2. The number of automobile accidents between 7am and 8am, in a particular city,

Question

2. The number of automobile accidents between 7am and 8am, in a particular city, on any given weekday is assumed to follow a Poisson distribution. The number of accidents is on average 5 during this time period.

A. Find the probability that there will be exactly 4 accidents during this time next Monday?

B. Find the probability that there will be at least 6 accidents during this time next Monday?

C. Find the probability that there will be between 22 and 26 accidents (inclusively) during 7am and 8am from Monday to Friday (5 days) next week?

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
a)
P( X = 4 ) = e ^-5 * 5^4 / 4! = 0.18
b)
P( X < 6) = P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= e^-5 * 4 ^ 5 / 5! + e^-5 * ^ 4 / 4! + e^-5 * ^ 3 / 3! + e^-5 * ^ 2 / 2! + e^-5 * ^ 1 / 1! + e^-5 * ^ 0 / 0!
= 0.62

P( X > = 6 ) = 1 - P (X < 6) = 0.38

c)
P( X = 22 ) = e ^-25 * 25^22 / 22! = 0.0702
P( X = 23 ) = e ^-25 * 25^23 / 23! = 0.0763
P( X = 24 ) = e ^-25 * 25^24 / 24! = 0.0795
P( X = 25 ) = e ^-25 * 25^25 / 25! = 0.0795
P( X = 26 ) = e ^-25 * 25^26 / 26! = 0.0765

P( 22<=X<=26) = 0.0702+0.0763+0.0795+0.0795+0.0765 = 0.382

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