Telephone calls arrive in an office at a rate of 25 calls pe

Telephone calls arrive in an office at a rate of 2.5 calls per hour. Find the probability that there are

a) Exactly two calls between 8am and 9am.

b) At least one call between 8am and 9:30am.

c) No calls between 8am and 11am.

d) At least one call between 9am and 11am given that there wasn’t any call between 8am and 9am.

Solution

Possion Distribution
PMF of P.D is = f ( k ) = e- x / x!
Where   
= parameter of the distribution.
x = is the number of independent trials
a)
P( X = 2 ) = e ^-2.5 * 2.5^2 / 2! = 0.2565
b)
Mean rate for 1.5 Hour is = 1.5*2.5 = 3.75
P( X > = 1 ) = 1 - P (X < 1) = 0.9765
P( X < 1) = P(X=0)
= e^-3.75 * 2 ^ 0 / 0!
= 0.0235
c)
Mean rate for 3 hours = 7.5
P( X = 0 ) = e ^-7.5 * 7.5^0 / 0! = 0.0006