The time SSS, in minutes, taken by an automated medical system to sterilise a surgical instrument is modelled by the random variable SSS with probability density function
f(s)={32000(20s−s2)0≤s≤100otherwise f(s) = \begin{cases} \frac{3}{2000}(20s - s^2) & 0 \le s \le 10 \\ 0 & \text{otherwise} \end{cases} f(s)={20003(20s−s2)00≤s≤10otherwiseUse algebraic integration to find, in minutes and seconds, the mean sterilisation time.
Show that P(2<S<8)=81125P(2 < S < 8) = \frac{81}{125}P(2<S<8)=12581.
A medical facility monitors a batch of 125 sterilisation sessions.
Use a suitable approximation to find the probability that fewer than 75 of these sessions take between 2 and 8 minutes.
