An old house in Fayetteville is inhabited by a variety of gh

An old house in Fayetteville is inhabited by a variety of ghosts. Ghost appearances occur in the house according to a Poisson process having a rate of 1.4 ghosts per hour. A professor from Transylvania University has developed a device which can be used to detect ghost appearances. Suppose we begin observing the facility at 8:00 a.m. What is the probability that 3 ghosts appear apear by 9:00AM On average, how many ghosts will appear by 10:30AM What is the probability that the 4th ghost appears more than 30 minutes after the 3rd ghost For parts (d) - (e), suppose 4 ghosts appeared between 7:00 p.m. and 10:00 p.m. What is the probability that more than 2 ghosts appear between 11:00 p.m. and 11:30 p.m.? What is the probability that 8 ghosts appear between 6:00 p.m. and midnight?

Solution

The process rate or the mean of the process = 1.4

poisson process equation is given as:

P (x = k) = e^-rt (rt)^k / k!

where r = rate = 1.4

thus,

a)

the probability that 3 ghosts appear in one hour = P (X =3)

= (e^-1.4*1)(1.4*1)³ / 3!

= 0.1127

b)

Average number of ghosts arriving would depend upon the length of time

= 1.4 ghosts/hour * 2.5 hours

= 3.5 ghosts

d)

Probability that more than two ghosts will appear in a half hour interval

= 1 - [ P(X=0) + P (X=1) + P (X=2) ]

= 1 - [ 0.4965 + 0.3476 + 0.1216]

= 1 - 0.9657

= 0.0342 is the required probability

e)

Probability of 8 ghost appearing in 6 hours

= e^(-1.4*6) * (1.4*6)⁸ / 8!

= 0.1382

Hope this helps.