Intensive care units (ICUs) generally treat the sickest patients in a hospital. ICUs are often the most expensive department in a hospital because of the specialized equipment and extensive training required to be an ICU doctor or nurse. Therefore, it is important to use ICUs as efficiently as possible in a hospital. Suppose that a large-scale study of elderly ICU patients shows that the average length of stay in the ICU is 3.7 days. Assume that this length of stay in the ICU has an exponential distribution. (Round your answers to four decimal places.)
(a) What is the probability that the length of stay in the ICU is one day or less?
(b) What is the probability that the length of stay in the ICU is between two and three days?
(c) What is the probability that the length of stay in the ICU is more than five days?

The probabilities of the Intensive Care Unit stays, using the exponential distribution, are given as follows:a) One day or less: 0.2368 = 23.68%.b) Between two and three days: 0.1379 = 13.79%.c) More than five days: 0.2589 = 25.89%.Exponential distributionThe exponential probability distribution, with mean given by m, is described by the following equation: [tex]f(x) = \mu e^{-\mu x}[/tex]In which [tex]\mu = \frac{1}{m}[/tex] is the decay parameter.The cumulative probability distribution function is:[tex]P(X \leq x) = \int\limits^a_0 {f(x)} \, dx[/tex]Hence:[tex]P(X \leq x) = 1 - e^{-\mu x}[/tex]For this problem, the mean and the decay parameter are given, respectively, by:[tex]m = 3.7, \mu = \frac{1}{3.7}[/tex]Hence the probability that an ICU stay is of one day or less is of:[tex]P(X \leq 1) = 1 - e^{-\frac{1}{3.7}} = 0.2368[/tex]The probability that it is between two and three days is obtained as follows:[tex]P(X \leq 2) = 1 - e^{-\frac{2}{3.7}} = 0.4176[/tex][tex]P(X \leq 3) = 1 - e^{-\frac{3}{3.7}} = 0.5555[/tex]0.5555 - 0.4176 = 0.1379.The probability that the stay is more than five days is:[tex]P(X > 5) = e^{-\frac{5}{3.7}} = 0.2589[/tex]More can be learned about the exponential distribution at https://brainly.com/question/14634921#SPJ1