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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. according to a large-scale study of elderly icu patients, the average length of stay in the icu is days (critical care medicine journal article). assume that this length of stay in the icu has an exponential distribution. do not round intermediate calculations.

Answer :

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a. Probability that the length of stay in the ICU is one day or less is:0.2614.

b. Probability that the length of stay in the ICU is between two and three days is:0.1426

c. Probability that the length of stay in the ICU is more than five days is:0.2198.

Probability

a. Probabilty of one day or less.

Using this formula

P(X≤1)=1-e^-ux

Since mean is 3.3 the decay parameter is:

u=1/3.3

Hence:

P(X≤1)=1-e^-1/3.3

P(X≤1)=1-e^-0.3030

P(X≤1)=1-0.7385991

P(X≤1)=0.2614

b. Probability of two and three days.

P(X≤2)=1-e^-ux

P(X≤2)=1-e^-2/3.3

P(X≤2)=1-e^-0.60606

P(X≤2)=1-0.54549589

P(X≤2)=0.45450

P(X≤3)=1-e^-ux

P(X≤3)=1-e^-3/3.3

P(X≤3)=1-e^-0.909091

P(X≤3)=1-0.402890

P(X≤3)=0.59711

Hence:

P(2≤X≤3)=0.59711-0.45450

P(2≤X≤3)=0.1426

c. Probability of more than 5 days.

P(X≤5)=e^-ux

P(X≤5)=e^-5/3.3

P(X≤5)=e^-1.515152

P(X≤5)=0.219775

P(X≤5)=0.2198 (Approximately)

Inconclusion the probability that the length of stay in the ICU is one day or less is:0.2614.

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