The continuous random variable X X\,X represents the time, in minutes, a patient spends in a waiting room. The probability density function of X X\,X is given by
f(x)={k(36−x2)0≤x≤60otherwise f(x) = \begin{cases} k(36 - x^2) & 0 \le x \le 6 \\ 0 & \text{otherwise} \end{cases} f(x)={k(36−x2)00≤x≤6otherwisewhere k k\,k is a constant.
Show that k=1144\displaystyle k = \frac{1}{144}k=1441.
Find the cumulative distribution function F(x)F(x)F(x) for 0≤x≤60 \le x \le 60≤x≤6.
Find the probability that a patient waits for longer than 2 minutes.
A patient has already been waiting for 2 minutes.
Find the probability that this patient will wait for at least 4 minutes in total.
Four patients are selected at random.
Find the probability that exactly 3 of them waited for longer than 2 minutes.
