The continuous random variable WWW is used to model the endurance of a high-performance industrial drill bit, where www represents the number of thousands of cycles completed before the bit requires replacement.
The probability that a drill bit lasts for more than www thousand cycles is given by
P(W>w)=16(w+4)2,w≥0 P(W > w) = \frac{16}{(w + 4)^2}, \quad w \ge 0 P(W>w)=(w+4)216,w≥0Show that the cumulative distribution function (CDF) of WWW is given by
F(w)={1−16(w+4)2w≥00otherwise F(w) = \begin{cases} 1 - \frac{16}{(w+4)^2} & w \ge 0 \\ 0 & \text{otherwise} \end{cases} F(w)={1−(w+4)2160w≥0otherwiseDetermine the probability that a randomly selected drill bit will fail within the first 4000 cycles.
Given that a drill bit has already completed 4000 cycles, find the probability that it will last for at least 8000 more cycles.
Calculate the number of cycles after which only 6.25%6.25\%6.25% of these drill bits are expected to remain functional.
