(t+10)21000t≥0otherwiseFind the probability that a randomly selected reactant will expire within 5 hours of activation.
Given that a reactant has already lasted for 5 hours, find the probability that it will last for at least 15 more hours.
Calculate the number of hours after which only 25%25\%25% of these reactants are expected to remain active.