Question

At an urgent care facility, patients arrive at an average rate of one patient every 6 minutes (that is λ-6). Assume that the duration between arrivals is exponentially distributed 1) (a) Find the probability that the time between two successive visits to the urgent care fa- cility is less than 4 minutes. (b) Find the 75th percentile. That is, determine To.75 (c) Find the probability that more than 6 patients arrive during a half-hour period.

Answer 1

a)P(X<4)=1-e^-x/$\lambda$ =1-e^-4/6 =0.4866

b)pth percentile =-$\lambda$ln(1-p)

hence 75th percentile =-6*ln(1-0.75)=8.318 minute

c)expected number of patients in 30 minutes=30/6=5

hence from Poisson distribution ; more than 6 patients arrive during a half-hour period= P(X>6)=1-P(X<=6)

=1-$\sum_{x=0}^{6}e^{-5}5^{x}/x!$ =1-0.7622 =0.2378