Question

2. Patients arrive at a hospital accident and emergency department at random follow a Poisson distribution at a rate of 6 per hour.
(a) Find the probability that, during any 90-minute period, the number of patients arriving at the hospital accident and emergency department is
(i) exactly 7 (2 marks) (ii) at least 3 (2 marks)
(b) What is the mean and standard deviation of the number of patients arriving at the hospital accident and emergency department in an hour? (2 marks)

Answer 1

Let $X=$ the number of patients: $X\sim Po(\lambda t)$

$$P(X=x)=\dfrac{e^{-\lambda t}(\lambda t)^x}{x!}$$

(a) Given $\lambda=6, t=1.5$

(i)

$$P(X=7)=\dfrac{e^{-9}(9)^7}{7!}\approx0.11712$$

(ii)

$$P(X\geq3)=1-P(X=0)-P(X=1)-P(X=2)=$$

$$=1-\dfrac{e^{-9}(9)^0}{0!}-\dfrac{e^{-9}(9)^1}{1!}-\dfrac{e^{-9}(9)^2}{2!}=$$

$$=1-50.5e^{-9}\approx0.9937678$$

(b)

$$mean=variance=\lambda=6$$

$$standard \ deviation=\sqrt{\lambda}=\sqrt{6}\approx2.44949$$