s / trucks depart from the station eery 4 hours When a truck departu, Bext truck

Question

s / trucks depart from the station eery 4 hours When a truck departu, Bext truck inmediately staria getting illed with A ruck has the capacity of 200 parcel A truck ls considered faly lhaded if it ie Slled up to at Packagen arrive to a portal station an Poisson process with a mean tine between arrivals of Laumato Postal u a least 98% of its capacity. a) Find the ity that a truck is fully loaded. (5 points) O. 9o 1 -12-(014) For a reporting period of 30 days (24/7, a total of 18 sucs), find the probability that at most one track is not (10 points) ayt J-y y fully loaded o.o.Ro., mre e) Fos the same reporting peried, find the expected number of fully louded trucks (10 poinbs) E17)

Explanation / Answer

Let X = number of packages filled into a truck…………………………………………….(1)

Trucks depart every 4 hours and when a truck departs, next truck immediately starts getting filled => a truck gets filled for 4 hours. ………………………………………………….(2)

(2) => number of packages getting filled into a truck = number of packages arriving in 4 hours ………………………………………………………………………………………(3)

Given packages arrive in a Poisson process with mean time between arrivals as 1.2 minutes

=> average number of packages arriving per hour = 60/1.2 = 50 ………………………..(4)

=> average number of packages arriving per 4 hours = 4 x 50 = 200…………………….(5)

Back-up Theory

If a random variable X ~ Poisson(), i.e., X has Poisson Distribution with mean then

probability mass function (pmf) of X is given by P(X = x) = e – .x/(x!) …………..(6)

where x = 0, 1, 2, ……. ,

Values of p(x) for various values of and x can be obtained by using Excel Function.

If X = number of times an event occurs during period t, Y = number of times the same event occurs during period kt, and X ~ Poisson(), then Y ~ Poisson (k) …………….. (7)

(3), (5), (1) and (7) => X ~ Poisson(200)………………………………………………(8)

Given, a truck’s capacity is 200 packages …………………………………………..(9)

And is considered fully loaded if it is filled up to at least 98% of its capacity ……...(10)

(10), (9) and (8) => P(a truck is fully loaded) = P(X 196) …………………………..(11)

Part (a)

Probability a truck is fully loaded = P(X 196) [vide (11) above]

= probability Poisson(200) 196

= 0.5934 [using Excel Function of Poisson Distribution] ANSWER

Part (b)

Reporting period is 30 days and 24/7 => total number of trucks during the reporting period

= (30 x 24)/4 = 180.

So, if Y = number of trucks which are fully loaded over the reporting period, then

Y ~ B(180, p), where p = probability a truck is fully loaded, which as per answer of Part (a) is 0.5934

Then, probability at most one truck is not fully loaded

= probability at least 179 trucks are fully loaded

= P(Y = 179) + P(Y = 180)

= 1.97E – 34 [using Excel Function of Binomial Distribution] ANSWER

Part (c)

Expected number of fully loaded trucks during the reporting period of 30 days

= np

= 180 x 0.5934

= 106.812

= 107 ANSWER

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