The arrival times of workers at a factory first aid room satisfy a Poisson proce

Question

The arrival times of workers at a factory first aid room satisfy a Poisson process with an average of 2.8 per hour.

a) What is the probability that there is at least 3 hours passed that the first worker arrives at the first aid room? What is the expected value of the time if the interarrival at the first aid room?

b) Let NT be the number of the workers arrived at the fiest aid room in the T hour of interval (t, t+T). What is the poisson PMF of NT?

c) What is the probability of during 4 hours period 3 workers visiting the first aid room?

Explanation / Answer

POSSION DISTRIBUTION
pmf of P.D is = f ( k ) = e- x / x!
where
= parameter of the distribution.   
x = is the number of independent trials
I.
mean =
= 2.8

a.
the probability that there is at least 3 hours passed that the first worker arrives at the first aid room
P( X < 3) = P(X=2) + P(X=1) + P(X=0)   
= e^-2.8 * 0 ^ 2 / 2! + e^-2.8 * ^ 1 / 1! + e^-2.8 * ^ 0 / 0!
= 0.46945,
P( X > = 3 ) = 1 - P (X < 3) = 0.53055

the expected value of the time if the interarrival at the first aid room = NT = 3*1 = 3

b.
Let NT be the number of the workers arrived at the fiest aid room in the T hour of interval (t, t+T)
mean = = NT
pmf of P.D is = f ( k ) = e-NT NT*x / x!
where
= parameter of the distribution =NT   
x = is the number of independent trials

C.
the probability of during 4 hours period 3 workers visiting the first aid room
mean = = NT = 3*4 =12
mean = = NT
pmf of P.D is = f ( k ) = e-NT NT*x / x!
where
= parameter of the distribution =NT   
x = is the number of independent trials

P( X = 4 ) = e ^-12 * 12^4 / 4! = 0.00531

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