Question

1) The number of calls received at a certain information desk has a Poisson Distribution with an average of 6 calls per hour. (15 points)

(a) Find the probability that there is at exactly one call during a 15 minute period. (You cannot use tables here - show all work)

(b) Find the probability that at least 6 calls are received during a 30 minute period. (you may use tables here) ********************************

2) Note that for the above problem, the waiting time, X (in hours) between the calls is a random variable that has the following density function:

f(x) = 6exp(-6x), x >0,

0, otherwise

a) Find the probability that the waiting time between calls exceeds half an hour.(7.5 points)

b) Find the expected waiting time between calls.

Answer 1

a ) expected calls in 15 minute perid =$\lambda$ =6*15/60=1.5

hence probability that there is at exactly one call during a 15 minute period =P(X=1)=e^-1.5*1.5¹/1! =0.3347

b)expected calls in 30 minute period =$\lambda$ =6*30/60 =3

probability that at least 6 calls are received during a 30 minute period =P(X>=6)

=1-P(X<=5) =1-$\sum_{x=0}^{5}e^{-3}3^{x}/x!$ =1-0.9161 =0.0839

2)

here this is exponential distirbution pdf for whcih paramter $\lambda$ =6

a) probability that the waiting time between calls exceeds half an hour =P(X>1/2) =e^-$\lambda$x =e^-6*(1/2) =e^-3 =0.0498

b) expected waiting time between calls =1/$\lambda$ =1/6 Hour