The continuous random variable T T\,T represents the operational lifespan, in days, of a bioluminescent underwater sensor. The probability density function of T T\,T is modeled by:
f(t)={c(25−t2)0≤t≤50otherwisef(t) = \begin{cases} c(25 - t^2) & 0 \le t \le 5 \\ 0 & \text{otherwise} \end{cases}f(t)={c(25−t2)00≤t≤5otherwise
where c c\,c is a constant.
Show that c=3250\displaystyle c = \frac{3}{250}c=2503.
Determine the cumulative distribution function F(t)F(t)F(t) for the interval 0≤t≤50 \le t \le 50≤t≤5.
Calculate the probability that a randomly selected sensor remains operational for more than 3 days.
Given that a sensor has already functioned for 3 days, determine the probability that it will last for at least 4 days in total.
Five such sensors are deployed independently. Find the probability that exactly 2 of them remain operational for more than 3 days.
