The number of customers entering a store in T hours, X(t), is a Poisson Process

Question

The number of customers entering a store in T hours, X(t), is a Poisson Process at rate 1 = 2 /hour on mild days, at rate 2 = 5 / hour on hot days, and at 3 = 3 / hour on cold days.

The proportion of mild days is 0.3; the proportion of hot days is 0.p; and of cold days is 0.2.

1) Find the probability that at least 3 customers enter the store between 8 AM and 12 PM.

2) Determine the mean and variance of the number of customers that enter the store between 8 AM and 12 PM.

Could you please be kind enough and show which formula is applied and how is calculated?

Thank you!

Explanation / Answer

Solution : Proportion of hot days i.e 0.p= 1-0.2-.03 = 0.5

Average poisson rate = 1*proportion of mild days +2*proportion of hot days + 3*propotion of cold days

= 2*0.3+5*0.5+3*0.2 = 3.7/hour , 8 AM to 12 PM = 4 hours so avg = 4*3.7 = 14.8

1) P(X<=3) = 1-P(X<3)

= 1-P(x=0)+P(x=1)+p(X=2)

=1-e^-14.8 (1+ 14.8 + (14.8^2)/2)

=0.99995

2) Poisson mean and variance is equal to

So mean = 14.8

variance= 14.8

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