A precision industrial laser's cooling system maintains a temperature deviation TTT, measured in millikelvins (mK) from a reference point. The random variable T T\,T follows a continuous distribution with cumulative distribution function F(t)F(t)F(t). The graph of F(t)F(t)F(t) consists of a single straight line segment from the point (−5,0)(-5, 0)(−5,0) to the point (15,1)(15, 1)(15,1). For t<−5t < -5t<−5, F(t)=0F(t) = 0F(t)=0, and for t>15t > 15t>15, F(t)=1F(t) = 1F(t)=1.
Specify fully the probability density function f(t)f(t)f(t) of TTT.
Write down the value of E(T)E(T)E(T).
Determine the value of k k\,k such that P(2.5≤T≤k)=0.35P(2.5 \le T \le k) = 0.35P(2.5≤T≤k)=0.35.
One operating hour is selected at random.
Calculate the probability that the temperature deviation is between 6 mK and 10 mK.
Given that the temperature deviation was between 6 mK and 10 mK, calculate the probability that it was greater than 9.1 mK.
A random sample of 40 operating hours is taken.
Calculate the probability that for at most 5 of these hours the temperature deviation is between 6 mK and 10 mK.
