The continuous random variable HHH is used to model the operational lifespan in months, hhh, of a specialized high-pressure seal used in deep-sea exploration probes.
The probability that a seal remains functional for more than hhh months is given by the survival function
P(H>h)=64(h+8)2,h≥0 P(H > h) = \frac{64}{(h+8)^2}, \quad h \ge 0 P(H>h)=(h+8)264,h≥0Show that the cumulative distribution function (CDF) of HHH is given by
F(h)={1−64(h+8)2h≥00otherwise F(h) = \begin{cases} 1 - \frac{64}{(h+8)^2} & h \ge 0 \\ 0 & \text{otherwise} \end{cases} F(h)={1−(h+8)2640h≥0otherwiseDetermine the probability that a randomly selected seal will fail within the first 2 months of operation.
Given that a seal has already remained functional for 2 months, find the probability that it will last for at least 30 more months.
Calculate the number of months after which only 25%25\%25% of these seals are expected to remain functional.
