Customers arrive to an ice-cream truck according to a Poisson process with const
ID: 2928260 • Letter: C
Question
Customers arrive to an ice-cream truck according to a Poisson process with constant rate of 10 per hour. The truck sells three flavours of ice-cream (vanilla, chocolate, and strawberry). Each customer buys vanilla, chocolate, strawberry with probabil- ities 0.3, 0.4, 0.2, respectively, and with probability 0.1 they do not buy anything. (a) On a given day, what is the probability that 17 chocolate ice-creams are sold between 10 am and 2 pm? (b) On a given day, what is the probability that 17 chocolate ice-creams are sold between 10 am and 2 pm, 13 vanilla ice-creams are sold between 8 am and 11 am, and 11 customers do not buy anything between 8 am and 4 pm? 1
Explanation / Answer
Since customers arrive to ice-cream truck according to Poisson process with constant rate of 10/hour. And also truck sells three flavors of ice creams (Vanilla, Chocolate and Strawberry).
The probability that customer buys Vanilla flavor of ice creams=0.3
The probability that customer buys Chocolate flavor of ice creams=0.4
The probability that customer buys Strawberry flavor of ice creams=0.2
The probability that customer buys no ice creams=0.1
the Vanilla flavor ice-cream buyer customer follows Poisson process with parameter= 0.3x10=3
The Chocolate flavor ice-cream buyer customer follows Poisson process with parameter= 0.4x10=4
The Strawberry flavor ice-cream buyer customer follows Poisson process with parameter= 0.2x10=2
No buyer customer also follows Poisson process with parameter= 0.1x10=1.
(a)
On a given day, the probability that 17 chocolate ice-creams are sold between 12am and 2pm (between 14 hours) is
17 chocolate ice-creams are sold between 12am and 2pm on a given day)== 0.00001239
(b)
Probability that on a given day 17 chocolate ice-creams are sold between 12am and 2pm (between 14 hours), 13 vanilla ice creams are sold between 8am to 11am and 11 customers do not buy anything between 8am to 4pm is given by
=0.00001239x0.00003824x0.00000007=0.0000000000000000349
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